Thesis
Submitted in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
by
Dedicated to the memory of
John von Neumann
(28 December 1903 - 08 February 1957)
DECLARATION
By the Ph.D. Research Scholar
I hereby declare that the research thesis entitled METRIC, SCHAUder AND OPERATOR-VALUED FRAMES which is being submitted to the National Institute of Technology Karnataka, Surathkal in partial fulfillment of the requirements for the award of the Degree of Doctor of Philosophy in Mathematical and Computational Sciences is a bonafide report of the research work carried out by me. The material contained in this research thesis has not been submitted to any University or Institution for the award of any degree.
Place: NITK, Surathkal
Date: February 2022
MAHESH KRISHNA K
165106MA16F02
Department of MACS
NITK, Surathkal
CERTIFICATE
This is to certify that the research thesis entitled METRIC, SCHAUDER AND OPERATOR-VALUED FRAMES submitted by Mr. MAHESH KRISHNA K (Register Number : 165106MA16F02) as the record of the research work carried out by him is accepted as the research thesis submission in partial fulfillment of the requirements for the award of degree of Doctor of Philosophy .
Place: NITK, Surathkal
Date: February 2022
Dr. P. Sam Johnson
Research Guide
Department of MACS
NITK Surathkal, Karnataka, India
Chairman - DRPC
(Signature with Date and Seal)
ACKNOWLEDGEMENTS
I begin by remembering my guide Dr. P. Sam Johnson. It is because of him of whatever epsilon today I am.
The kind reply by Prof. Victor Kaftal, University of Cincinnati, Ohio developed confidence in me to do research. I am thankful to Prof. Victor Kaftal very much. Also, I thank Prof. P. G. Casazza, Missouri University, USA, Prof. Radu Balan, University of Maryland, USA, Prof. D. Freeman, St. Louis University, USA, Prof. Orr Moshe Shalit, Technion, Israel, Prof. Terence Tao, University of California, Los Angeles, USA and Prof. B. V. Rajarama Bhat, ISI Bangalore for their replies to my mails. I am thankful to all of them.
I am happy to thank my friends Ramu G., Vinoth A., Kanagaraj K., Palanivel R., Niranjan P.K., Sumukha S., Saumya Y. M., Shivarama K. N., Sachin M., Mahesh Mayya, Chaitanya G. K., Manasa M., Rashmi K., Megha P. for their help in my life at NITK.
My Research Progress Assessment Committee (RPAC) members Prof. Subhash C. Yaragal, Department of Civil Engineering and Prof. B. R. Shankar, Department of MACS helped me at various stages of research. I thank the RPAC members.
I would like to thank former heads of MACS department Prof. Santhosh George, Prof. B. R. Shankar, Prof. S. S. Kamath and the present head Dr. R. Madhusudhan for their help at different times. Further, I thank all faculty members of MACS department and office staff.
It was in MTTS where I started thinking seriously about results. I am thankful to Prof. S. Kumaresan, Central University of Hyderabad for the initiation of MTTS which transformed Mathematics of India to a greater extent.
The concept of Advanced Training in Mathematics Schools (ATM Schools) is a very useful initiative of National Centre for Mathematics (NCM) India which helps a lot for Ph.D. Scholars. I am fortunate to attend a dozen of these schools. Thanks to initiators and organizers of various ATM schools.
Ph.D. course work papers of mine were given by Dr. Murugan V., Mrs. Sujatha D. Achar, Prof. B. R. Shankar and Prof. A. H. Sequeira. I am thankful to them.
When I was teaching at Centre for Post-Graduate Studies and Research, St. Philomena College at Puttur, my classmates and colleagues Vaishnavi C. and Prasad H. M. helped at various times. My students also helped a lot understand the subject better. It is necessary to remember all of them.
Prof. M. S. Balasubramani, Prof. S. Parameshwara Bhatta, Mohana K. S., Dr. Chandru Hegde taught me at Mangalore University. I particularly remember them for their teaching.
I end the acknowledgements by remembering my father Narasimha D. K., my mother Sundari V., my sisters Shraddha K., Pavithra K., Pavana K., my brother late Ganesh K. R. and my primary teacher Nataliya K.
Place: NITK, Surathkal MAHESH KRISHNA K
Date: 17 February 2022
INTRODUCTION
GENERAL INTRODUCTION
A vector in a vector space is usually obtained as a linear combination of elements of a basis for the vector space. Thus a vector is fully known if we know the coefficients in the representation of it using basis elements. However, as dimension of the space increases, it is difficult to get the coefficients. Hence we look for nice spaces and certain bases which give coefficients of a given vector easily. For this purpose, Hilbert spaces and orthonormal bases become a very handy tool to obtain representation of a vector.
Orthonormal bases for Hilbert spaces have practical disadvantages. Since each coefficient in the expansion is very important, a small error in one of the coefficient leads to significant variation in the resultant vector and the actual vector. Thus we seek a collection in Hilbert space which gives representation as well as a small change in coefficient need not effect much to the original vector. This is where the theory of frames becomes important.
Historically it was Gabor [114 ] , who first studied representation of functions using translations and modulations of a single function ([103 , 104 , 106 , 123 ] (); Feichtinger and Strohmer 2003; ); Sondergaard 2007). In 1947, Sz. Nagy studied sequences which are close to orthonormal bases using Paley-Wiener type results ([81 ] ). Modern definition of frames was set by Duffin and Schaeffer [89 ] in the study of sequences of type { e i λ n x } n ∈ Z , λ n ∈ C , x ∈ ( − r , r ) , r > 0 \{e^{i\lambda_n x}\}_{n\in\mathbb{Z}}, \lambda_n\in\mathbb{C}, x\in(-r,r), r>0 { e i λ n x } n ∈ Z , λ n ∈ C , x ∈ ( − r , r ) , r > 0
Paper of Daubechies, Grossmann, and Meyer ([79 ] ) triggered the area of frames for Hilbert spaces. Later, the paper of Benedetto and Fickus [31 ] influenced the development of frame theory for finite dimensional Hilbert spaces. Today theory of frames find its uses in many areas such as wireless communication (Strohmer (2001)), signal processing [162 ] , image processing [82 ] , sampling theory [30 ] , filter banks [108 ] , psycho-acoustics [28 ] , quantum design [40 ] , quantum channels [127 ] , quantum optics (Jamioikowski (2010)), quantum measurement [93 ] , numerical approximation [2 ] , Sigma-Delta quantization [34 ] , coding (Strohmer and Heath (2003)) and graph theory [41 ] . For a comprehensive look on the theory of frames, we refer [65 , 129 , 128 , 56 , 130 ] (), Waldron 2018; ), Okoudjou 2016; ), Pesenson et al. 2017).
Since many spaces appearing both in theoretical and practicals are Banach spaces which may not be Hilbert spaces, there is a need for extending the notion of frames to Banach spaces. This was first done by Gröchenig [122 ] . After the study of several function spaces [99 ] , Gröchenig first studied the notion of an atomic decomposition for Banach spaces and then defined the notion of a Banach frame. Feichtinger and Gröchenig (1988, 1989a,b) in 90’s developed a theory of atomic decompositions and frames for a large class of function spaces such as modulation spaces [98 ] and coorbit spaces [35 ] , via, group representations and projective representations.
Abstract study of atomic decompositions and frames for Banach spaces started from the fundamental paper [53 ] . Further study and variations of the frames for Banach spaces are done in [44 ] , Terekhin (2010), Terekhin (2009), Terekhin (2004), [110 ] , [45 ] , Stoeva (2009), Stoeva (2012), [7 ] , [124 ] and so on.
ORTHONORMAL BASES, RIESZ BASES, FRAMES AND BESSEL SEQUENCES FOR HILBERT SPACES
In the study of integral equations, Hilbert studied the space of square integrable sequences [39 ] . Later, John von Neumann (1930) formulated the notion of Hilbert spaces.
Definition 1.2.1. ([cf. 158 ] ) A vector space H \mathcal{H} H K \mathbb{K} K R \mathbb{R} R C \mathbb{C} C ⟨ ⋅ , ⋅ ⟩ : H × H → K \langle\cdot,\cdot\rangle:\mathcal{H}\times\mathcal{H}\to\mathbb{K} ⟨ ⋅ , ⋅ ⟩ : H × H → K
(i) ⟨ h , h ⟩ ≥ 0 \langle h,h\rangle\ge0 ⟨ h , h ⟩ ≥ 0 ∀ h ∈ H \forall h \in\mathcal{H} ∀ h ∈ H
(ii) If h ∈ H h \in\mathcal{H} h ∈ H ⟨ h , h ⟩ = 0 \langle h,h\rangle= 0 ⟨ h , h ⟩ = 0 h = 0 h = 0 h = 0
(iii) ⟨ h , h 1 ⟩ = ⟨ h 1 , h ⟩ ‾ \langle h,h_1\rangle= \overline{\langle h_1,h\rangle} ⟨ h , h 1 ⟩ = ⟨ h 1 , h ⟩ ∀ h , h 1 ∈ H \forall h,h_1 \in\mathcal{H} ∀ h , h 1 ∈ H
(iv) ⟨ α h + h 1 , h 2 ⟩ = α ⟨ h , h 2 ⟩ + ⟨ h 1 , h 2 ⟩ \langle\alpha h+h_1,h_2\rangle= \alpha\langle h,h_2\rangle+\langle h_1,h_2\rangle ⟨ α h + h 1 , h 2 ⟩ = α ⟨ h , h 2 ⟩ + ⟨ h 1 , h 2 ⟩ ∀ h , h 1 , h 2 ∈ H \forall h,h_1,h_2 \in\mathcal{H} ∀ h , h 1 , h 2 ∈ H ∀ α ∈ K \forall\alpha\in\mathbb{K} ∀ α ∈ K
(v) H \mathcal{H} H ∥ h ∥ : = ⟨ h , h ⟩ \lVert h\rVert:= \sqrt{\langle h,h\rangle} ∥ h ∥ := ⟨ h , h ⟩
We now mention two important examples of Hilbert spaces.
Example 1.2.2. (cf. Limaye [158 ] )
(i) Let n ∈ N n \in\mathbb{N} n ∈ N K n \mathbb{K}^{n} K n
⟨ ( a k ) k = 1 n , ( b k ) k = 1 n ⟩ : = ∑ k = 1 n a k b k ‾ , ∀ ( a k ) k = 1 n , ( b k ) k = 1 n ∈ K n \langle(a_k)_{k=1}^{n},(b_k)_{k=1}^{n}\rangle:= \sum_{k=1}^{n}a_k\overline{b_k}, \quad\forall(a_k)_{k=1}^{n},(b_k)_{k=1}^{n} \in\mathbb{K}^{n} ⟨( a k ) k = 1 n , ( b k ) k = 1 n ⟩ := k = 1 ∑ n a k b k , ∀ ( a k ) k = 1 n , ( b k ) k = 1 n ∈ K n is a finite dimensional separable Hilbert space.
(ii) The space ℓ 2 ( N ) : = { { a n } n : a n ∈ K , ∀ n ∈ N , ∑ n = 1 ∞ ∣ a n ∣ 2 < ∞ } \ell^{2}(\mathbb{N}) := \{\{a_n\}_{n}: a_n \in\mathbb{K},\forall n \in\mathbb{N},\sum_{n=1}^{\infty}|a_n|^{2} < \infty\} ℓ 2 ( N ) := {{ a n } n : a n ∈ K , ∀ n ∈ N , ∑ n = 1 ∞ ∣ a n ∣ 2 < ∞ }
⟨ { a n } n , { b n } n ⟩ : = ∑ n = 1 ∞ a n b n ‾ , ∀ { a n } n , { b n } n ∈ ℓ 2 ( N ) \langle\{a_n\}_{n},\{b_n\}_{n}\rangle:= \sum_{n=1}^{\infty}a_n\overline{b_n}, \quad\forall\{a_n\}_{n},\{b_n\}_{n} \in\ell^{2}(\mathbb{N}) ⟨{ a n } n , { b n } n ⟩ := n = 1 ∑ ∞ a n b n , ∀ { a n } n , { b n } n ∈ ℓ 2 ( N ) is an infinite dimensional separable Hilbert space. The space ℓ 2 ( N ) \ell^{2}(\mathbb{N}) ℓ 2 ( N )
Throughout this thesis, we assume that all of Hilbert spaces are separable. Among all kinds of sets in a Hilbert space, orthonormal sets are the easiest to handle, whose definition reads as follows.
Definition 1.2.3. (cf. Limaye [158 ] ) A collection { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H orthonormal set in H \mathcal{H} H ⟨ τ j , τ k ⟩ = δ j , k , ∀ j , k ∈ N \langle\tau_j,\tau_k\rangle= \delta_{j,k},\forall j,k \in\mathbb{N} ⟨ τ j , τ k ⟩ = δ j , k , ∀ j , k ∈ N
Following theorem, known as Gram-Schmidt orthonormalization (cf. Leon et al. [152 ] ) shows that a linearly independent sequence of vectors can be converted into an orthonormal set such that at each stage of conversion the spaces spanned by the original set and transformed set are the same.
Theorem 1.2.4. (cf. Limaye [158 ] ) (Gram-Schmidt orthonormalization ) Let { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H ω 1 : = τ 1 \omega_1 := \tau_1 ω 1 := τ 1 ρ 1 : = ω 1 / ∥ ω 1 ∥ \rho_1 := \omega_1/\lVert\omega_1\rVert ρ 1 := ω 1 / ∥ ω 1 ∥
ω n : = τ n − ∑ k = 1 n − 1 ⟨ τ n , ρ k ⟩ ρ k , ρ n : = ω n ∥ ω n ∥ , ∀ n ≥ 2. \omega_n := \tau_n - \sum_{k=1}^{n-1}\langle\tau_n,\rho_k\rangle\rho_k,\quad\rho_n := \frac{\omega_n}{\lVert\omega_n\rVert},\quad\forall n \ge2. ω n := τ n − k = 1 ∑ n − 1 ⟨ τ n , ρ k ⟩ ρ k , ρ n := ∥ ω n ∥ ω n , ∀ n ≥ 2. :::`
Then { ρ n } n \{\rho_n\}_n { ρ n } n
span { ρ k } k = 1 n = span { τ k } k = 1 n , ∀ n ≥ 1. \operatorname{span}\{\rho_k\}_{k=1}^{n}=\operatorname{span}\{\tau_k\}_{k=1}^{n},\qquad\forall n\ge1. span { ρ k } k = 1 n = span { τ k } k = 1 n , ∀ n ≥ 1. One of the most important inequalities associated with an orthonormal sequence is the Bessel’s inequality. It is a generalization of Cauchy-Schwarz inequality.
Theorem 1.2.5. ([cf. 158 ] ) (Bessel’s inequality ) If { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 \sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2 ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 h ∈ H h\in\mathcal{H} h ∈ H
∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2\le\|h\|^2,\qquad\forall h\in\mathcal{H}. n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ ∥ h ∥ 2 , ∀ h ∈ H . Next theorem characterizes convergence of series in a Hilbert space with that of sequence of scalars.
Theorem 1.2.6. ([cf. 158 ] ) (Riesz-Fisher theorem ) Let { a n } n \{a_n\}_n { a n } n { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
∑ n = 1 ∞ a n τ n converges in H if and only if ∑ n = 1 ∞ ∣ a n ∣ 2 converges in R . \sum_{n=1}^{\infty}a_n\tau_n\text{ converges in }\mathcal{H}\text{ if and only if }\sum_{n=1}^{\infty}|a_n|^2\text{ converges in }\mathbb{R}. n = 1 ∑ ∞ a n τ n converges in H if and only if n = 1 ∑ ∞ ∣ a n ∣ 2 converges in R . Next theorem shows that given any orthonormal set and an element in a Hilbert space, the inner product of the element with the members of an orthonormal set can be non zero at most countably many times.
Theorem 1.2.7. ([cf. 158 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H h ∈ H h\in\mathcal{H} h ∈ H E h : = { τ n : ⟨ h , τ n ⟩ ≠ 0 , n ∈ N } E_h:=\{\tau_n:\langle h,\tau_n\rangle\ne0,n\in\mathbb{N}\} E h := { τ n : ⟨ h , τ n ⟩ = 0 , n ∈ N }
A natural analogue of basis for finite dimensional vector spaces to that of infinite dimensional Hilbert spaces is the notion of Schauder basis and orthonormal basis.
Definition 1.2.8. ([cf. 65 ] ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) a Schauder basis for H \mathcal{H} H if for each h ∈ H h\in\mathcal{H} h ∈ H { a n ( h ) } n \{a_n(h)\}_n { a n ( h ) } n ∑ n = 1 ∞ a n ( h ) τ n \sum_{n=1}^{\infty}a_n(h)\tau_n ∑ n = 1 ∞ a n ( h ) τ n H \mathcal{H} H h = ∑ n = 1 ∞ a n ( h ) τ n h=\sum_{n=1}^{\infty}a_n(h)\tau_n h = ∑ n = 1 ∞ a n ( h ) τ n
(ii) an orthonormal basis for H \mathcal{H} H if it is a Schauder basis for H \mathcal{H} H
We now give various examples of orthonormal bases for Hilbert spaces.
Example 1.2.9. ([cf. 65 ] )
(i) Define e n : = { δ n , k } k e_n := \{\delta_{n,k}\}_k e n := { δ n , k } k δ ⋅ , ⋅ \delta_{\cdot,\cdot} δ ⋅ , ⋅ { e n } n \{e_n\}_n { e n } n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
(ii) Define
L 2 [ 0 , 1 ] : = { f : [ 0 , 1 ] → C is measurable and ∫ 0 1 ∣ f ( x ) ∣ 2 d x < ∞ } \mathcal{L}^2[0,1] := \left\{f : [0,1] \to\mathbb{C}\text{ is measurable and }\int_0^1 |f(x)|^2\,\mathrm{d}x < \infty\right\} L 2 [ 0 , 1 ] := { f : [ 0 , 1 ] → C is measurable and ∫ 0 1 ∣ f ( x ) ∣ 2 d x < ∞ } equipped with the inner product
⟨ f , g ⟩ : = ∫ 0 1 f ( x ) g ( x ) ‾ d x . \langle f,g\rangle:= \int_0^1 f(x)\overline{g(x)}\,\mathrm{d}x. ⟨ f , g ⟩ := ∫ 0 1 f ( x ) g ( x ) d x . Let n ∈ Z n \in\mathbb{Z} n ∈ Z e n : [ 0 , 1 ] ∋ x ↦ e 2 π i n x ∈ C e_n : [0,1] \ni x \mapsto e^{2\pi i n x} \in\mathbb{C} e n : [ 0 , 1 ] ∋ x ↦ e 2 π in x ∈ C { e n } n = − ∞ ∞ \{e_n\}_{n=-\infty}^{\infty} { e n } n = − ∞ ∞ L 2 [ 0 , 1 ] \mathcal{L}^2[0,1] L 2 [ 0 , 1 ]
(iii) (Gabor basis ) Define
L 2 ( R ) : = { f : R → C is measurable and ∫ − ∞ ∞ ∣ f ( x ) ∣ 2 d x < ∞ } \mathcal{L}^2(\mathbb{R}) := \left\{f : \mathbb{R} \to\mathbb{C}\text{ is measurable and }\int_{-\infty}^{\infty} |f(x)|^2\,\mathrm{d}x < \infty\right\} L 2 ( R ) := { f : R → C is measurable and ∫ − ∞ ∞ ∣ f ( x ) ∣ 2 d x < ∞ } equipped with the inner product
⟨ f , g ⟩ : = ∫ − ∞ ∞ f ( x ) g ( x ) ‾ d x . \langle f,g\rangle:= \int_{-\infty}^{\infty} f(x)\overline{g(x)}\,\mathrm{d}x. ⟨ f , g ⟩ := ∫ − ∞ ∞ f ( x ) g ( x ) d x . Let χ [ 0 , 1 ] \chi_{[0,1]} χ [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] j , k ∈ Z j,k \in\mathbb{Z} j , k ∈ Z f j , k ( x ) : = e 2 π i j x χ [ 0 , 1 ] ( x − k ) , ∀ x ∈ R f_{j,k}(x) := e^{2\pi i j x}\chi_{[0,1]}(x-k),\forall x \in\mathbb{R} f j , k ( x ) := e 2 π ij x χ [ 0 , 1 ] ( x − k ) , ∀ x ∈ R { f j , k } j , k ∈ Z \{f_{j,k}\}_{j,k\in\mathbb{Z}} { f j , k } j , k ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(iv) (Haar system ) Let ψ \psi ψ R \mathbb{R} R
ψ ( x ) : = { 1 if 0 ≤ x < 1 2 , − 1 if 1 2 ≤ x ≤ 1 , 0 otherwise . \psi(x) :=
\begin{cases}
1 & \text{if }0 \le x < \frac{1}{2},\\
-1 & \text{if }\frac{1}{2} \le x \le1,\\
0 & \text{otherwise}.
\end{cases} ψ ( x ) := ⎩ ⎨ ⎧ 1 − 1 0 if 0 ≤ x < 2 1 , if 2 1 ≤ x ≤ 1 , otherwise . For j , k ∈ Z j,k \in\mathbb{Z} j , k ∈ Z ψ j , k ( x ) : = 2 j / 2 ψ ( 2 j x − k ) , ∀ x ∈ R \psi_{j,k}(x) := 2^{j/2}\psi(2^j x-k),\forall x \in\mathbb{R} ψ j , k ( x ) := 2 j /2 ψ ( 2 j x − k ) , ∀ x ∈ R { ψ j , k } j , k ∈ Z \{\psi_{j,k}\}_{j,k\in\mathbb{Z}} { ψ j , k } j , k ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(v) (cf. Christensen [65 ] ) For a , b > 0 a,b > 0 a , b > 0
T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) : = f ( x − a ) ∈ C T_a : \mathcal{L}^2(\mathbb{R}) \ni f \mapsto T_a f \in\mathcal{L}^2(\mathbb{R}),\quad T_a f : \mathbb{R} \mapsto(T_a f)(x) := f(x-a) \in\mathbb{C} T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) := f ( x − a ) ∈ C and
E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) : = e 2 π i b x f ( x ) ∈ C . E_b:\mathcal{L}^2(\mathbb{R})\ni f\mapsto E_bf\in\mathcal{L}^2(\mathbb{R}),\quad E_bf:\mathbb{R}\mapsto(E_bf)(x):=e^{2\pi ibx}f(x)\in\mathbb{C}. E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) := e 2 π ib x f ( x ) ∈ C . Let g : R → C g:\mathbb{R}\to\mathbb{C} g : R → C a , b > 0 a,b>0 a , b > 0 { E m b T n a g } n , m ∈ Z \{E_{mb}T_{na}g\}_{n,m\in\mathbb{Z}} { E mb T na g } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
Following theorem shows that given an orthonormal set, we can check whether it is an orthonormal basis by checking several equivalent conditions rather appealing to Definition 1.2.8, which is difficult in many cases.
Theorem 1.2.10. ([cf. 65 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(ii) (Fourier expansion ) h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n , ∀ h ∈ H h=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n,\ \forall h\in\mathcal{H} h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n , ∀ h ∈ H
(iii) (Parseval identity for the inner product ) ⟨ h , g ⟩ = ∑ n = 1 ∞ ⟨ h , τ n ⟩ ⟨ τ n , g ⟩ , ∀ h , g ∈ H \langle h,g\rangle=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\langle\tau_n,g\rangle,\ \forall h,g\in\mathcal{H} ⟨ h , g ⟩ = ∑ n = 1 ∞ ⟨ h , τ n ⟩ ⟨ τ n , g ⟩ , ∀ h , g ∈ H
(iv) (Parseval identity for the norm ) ∥ h ∥ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 , h ∈ H \lVert h\rVert^2=\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2,\ h\in\mathcal{H} ∥ h ∥ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 , h ∈ H
(v) span ‾ { τ n } n = H \overline{\operatorname{span}}\{\tau_n\}_n=\mathcal{H} span { τ n } n = H
(vi) If h ∈ H h\in\mathcal{H} h ∈ H ⟨ h , τ n ⟩ = 0 , ∀ n ∈ N \langle h,\tau_n\rangle=0,\ \forall n\in\mathbb{N} ⟨ h , τ n ⟩ = 0 , ∀ n ∈ N h = 0 h=0 h = 0
Theorem 1.2.11. ([cf. 65 ] ) A Hilbert space is separable if and only if it has a countable orthonormal basis.
Hilbert spaces have the remarkable property that every bounded linear functional from the space to the scalar field is given by the inner product with a unique element in the space.
Theorem 1.2.12. ([cf. 158 ] ) (Riesz representation theorem ) Let f : H → K f:\mathcal{H}\to\mathbb{K} f : H → K τ f ∈ H \tau_f\in\mathcal{H} τ f ∈ H
f ( h ) = ⟨ h , τ f ⟩ , ∀ h ∈ H and ∥ f ∥ = ∥ τ f ∥ . f(h)=\langle h,\tau_f\rangle,\quad\forall h\in\mathcal{H}\quad\text{and}\quad\lVert f\rVert=\lVert\tau_f\rVert. f ( h ) = ⟨ h , τ f ⟩ , ∀ h ∈ H and ∥ f ∥ = ∥ τ f ∥ . Riesz representation theorem opens the door to the following definition.
Definition 1.2.13. ([cf. 158 ] ) Let T : H → H 0 T:\mathcal{H}\to\mathcal{H}_0 T : H → H 0 T ∗ : H 0 → H T^*:\mathcal{H}_0\to\mathcal{H} T ∗ : H 0 → H
⟨ T h , h 0 ⟩ = ⟨ h , T ∗ h 0 ⟩ , ∀ h ∈ H , ∀ h 0 ∈ H 0 \langle Th,h_0\rangle=\langle h,T^*h_0\rangle,\quad\forall h\in\mathcal{H},\ \forall h_0\in\mathcal{H}_0 ⟨ T h , h 0 ⟩ = ⟨ h , T ∗ h 0 ⟩ , ∀ h ∈ H , ∀ h 0 ∈ H 0 is called as the adjoint of T T T
Hilbert spaces are studied along with various kinds of operators. These are defined as follows.
Definition 1.2.14. [cf. 158 ] Let T : H → H 0 T : \mathcal{H} \to\mathcal{H}_0 T : H → H 0 T T T
invertible if there exists a bounded linear operator S : H 0 → H S : \mathcal{H}_0 \to\mathcal{H} S : H 0 → H S T = I H ST = I_{\mathcal{H}} S T = I H T S = I H 0 TS = I_{\mathcal{H}_0} T S = I H 0
isometry if ∥ T h ∥ = ∥ h ∥ \lVert Th\rVert= \lVert h\rVert ∥ T h ∥ = ∥ h ∥ ∀ h ∈ H \forall h \in\mathcal{H} ∀ h ∈ H
unitary if T T ∗ = I H 0 TT^* = I_{\mathcal{H}_0} T T ∗ = I H 0 T ∗ T = I H T^*T = I_{\mathcal{H}} T ∗ T = I H
Definition 1.2.15. [cf. 158 ] Let T : H → H T : \mathcal{H} \to\mathcal{H} T : H → H
Operator T T T normal operator if T T ∗ = T ∗ T TT^* = T^*T T T ∗ = T ∗ T
Operator T T T projection if T 2 = T = T ∗ T^2 = T = T^* T 2 = T = T ∗
Operator T T T self-adjoint operator if T = T ∗ T = T^* T = T ∗
Operator T T T positive operator if T = T ∗ T = T^* T = T ∗ ⟨ T h , h ⟩ ≥ 0 \langle Th,h\rangle\ge0 ⟨ T h , h ⟩ ≥ 0 ∀ h ∈ H \forall h \in\mathcal{H} ∀ h ∈ H
Theorem 1.2.16. [cf. 158 ] Let H \mathcal{H} H
If H \mathcal{H} H H \mathcal{H} H K n \mathbb{K}^n K n n n n
If H \mathcal{H} H H \mathcal{H} H ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
Orthonormal bases have the nice property that given a single orthonormal basis, we can generate all of them just by acting unitary operators.
Theorem 1.2.17. [cf. 65 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H \mathcal{H} H { U τ n } n \{U\tau_n\}_n { U τ n } n U ∈ B ( H ) U \in\mathcal{B}(\mathcal{H}) U ∈ B ( H )
Hilbert spaces have another nice property that closed subspaces decompose the original space.
Theorem 1.2.18. [cf. 158 ] (Orthogonal complement theorem ) If W \mathcal{W} W H \mathcal{H} H H = W ⊕ W ⊥ \mathcal{H} = \mathcal{W} \oplus\mathcal{W}^{\perp} H = W ⊕ W ⊥ W ⊥ \mathcal{W}^{\perp} W ⊥ W \mathcal{W} W H \mathcal{H} H
First level of generalization of orthonormal basis is that of Riesz basis. These are defined as follows.
Definition 1.2.19. ([cf. 65 ] ) A collection { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H H \mathcal{H} H { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H T ∈ B ( H ) T \in\mathcal{B}(\mathcal{H}) T ∈ B ( H ) ω n = T τ n , ∀ n \omega_n = T\tau_n,\ \forall n ω n = T τ n , ∀ n
As written by Simon (2015), the origin of the term "Riesz basis" is unknown. By taking T T T
Example 1.2.20. (i) Let { λ n } n = 1 ∞ \{\lambda_n\}_{n=1}^{\infty} { λ n } n = 1 ∞ a , b > 0 a,b > 0 a , b > 0 a ≤ ∣ λ n ∣ ≤ b , ∀ n ∈ N a \leq|\lambda_n| \leq b,\ \forall n \in\mathbb{N} a ≤ ∣ λ n ∣ ≤ b , ∀ n ∈ N { λ n e n } n = 1 ∞ \{\lambda_n e_n\}_{n=1}^{\infty} { λ n e n } n = 1 ∞ ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) { e n } n = 1 ∞ \{e_n\}_{n=1}^{\infty} { e n } n = 1 ∞
T : ℓ 2 ( N ) ∋ { x n } n = 1 ∞ ↦ { λ n x n } n = 1 ∞ ∈ ℓ 2 ( N ) . T:\ell^2(\mathbb{N}) \ni\{x_n\}_{n=1}^{\infty} \mapsto\{\lambda_n x_n\}_{n=1}^{\infty} \in\ell^2(\mathbb{N}). T : ℓ 2 ( N ) ∋ { x n } n = 1 ∞ ↦ { λ n x n } n = 1 ∞ ∈ ℓ 2 ( N ) . We note further that if ∣ λ n ∣ ≠ 1 |\lambda_n| \neq1 ∣ λ n ∣ = 1 n n n { λ n e n } n = 1 ∞ \{\lambda_n e_n\}_{n=1}^{\infty} { λ n e n } n = 1 ∞ ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
(ii) ([141 ] ) (Kadec 1/4 theorem ) Let { λ n } n ∈ Z \{\lambda_n\}_{n\in\mathbb{Z}} { λ n } n ∈ Z
sup n ∈ N ∣ λ n − n ∣ < 1 4 , ∀ n ∈ Z . \sup_{n\in\mathbb{N}}|\lambda_n-n| < \frac{1}{4}, \quad\forall n \in\mathbb{Z}. n ∈ N sup ∣ λ n − n ∣ < 4 1 , ∀ n ∈ Z . Define f n : ( − π , π ) ∋ x ↦ e i λ n x ∈ C , ∀ n ∈ Z f_n:(-\pi,\pi)\ni x\mapsto e^{i\lambda_n x}\in\mathbb{C},\ \forall n\in\mathbb{Z} f n : ( − π , π ) ∋ x ↦ e i λ n x ∈ C , ∀ n ∈ Z { f n } n ∈ Z \{f_n\}_{n\in\mathbb{Z}} { f n } n ∈ Z L 2 ( − π , π ) L^2(-\pi,\pi) L 2 ( − π , π ) 1 / 4 1/4 1/4 [153 ] , [cf. 64 ] ).
(iii) ([46 ] ) (Kalton-Casazza theorem ) A linear combination of two orthonormal bases is a Riesz basis.
(iv) ([cf. 65 ] ) For a , b > 0 a,b>0 a , b > 0
T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) : = f ( x − a ) ∈ C T_a:L^2(\mathbb{R})\ni f\mapsto T_a f\in L^2(\mathbb{R}),\qquad T_a f:\mathbb{R}\mapsto(T_a f)(x):=f(x-a)\in\mathbb{C} T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) := f ( x − a ) ∈ C and
E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) : = e 2 π i b x f ( x ) ∈ C . E_b:L^2(\mathbb{R})\ni f\mapsto E_b f\in L^2(\mathbb{R}),\qquad E_b f:\mathbb{R}\mapsto(E_b f)(x):=e^{2\pi i b x}f(x)\in\mathbb{C}. E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) := e 2 π ib x f ( x ) ∈ C . Let g : R → C g:\mathbb{R}\to\mathbb{C} g : R → C a , b > 0 a,b>0 a , b > 0 { E m b T n a g } n , m ∈ Z \{E_{mb}T_{na}g\}_{n,m\in\mathbb{Z}} { E mb T na g } n , m ∈ Z L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
Like orthonormal basis, Riesz basis will also give a series representation of every element in a Hilbert space.
Theorem 1.2.22. [cf. 65 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) There exists a unique collection { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H
h = ∑ n = 1 ∞ ⟨ h , ω n ⟩ τ n , ∀ h ∈ H . (1) h = \sum_{n=1}^{\infty} \langle h,\omega_n\rangle\tau_n,\quad\forall h \in\mathcal{H}. \tag*{(1)} h = n = 1 ∑ ∞ ⟨ h , ω n ⟩ τ n , ∀ h ∈ H . (1) Moreover, { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H (1) converges unconditionally for all h ∈ H h \in\mathcal{H} h ∈ H
(ii) There exist a , b > 0 a,b > 0 a , b > 0 a ∥ h ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H a\|h\|^2 \leq\sum_{n=1}^{\infty} |\langle h,\tau_n\rangle|^2 \leq b\|h\|^2,\ \forall h \in\mathcal{H} a ∥ h ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H
Next result says that there is a characterization of Riesz basis which is free from orthonormal basis. It also gives a tool to check whether a collection is a Riesz basis for a given Hilbert space. To state the result, we need two definitions.
Definition 1.2.23. [cf. 65 ] A sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H complete if span ‾ n ∈ N { τ n } = H \overline{\operatorname{span}}_{n\in\mathbb{N}}\{\tau_n\} = \mathcal{H} span n ∈ N { τ n } = H
Definition 1.2.24. [cf. 65 ] A sequence { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H biorthogonal to a sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H ⟨ ω n , τ m ⟩ = δ n , m \langle\omega_n,\tau_m\rangle= \delta_{n,m} ⟨ ω n , τ m ⟩ = δ n , m n , m n,m n , m
Theorem 1.2.25. [cf. 65 , 130 ] (); Stoeva 2020) For a sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(ii) span ‾ { τ n } n = H \overline{\operatorname{span}}\{\tau_n\}_n = \mathcal{H} span { τ n } n = H a , b > 0 a,b > 0 a , b > 0 S S S N \mathbb{N} N
a ∑ n ∈ S ∣ c n ∣ 2 ≤ ∥ ∑ n ∈ S c n τ n ∥ 2 ≤ b ∑ n ∈ S ∣ c n ∣ 2 , ∀ c n ∈ K . (2) a\sum_{n\in S}|c_n|^2 \leq\left\|\sum_{n\in S}c_n\tau_n\right\|^2 \leq b\sum_{n\in S}|c_n|^2,\quad\forall c_n \in\mathbb{K}. \tag*{(2)} a n ∈ S ∑ ∣ c n ∣ 2 ≤ n ∈ S ∑ c n τ n 2 ≤ b n ∈ S ∑ ∣ c n ∣ 2 , ∀ c n ∈ K . (2) (iii) { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H infinite Gram matrix [ ⟨ τ m , τ n ⟩ ] 1 ≤ n , m < ∞ [\langle\tau_m,\tau_n\rangle]_{1\leq n,m<\infty} [⟨ τ m , τ n ⟩ ] 1 ≤ n , m < ∞
ℓ 2 ( N ) ∋ { c m } m ↦ { ∑ m = 1 ∞ ⟨ τ n , τ m ⟩ c m } n ∈ ℓ 2 ( N ) \ell^2(\mathbb{N}) \ni\{c_m\}_m \mapsto\left\{\sum_{m=1}^{\infty}\langle\tau_n,\tau_m\rangle c_m\right\}_n \in\ell^2(\mathbb{N}) ℓ 2 ( N ) ∋ { c m } m ↦ { m = 1 ∑ ∞ ⟨ τ n , τ m ⟩ c m } n ∈ ℓ 2 ( N ) (iv) { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H
(v) { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H ∑ n = 1 ∞ c n τ n \sum_{n=1}^{\infty} c_n\tau_n ∑ n = 1 ∞ c n τ n H \mathcal{H} H ∑ n = 1 ∞ ∣ c n ∣ 2 < ∞ \sum_{n=1}^{\infty} \lvert c_n\rvert^2 < \infty ∑ n = 1 ∞ ∣ c n ∣ 2 < ∞
Theorem 1.2.25 leads to the following definition.
Definition 1.2.27. (cf. Christensen [65 ] ) A sequence { ω n } n \{\omega_n\}_{n} { ω n } n H \mathcal{H} H Riesz sequence for H \mathcal{H} H a , b > 0 a,b>0 a , b > 0 S S S N \mathbb{N} N
a ∑ n ∈ S ∣ c n ∣ 2 ≤ ∥ ∑ n ∈ S c n τ n ∥ 2 ≤ b ∑ n ∈ S ∣ c n ∣ 2 , ∀ c n ∈ K . a\sum_{n\in S}\lvert c_n\rvert^2\leq\left\lVert\sum_{n\in S}c_n\tau_n\right\rVert^2\leq b\sum_{n\in S}\lvert c_n\rvert^2,\qquad\forall c_n\in\mathbb{K}. a n ∈ S ∑ ∣ c n ∣ 2 ≤ n ∈ S ∑ c n τ n 2 ≤ b n ∈ S ∑ ∣ c n ∣ 2 , ∀ c n ∈ K . Theorem 1.2.25 says that every Riesz basis is a Riesz sequence. It is easy to see that a Riesz sequence need not be a Riesz basis. Following theorem gives another characterization of Riesz basis which is also free from orthonormal basis. It also helps to check whether a collection is a Riesz basis.
Theorem 1.2.28. (cf. Gohberg and Krein [121 ] ) (Kothe-Lorch theorem ) A sequence { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H
(i) { τ n } n \{\tau_n\}_{n} { τ n } n H \mathcal{H} H
(ii) 0 < inf n ∈ N ∥ τ n ∥ ≤ sup n ∈ N ∥ τ n ∥ < ∞ 0<\inf_{n\in\mathbb{N}}\lVert\tau_n\rVert\leq\sup_{n\in\mathbb{N}}\lVert\tau_n\rVert<\infty 0 < inf n ∈ N ∥ τ n ∥ ≤ sup n ∈ N ∥ τ n ∥ < ∞
It is clear that whenever we perturb an orthonormal basis, we may not get an orthonormal basis. However, it is a classical theorem of Paley and Wiener which says whenever we perturb an orthonormal basis, we get a Riesz basis.
Theorem 1.2.30. (Paley and Wiener (1987), cf. Young, 1980) (Paley-Wiener theorem ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H 0 < α < 1 0 < \alpha< 1 0 < α < 1 m = 1 , 2 , … m = 1,2,\ldots m = 1 , 2 , …
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ( ∑ n = 1 m ∣ c n ∣ 2 ) 1 2 , ∀ c n ∈ K , \left\Vert\sum_{n=1}^{m} c_n(\tau_n-\omega_n) \right\Vert\leq\alpha\left(\sum_{n=1}^{m} |c_n|^2\right)^{\frac{1}{2}}, \qquad\forall c_n \in\mathbb{K}, n = 1 ∑ m c n ( τ n − ω n ) ≤ α ( n = 1 ∑ m ∣ c n ∣ 2 ) 2 1 , ∀ c n ∈ K , then { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H
Next level of generalization of Riesz basis for Hilbert spaces is the notion of frame.
Definition 1.2.31. (Duffin and Schaeffer [89 ] ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H \mathcal{H} H a , b > 0 a,b > 0 a , b > 0
a ∥ h ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . (3) a\lVert h\rVert^2 \leq\sum_{n=1}^{\infty} \lvert\langle h,\tau_n\rangle\rvert^2 \leq b\lVert h\rVert^2,\qquad\forall h\in\mathcal{H}. \tag*{(3)} a ∥ h ∥ 2 ≤ n = 1 ∑ ∞ ∣⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . (3) Constants a a a b b b lower frame bound and upper frame bound , respectively. Supremum (resp. infimum) of the set of all lower (resp. upper) frame bounds is called optimal lower frame bound (resp. optimal upper frame bound ). If the optimal frame bounds are equal, then the frame is called as tight frame . A tight frame whose optimal frame bound is one is termed as Parseval frame .
As recorded by Kovacevic and Chebira [150 ] , and Heil [131 ] , the reason for using the term “frame” is unknown.
Note that in Definition 1.2.31 we indexed the frame by natural numbers. Since the convergence of series in Definition 1.2.31 is unconditional, any rearrangement of a frame is again a frame. We also note that Definition 1.2.31 can be formulated for arbitrary indexing set J \mathbb{J} J J \mathbb{J} J
From (ii) in Theorem 1.2.22 we can conclude that every Riesz basis is a frame. On the other hand, every frame can be written as a finite union of Riesz sequences (which is called as Feichtinger conjecture (Casazza et al. [50 ] ) and is known as Marcus-Spielman-Srivastava Theorem (cf. Casazza and Edidin [52 ] ; Casazza and Tremain [58 ] ). We now give various examples of frames.
Example 1.2.32. (i) Let { e n } n = 1 ∞ \{e_n\}_{n=1}^{\infty} { e n } n = 1 ∞ H \mathcal{H} H m ∈ N m \in\mathbb{N} m ∈ N { e 1 , … , e m } ∪ { e n } n = 1 ∞ \{e_1,\ldots,e_m\}\cup\{e_n\}_{n=1}^{\infty} { e 1 , … , e m } ∪ { e n } n = 1 ∞ H \mathcal{H} H h ∈ H h\in\mathcal{H} h ∈ H
1 ⋅ ∥ h ∥ 2 = ∑ n = 1 ∞ ∣ ⟨ h , e n ⟩ ∣ 2 ≤ ∑ k = 1 m ∣ ⟨ h , e k ⟩ ∣ 2 + ∑ n = 1 ∞ ∣ ⟨ h , e n ⟩ ∣ 2 ≤ 2 ∑ n = 1 ∞ ∣ ⟨ h , e n ⟩ ∣ 2 = 2 ∥ h ∥ 2 . 1\cdot\lVert h\rVert^2=\sum_{n=1}^{\infty}|\langle h,e_n\rangle|^2\leq\sum_{k=1}^{m}|\langle h,e_k\rangle|^2+\sum_{n=1}^{\infty}|\langle h,e_n\rangle|^2\leq2\sum_{n=1}^{\infty}|\langle h,e_n\rangle|^2=2\lVert h\rVert^2. 1 ⋅ ∥ h ∥ 2 = n = 1 ∑ ∞ ∣ ⟨ h , e n ⟩ ∣ 2 ≤ k = 1 ∑ m ∣ ⟨ h , e k ⟩ ∣ 2 + n = 1 ∑ ∞ ∣ ⟨ h , e n ⟩ ∣ 2 ≤ 2 n = 1 ∑ ∞ ∣ ⟨ h , e n ⟩ ∣ 2 = 2 ∥ h ∥ 2 . In particular, the collection in Remark 1.2.26 is a frame for ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
(ii) [cf. 65 ] (Harmonic frame ) Let m , n ∈ N m,n\in\mathbb{N} m , n ∈ N n ≤ m n\leq m n ≤ m ω 1 , … , ω m \omega_1,\ldots,\omega_m ω 1 , … , ω m m t h m^{\mathrm{th}} m th
η k : = 1 m ( ω 1 k , … , ω n k ) , 1 ≤ k ≤ m . \eta_k:=\frac{1}{\sqrt{m}}(\omega_1^k,\ldots,\omega_n^k),\qquad1\leq k\leq m. η k := m 1 ( ω 1 k , … , ω n k ) , 1 ≤ k ≤ m . Then { η k } k = 1 m \{\eta_k\}_{k=1}^{m} { η k } k = 1 m C n \mathbb{C}^n C n
(iii) [cf. 150 ] (), Shor 2004) (Mercedes-Benz frame or Peres-Wooters states ) { ( 0 , 1 ) , ( − 3 / 2 , − 1 / 2 ) , ( 3 / 2 , − 1 / 2 ) } \{(0,1),(-\sqrt{3}/2,-1/2),(\sqrt{3}/2,-1/2)\} {( 0 , 1 ) , ( − 3 /2 , − 1/2 ) , ( 3 /2 , − 1/2 )} R 2 \mathbb{R}^2 R 2 3 / 2 3/2 3/2
(iv) [cf. 128 ] For n ≥ 3 n\geq3 n ≥ 3 { ( cos ( 2 π j / n ) , sin ( 2 π j / n ) ) } j = 0 n − 1 \{(\cos(2\pi j/n),\sin(2\pi j/n))\}_{j=0}^{n-1} {( cos ( 2 π j / n ) , sin ( 2 π j / n )) } j = 0 n − 1 R 2 \mathbb{R}^2 R 2 n = 2 n=2 n = 2 { ( 1 , 0 ) , ( − 1 , 0 ) } \{(1,0),(-1,0)\} {( 1 , 0 ) , ( − 1 , 0 )} R 2 \mathbb{R}^2 R 2
(v) [cf. 65 ] (Gabor frame or Weyl-Heisenberg frame ) Let g g g g : R ∋ x ↦ g ( x ) : = e − x 2 ∈ R g:\mathbb{R}\ni x\mapsto g(x):=e^{-x^2}\in\mathbb{R} g : R ∋ x ↦ g ( x ) := e − x 2 ∈ R a , b > 0 a,b>0 a , b > 0 f n , m : R ∋ x ↦ e 2 π i m b x g ( x − n a ) ∈ R f_{n,m}:\mathbb{R}\ni x\mapsto e^{2\pi imb x}g(x-na)\in\mathbb{R} f n , m : R ∋ x ↦ e 2 π imb x g ( x − na ) ∈ R ∀ n , m ∈ Z \forall n,m\in\mathbb{Z} ∀ n , m ∈ Z { f n , m } n , m ∈ Z \{f_{n,m}\}_{n,m\in\mathbb{Z}} { f n , m } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R ) a b < 1 ab<1 ab < 1 { f n , m } n , m ∈ Z \{f_{n,m}\}_{n,m\in\mathbb{Z}} { f n , m } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R ) a b = 1 ab=1 ab = 1 { f n , m } n , m ∈ Z \{f_{n,m}\}_{n,m\in\mathbb{Z}} { f n , m } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(vi) (Duffin and Schaeffer [89 ] ) Let { λ n } n ∈ Z \{\lambda_n\}_{n\in\mathbb{Z}} { λ n } n ∈ Z d , L , δ > 0 d,L,\delta>0 d , L , δ > 0
∣ λ n − n d ∣ ≤ L , ∀ n ∈ Z and ∣ λ n − λ m ∣ ≥ δ , ∀ n , m ∈ Z , n ≠ m . \left|\lambda_n-\frac{n}{d}\right|\leq L,\quad\forall n\in\mathbb{Z}\qquad\text{and}\qquad|\lambda_n-\lambda_m|\geq\delta,\quad\forall n,m\in\mathbb{Z},n\neq m. λ n − d n ≤ L , ∀ n ∈ Z and ∣ λ n − λ m ∣ ≥ δ , ∀ n , m ∈ Z , n = m . Let 0 < r < d π 0<r<d\pi 0 < r < d π f n : ( − r , r ) ∋ x ↦ e i λ n x ∈ C f_n:(-r,r)\ni x\mapsto e^{i\lambda_nx}\in\mathbb{C} f n : ( − r , r ) ∋ x ↦ e i λ n x ∈ C ∀ n ∈ Z \forall n\in\mathbb{Z} ∀ n ∈ Z { f n } n ∈ Z \{f_n\}_{n\in\mathbb{Z}} { f n } n ∈ Z L 2 ( − r , r ) \mathcal{L}^2(-r,r) L 2 ( − r , r )
(vii) (cf. Feichtinger and Strohmer (2003)) For a , b > 0 a,b>0 a , b > 0
T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) : = f ( x − a ) ∈ C T_a:\mathcal{L}^2(\mathbb{R})\ni f\mapsto T_af\in\mathcal{L}^2(\mathbb{R}),\qquad T_af:\mathbb{R}\mapsto(T_af)(x):=f(x-a)\in\mathbb{C} T a : L 2 ( R ) ∋ f ↦ T a f ∈ L 2 ( R ) , T a f : R ↦ ( T a f ) ( x ) := f ( x − a ) ∈ C and
E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) : = e 2 π i b x f ( x ) ∈ C . E_b:\mathcal{L}^2(\mathbb{R})\ni f\mapsto E_bf\in\mathcal{L}^2(\mathbb{R}),\quad E_bf:\mathbb{R}\mapsto(E_bf)(x):=e^{2\pi ibx}f(x)\in\mathbb{C}. E b : L 2 ( R ) ∋ f ↦ E b f ∈ L 2 ( R ) , E b f : R ↦ ( E b f ) ( x ) := e 2 π ib x f ( x ) ∈ C .
For c > 0 c>0 c > 0 χ [ 0 , c ] \chi_{[0,c]} χ [ 0 , c ] [ 0 , c ] [0,c] [ 0 , c ] a ≤ c ≤ 1 a\leq c\leq1 a ≤ c ≤ 1 { E m b T n a χ [ 0 , c ] } n , m ∈ Z \{E_{mb}T_{na}\chi_{[0,c]}\}_{n,m\in\mathbb{Z}} { E mb T na χ [ 0 , c ] } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R ) [78 ] ).
(viii) (Janssen and Strohmer [138 ] ) Let T a T_a T a E b E_b E b g ( x ) : = cosh ( π x ) = 2 e π x + e − π x g(x):=\cosh(\pi x)=\frac{2}{e^{\pi x}+e^{-\pi x}} g ( x ) := cosh ( π x ) = e π x + e − π x 2 ∀ x ∈ R \forall x\in\mathbb{R} ∀ x ∈ R a b < 1 ab<1 ab < 1 { E m b T n a g } n , m ∈ Z \{E_{mb}T_{na}g\}_{n,m\in\mathbb{Z}} { E mb T na g } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(ix) (Janssen [137 ] ) Let T a T_a T a E b E_b E b g ( x ) : = e − ∣ x ∣ g(x):=e^{-|x|} g ( x ) := e − ∣ x ∣ ∀ x ∈ R \forall x\in\mathbb{R} ∀ x ∈ R a b < 1 ab<1 ab < 1 { E m b T n a g } n , m ∈ Z \{E_{mb}T_{na}g\}_{n,m\in\mathbb{Z}} { E mb T na g } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(x) (Janssen [136 ] ) Let T a T_a T a E b E_b E b g ( x ) : = e − x χ [ 0 , ∞ ) ( x ) g(x):=e^{-x}\chi_{[0,\infty)}(x) g ( x ) := e − x χ [ 0 , ∞ ) ( x ) ∀ x ∈ R \forall x\in\mathbb{R} ∀ x ∈ R { E m b T n a g } n , m ∈ Z \{E_{mb}T_{na}g\}_{n,m\in\mathbb{Z}} { E mb T na g } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R ) a b ≤ 1 ab\leq1 ab ≤ 1
(xi) (Casazza [46 ] ) (Kalton-Casazza theorem ) Every frame is a sum of three orthonormal bases.
(xii) (Christensen [65 ] ) (Wavelet frame ) Let 0 < b < 0.0084 0<b<0.0084 0 < b < 0.0084 Mexican hat function
ψ ( x ) : = 2 3 π − 1 / 4 ( 1 − x 2 ) e − x 2 / 2 , ∀ x ∈ R . \psi(x):=\frac{2}{\sqrt{3}}\pi^{-1/4}(1-x^2)e^{-x^2/2},\quad\forall x\in\mathbb{R}. ψ ( x ) := 3 2 π − 1/4 ( 1 − x 2 ) e − x 2 /2 , ∀ x ∈ R . For j , k ∈ Z j,k\in\mathbb{Z} j , k ∈ Z
ψ j , k ( x ) : = ψ ( 2 j x − k b ) , ∀ x ∈ R . \psi_{j,k}(x):=\psi(2^jx-kb),\quad\forall x\in\mathbb{R}. ψ j , k ( x ) := ψ ( 2 j x − k b ) , ∀ x ∈ R . Then { 2 j / 2 ψ j , k } j , k ∈ Z \{2^{j/2}\psi_{j,k}\}_{j,k\in\mathbb{Z}} { 2 j /2 ψ j , k } j , k ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(xiii) (Benedetto and Kolesar [32 ] ; ); Strohmer and Heath (2003)) (Grassmannian frame ) The n n n R 2 \mathbb{R}^2 R 2 { ( cos ( π j / n ) , sin ( π j / n ) ) } j = 0 n − 1 \{(\cos(\pi j/n),\sin(\pi j/n))\}_{j=0}^{n-1} {( cos ( π j / n ) , sin ( π j / n )) } j = 0 n − 1 R 2 \mathbb{R}^2 R 2
(xiv) (Benedetto and Fickus [31 ] ) (Group frame ) Vertices of each of (five) Platonic solids is a tight frame for R 3 \mathbb{R}^3 R 3
(xv) (cf. Waldron (2018)) (Equiangular frame ) For each d ∈ N d\in\mathbb{N} d ∈ N d + 1 d+1 d + 1 R d \mathbb{R}^d R d R d \mathbb{R}^d R d
We now give various examples which are not frames.
Example 1.2.33. (i) (cf. Christensen [65 ] ) If { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H { τ n + τ n + 1 } n = 1 ∞ \{\tau_n+\tau_{n+1}\}_{n=1}^{\infty} { τ n + τ n + 1 } n = 1 ∞ H \mathcal{H} H
(ii) (cf. Bachman et al. [18 ] ) If { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H { τ n n } n = 1 ∞ \{\frac{\tau_n}{n}\}_{n=1}^{\infty} { n τ n } n = 1 ∞ H \mathcal{H} H
(iii) (cf. Christensen [65 ] ) If { τ n } n = − ∞ ∞ \{\tau_n\}_{n=-\infty}^{\infty} { τ n } n = − ∞ ∞ H \mathcal{H} H { τ n + τ n + 1 } n = − ∞ ∞ \{\tau_n+\tau_{n+1}\}_{n=-\infty}^{\infty} { τ n + τ n + 1 } n = − ∞ ∞ H \mathcal{H} H
(iv) (cf. Christensen [65 ] ) For n ∈ Z n \in\mathbb{Z} n ∈ Z T n : L 2 ( R ) ∋ f ↦ T n f ∈ L 2 ( R ) T_n:\mathcal{L}^2(\mathbb{R}) \ni f \mapsto T_n f \in\mathcal{L}^2(\mathbb{R}) T n : L 2 ( R ) ∋ f ↦ T n f ∈ L 2 ( R ) T n f : R ∋ x ↦ f ( x − n ) ∈ C T_n f:\mathbb{R} \ni x \mapsto f(x-n) \in\mathbb{C} T n f : R ∋ x ↦ f ( x − n ) ∈ C ϕ ∈ L 2 ( R ) \phi\in\mathcal{L}^2(\mathbb{R}) ϕ ∈ L 2 ( R ) { T n ϕ } n ∈ Z \{T_n\phi\}_{n\in\mathbb{Z}} { T n ϕ } n ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(v) (Aldroubi and Petrosyan [11 ] ; Christensen et al. [67 ] ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H τ ∈ H \tau\in\mathcal{H} τ ∈ H { T n τ } n = 0 ∞ \{T^n\tau\}_{n=0}^{\infty} { T n τ } n = 0 ∞ H \mathcal{H} H [8 ] )).
(vi) (cf. Feichtinger and Strohmer (2003)) Let T a T_a T a E b E_b E b c > 0 c>0 c > 0 χ [ 0 , c ] \chi_{[0,c]} χ [ 0 , c ] [ 0 , c ] [0,c] [ 0 , c ] c ≤ a c\leq a c ≤ a a > 1 a>1 a > 1 { E m T n a χ [ 0 , c ] } n , m ∈ Z \{E_mT_{na}\chi_{[0,c]}\}_{n,m\in\mathbb{Z}} { E m T na χ [ 0 , c ] } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
There is a simple criterion to check for frames in finite dimensional Hilbert spaces. This reads as follows.
Theorem 1.2.34. (cf. Han et al. [128 ] ) A finite set of vectors for a finite dimensional Hilbert space is a frame if and only if it spans the space.
(iii) Suppose dim ( H ) = n \dim(\mathcal{H}) = n dim ( H ) = n Theorem 1.2.34 we see that a spanning set having at least n + 1 n+1 n + 1 H \mathcal{H} H H \mathcal{H} H
(iv) A spanning set need not be a frame. For instance, { n } n = 1 ∞ \{n\}_{n=1}^{\infty} { n } n = 1 ∞ C \mathbb{C} C ∑ n = 1 ∞ ∣ ⟨ 1 , n ⟩ ∣ 2 = ∑ n = 1 ∞ n 2 = ∞ \sum_{n=1}^{\infty} \lvert\langle1,n\rangle\rvert^{2} = \sum_{n=1}^{\infty} n^{2} = \infty ∑ n = 1 ∞ ∣⟨ 1 , n ⟩ ∣ 2 = ∑ n = 1 ∞ n 2 = ∞ { n } n = 1 ∞ \{n\}_{n=1}^{\infty} { n } n = 1 ∞ C \mathbb{C} C
Following theorem is the most important result in the theory of frames.
Theorem 1.2.36. (Duffin and Schaeffer [89 ] , Christensen [65 ] , Han and Larson [129 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H a a a b b b
(i) span { τ n } n = H \operatorname{span}\{\tau_n\}_n = \mathcal{H} span { τ n } n = H
(ii) The map θ τ : H ∋ h ↦ θ τ h : = { ⟨ h , τ n ⟩ } n ∈ ℓ 2 ( N ) \theta_\tau: \mathcal{H} \ni h \mapsto\theta_\tau h := \{\langle h,\tau_n\rangle\}_n \in\ell^2(\mathbb{N}) θ τ : H ∋ h ↦ θ τ h := {⟨ h , τ n ⟩ } n ∈ ℓ 2 ( N ) a ∥ h ∥ ≤ ∥ θ τ h ∥ ≤ b ∥ h ∥ \sqrt{a}\lVert h\rVert\le\lVert\theta_\tau h\rVert\le\sqrt{b}\lVert h\rVert a ∥ h ∥ ≤ ∥ θ τ h ∥ ≤ b ∥ h ∥ ∀ h ∈ H \forall h \in\mathcal{H} ∀ h ∈ H θ τ \theta_\tau θ τ
(iii) The map S τ : H ∋ h ↦ S τ h : = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H S_\tau: \mathcal{H} \ni h \mapsto S_\tau h := \sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n \in\mathcal{H} S τ : H ∋ h ↦ S τ h := ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H
a ∥ h ∥ 2 ≤ ⟨ S τ h , h ⟩ ≤ b ∥ h ∥ 2 , ∀ h ∈ H , a ∥ h ∥ ≤ ∥ S τ h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . a\lVert h\rVert^2 \le\langle S_\tau h,h\rangle\le b\lVert h\rVert^2,\quad\forall h\in\mathcal{H},\qquad a\lVert h\rVert\le\lVert S_\tau h\rVert\le b\lVert h\rVert,\quad\forall h\in\mathcal{H}. a ∥ h ∥ 2 ≤ ⟨ S τ h , h ⟩ ≤ b ∥ h ∥ 2 , ∀ h ∈ H , a ∥ h ∥ ≤ ∥ S τ h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . (iv) (General Fourier expansion or frame decomposition )
h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ S τ − 1 τ n = ∑ n = 1 ∞ ⟨ h , S τ − 1 τ n ⟩ τ n , ∀ h ∈ H . (4) h = \sum_{n=1}^{\infty}\langle h,\tau_n\rangle S_\tau^{-1}\tau_n = \sum_{n=1}^{\infty}\langle h,S_\tau^{-1}\tau_n\rangle\tau_n,\quad\forall h\in\mathcal{H}. \tag*{(4)} h = n = 1 ∑ ∞ ⟨ h , τ n ⟩ S τ − 1 τ n = n = 1 ∑ ∞ ⟨ h , S τ − 1 τ n ⟩ τ n , ∀ h ∈ H . (4) (v) θ τ ∗ ( { a n } n ) = ∑ n a n τ n \theta_\tau^*(\{a_n\}_n) = \sum_n a_n\tau_n θ τ ∗ ({ a n } n ) = ∑ n a n τ n ∀ { a n } n ∈ ℓ 2 ( N ) \forall\{a_n\}_n\in\ell^2(\mathbb{N}) ∀ { a n } n ∈ ℓ 2 ( N ) θ τ ∗ e n = τ n \theta_\tau^*e_n=\tau_n θ τ ∗ e n = τ n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N
(vi) S τ S_\tau S τ S τ = θ τ ∗ θ τ S_\tau= \theta_\tau^*\theta_\tau S τ = θ τ ∗ θ τ
(vii) θ τ ∗ \theta_\tau^* θ τ ∗
(viii) ∥ S τ − 1 ∥ − 1 \lVert S_\tau^{-1}\rVert^{-1} ∥ S τ − 1 ∥ − 1 ∥ S τ ∥ = ∥ θ τ ∥ 2 \lVert S_\tau\rVert= \lVert\theta_\tau\rVert^2 ∥ S τ ∥ = ∥ θ τ ∥ 2
(ix) P τ : = θ τ S τ − 1 θ τ ∗ P_\tau:= \theta_\tau S_\tau^{-1}\theta_\tau^* P τ := θ τ S τ − 1 θ τ ∗ θ τ ( H ) \theta_\tau(\mathcal{H}) θ τ ( H )
(x) { τ n } n \{\tau_n\}_n { τ n } n θ τ \theta_\tau θ τ θ τ θ τ ∗ \theta_\tau\theta_\tau^* θ τ θ τ ∗
(xi) { S τ − 1 τ n } n \{S_\tau^{-1}\tau_n\}_n { S τ − 1 τ n } n H \mathcal{H} H b − 1 b^{-1} b − 1 a − 1 a^{-1} a − 1
(xii) { S τ − 1 / 2 τ n } n \{S_\tau^{-1/2}\tau_n\}_n { S τ − 1/2 τ n } n H \mathcal{H} H
(xiii) (Best approximation) If h ∈ H h \in\mathcal{H} h ∈ H h = ∑ n = 1 ∞ c n τ n h = \sum_{n=1}^{\infty} c_n\tau_n h = ∑ n = 1 ∞ c n τ n { c n } n ∈ ℓ 2 ( N ) \{c_n\}_n \in\ell^2(\mathbb{N}) { c n } n ∈ ℓ 2 ( N )
∑ n = 1 ∞ ∣ c n ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h , S τ − 1 τ n ⟩ ∣ 2 + ∑ n = 1 ∞ ∣ c n − ⟨ h , S τ − 1 τ n ⟩ ∣ 2 . \sum_{n=1}^{\infty} |c_n|^2 = \sum_{n=1}^{\infty} |\langle h,S_{\tau}^{-1}\tau_n\rangle|^2 + \sum_{n=1}^{\infty} |c_n-\langle h,S_{\tau}^{-1}\tau_n\rangle|^2. n = 1 ∑ ∞ ∣ c n ∣ 2 = n = 1 ∑ ∞ ∣ ⟨ h , S τ − 1 τ n ⟩ ∣ 2 + n = 1 ∑ ∞ ∣ c n − ⟨ h , S τ − 1 τ n ⟩ ∣ 2 . Theorem 1.2.36 says several things. First, it says that every vector in the Hilbert space admits an expansion, called as general Fourier expansion, similar to Fourier expansion coming from an orthonormal basis for a Hilbert space. Second, it says that coefficients in the expansion of a vector need not be unique. This is particularly important in applications, since loss in the information of a vector is less if some of the coefficients are missing. Third, given a frame, it naturally generates other frames. Fourth, a frame gives a bounded linear injective operator from the less known inner product on the Hilbert space H \mathcal{H} H ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) H \mathcal{H} H ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) H \mathcal{H} H ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) θ τ \theta_\tau θ τ θ τ ∗ \theta_\tau^* θ τ ∗ ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
The operators θ τ \theta_\tau θ τ θ τ ∗ \theta_\tau^* θ τ ∗ S τ S_\tau S τ analysis operator, synthesis operator and frame operator , respectively (cf. [65 ] ).
Dilation theory usually tries to extend operator on Hilbert space to larger Hilbert space which are easier to handle as well as well-understood and study the original operator as a slice of it [17 , 154 ] (); Sz.-Nagy et al. 2010). As long as frame theory for Hilbert spaces is considered, following theorem is known as Naimark-Han-Larson dilation theorem. This was proved independently by Han and Larson [129 ] and by Kashin and Kulikova [148 ] . History of this theorem is nicely presented in the paper [74 ] .
Theorem 1.2.37. ([129 , 148 ] (Naimark-Han-Larson dilation theorem ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) frame for H \mathcal{H} H H 1 ⊇ H \mathcal{H}_1 \supseteq\mathcal{H} H 1 ⊇ H { ω n } n \{\omega_n\}_n { ω n } n H 1 \mathcal{H}_1 H 1 P : H 1 → H P:\mathcal{H}_1 \to\mathcal{H} P : H 1 → H τ n = P ω n , ∀ n ∈ N \tau_n=P\omega_n,\forall n \in\mathbb{N} τ n = P ω n , ∀ n ∈ N
(ii) Parseval frame for H \mathcal{H} H H 1 ⊇ H \mathcal{H}_1 \supseteq\mathcal{H} H 1 ⊇ H { ω n } n \{\omega_n\}_n { ω n } n H 1 \mathcal{H}_1 H 1 P : H 1 → H P:\mathcal{H}_1 \to\mathcal{H} P : H 1 → H τ n = P ω n , ∀ n ∈ N \tau_n=P\omega_n,\forall n \in\mathbb{N} τ n = P ω n , ∀ n ∈ N
In order to construct an element of the Hilbert space using frames using Equation (4) , we have to first determine inverse of the frame operator which is difficult in general. Thus we seek a way to approximate an element using a sequence which does not involve calculating inverse of frame operator. This is given in the following theorem.
Proposition 1.2.38. ([89 ] ) (Frame algorithm ) Let { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H a a a b b b h ∈ H h \in\mathcal{H} h ∈ H
h 0 : = 0 , h n : = h n − 1 + 2 a + b S τ ( h − h n − 1 ) , ∀ n ≥ 1. h_0 := 0,\qquad h_n := h_{n-1}+\frac{2}{a+b}S_{\tau}(h-h_{n-1}),\qquad\forall n \ge1. h 0 := 0 , h n := h n − 1 + a + b 2 S τ ( h − h n − 1 ) , ∀ n ≥ 1. Then
∥ h n − h ∥ ≤ ( b − a b + a ) n ∥ h ∥ , ∀ n ≥ 1. \lVert h_n-h\rVert\le\left(\frac{b-a}{b+a}\right)^n\lVert h\rVert,\qquad\forall n \ge1. ∥ h n − h ∥ ≤ ( b + a b − a ) n ∥ h ∥ , ∀ n ≥ 1. In particular, h n → h h_n \to h h n → h n → ∞ n \to\infty n → ∞
Given a collection { τ n } n \{\tau_n\}_n { τ n } n a a a b b b ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) { e n } n \{e_n\}_n { e n } n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) { e n } n \{e_n\}_n { e n } n [133 ] as follows.
Theorem 1.2.39. ([133 ] ) (Holub’s theorem ) A sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H \mathcal{H} H T : ℓ 2 ( N ) → H T : \ell^2(\mathbb{N}) \to\mathcal{H} T : ℓ 2 ( N ) → H T e n = τ n Te_n = \tau_n T e n = τ n n ∈ N n \in\mathbb{N} n ∈ N
There is a slight variation of Theorem 1.2.39 given by Christensen [65 ] .
Theorem 1.2.40. ([65 ] ) Let { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H \mathcal{H} H T : H → H T : \mathcal{H} \to\mathcal{H} T : H → H T ω n = τ n T\omega_n = \tau_n T ω n = τ n n ∈ N n \in\mathbb{N} n ∈ N
Given a frame { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { S τ − 1 τ n } n \{S_{\tau}^{-1}\tau_n\}_n { S τ − 1 τ n } n (4) . However, in general there may be other frames satisfying the Equation (4) like { S τ − 1 τ n } n \{S_{\tau}^{-1}\tau_n\}_n { S τ − 1 τ n } n
Definition 1.2.41. ([cf. 65 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
for H \mathcal{H} H dual frame for { τ n } n \{\tau_n\}_n { τ n } n
h = ∑ n = 1 ∞ ⟨ h , ω n ⟩ τ n = ∑ n = 1 ∞ ⟨ h , τ n ⟩ ω n , ∀ h ∈ H . h=\sum_{n=1}^{\infty}\langle h,\omega_n\rangle\tau_n=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\omega_n,\quad\forall h\in\mathcal{H}. h = n = 1 ∑ ∞ ⟨ h , ω n ⟩ τ n = n = 1 ∑ ∞ ⟨ h , τ n ⟩ ω n , ∀ h ∈ H . Just like characterization of frames, given a frame, we seek a description of each of its dual frame. This problem was solved by Li [156 ] in the following two lemmas and a theorem.
Lemma 1.2.42. (Li [156 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { e n } n \{e_n\}_n { e n } n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) { ω n } n \{\omega_n\}_n { ω n } n { τ n } n \{\tau_n\}_n { τ n } n
ω n = U e n , ∀ n ∈ N , \omega_n=Ue_n,\quad\forall n\in\mathbb{N}, ω n = U e n , ∀ n ∈ N , where U : ℓ 2 ( N ) → H U:\ell^2(\mathbb{N})\to\mathcal{H} U : ℓ 2 ( N ) → H θ τ \theta_\tau θ τ
Lemma 1.2.43. (Li [156 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H L : ℓ 2 ( N ) → H L:\ell^2(\mathbb{N})\to\mathcal{H} L : ℓ 2 ( N ) → H θ τ \theta_\tau θ τ
L = S τ − 1 θ τ ∗ + V ( I ℓ 2 ( N ) − θ τ S τ − 1 θ τ ∗ ) , L=S_\tau^{-1}\theta_\tau^*+V(I_{\ell^2(\mathbb{N})}-\theta_\tau S_\tau^{-1}\theta_\tau^*), L = S τ − 1 θ τ ∗ + V ( I ℓ 2 ( N ) − θ τ S τ − 1 θ τ ∗ ) , where V : ℓ 2 ( N ) → H V:\ell^2(\mathbb{N})\to\mathcal{H} V : ℓ 2 ( N ) → H
Theorem 1.2.44. (Li [156 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { ω n } n \{\omega_n\}_n { ω n } n { τ n } n \{\tau_n\}_n { τ n } n
ω n = S τ − 1 τ n + ρ n − ∑ k = 1 ∞ ⟨ S τ − 1 τ n , τ k ⟩ ρ k , ∀ n ∈ N , \omega_n=S_\tau^{-1}\tau_n+\rho_n-\sum_{k=1}^{\infty}\langle S_\tau^{-1}\tau_n,\tau_k\rangle\rho_k,\quad\forall n\in\mathbb{N}, ω n = S τ − 1 τ n + ρ n − k = 1 ∑ ∞ ⟨ S τ − 1 τ n , τ k ⟩ ρ k , ∀ n ∈ N , where { ρ n } n \{\rho_n\}_n { ρ n } n H \mathcal{H} H b > 0 b>0 b > 0
∑ n = 1 ∞ ∣ ⟨ h , ρ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n=1}^{\infty}|\langle h,\rho_n\rangle|^2\le b\lVert h\rVert^2,\quad\forall h\in\mathcal{H}. n = 1 ∑ ∞ ∣ ⟨ h , ρ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . We again consider the frame { S τ − 1 τ n } n \{S_\tau^{-1}\tau_n\}_n { S τ − 1 τ n } n S τ − 1 S_\tau^{-1} S τ − 1 { τ n } n \{\tau_n\}_n { τ n } n
Definition 1.2.45. (Balan [20 ] ) Two frames { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H similar or equivalent if there exists a bounded invertible operator T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H
that
ω n = T τ n , ∀ n ∈ N . (5) \omega_n = T\tau_n,\quad\forall n\in\mathbb{N}. \tag*{(5)} ω n = T τ n , ∀ n ∈ N . (5) Given frames { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n (5) for every natural number. Thus it is better if there is a characterization which does not involve natural numbers and involves only operators. Further, it is natural to ask whether there is a formula for the operator T T T [20 ] and independently by Han and Larson [129 ] which states as follows.
Theorem 1.2.46. [20 , 129 ] For two frames { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H
(i) { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H ω n = T τ n \omega_n=T\tau_n ω n = T τ n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N
(ii) θ ω = θ τ T \theta_\omega=\theta_\tau T θ ω = θ τ T T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H
(iii) P ω = P τ P_\omega=P_\tau P ω = P τ
If one of the above conditions is satisfied, then the invertible operator in (i) and (ii) is unique and is given by T = S τ − 1 θ τ ∗ θ ω T=S_\tau^{-1}\theta_\tau^*\theta_\omega T = S τ − 1 θ τ ∗ θ ω
For a given subset M \mathbb{M} M N \mathbb{N} N S M : H ∋ h ↦ ∑ n ∈ M ⟨ h , τ n ⟩ τ n ∈ H S_{\mathbb{M}}:\mathcal{H}\ni h\mapsto\sum_{n\in\mathbb{M}}\langle h,\tau_n\rangle\tau_n\in\mathcal{H} S M : H ∋ h ↦ ∑ n ∈ M ⟨ h , τ n ⟩ τ n ∈ H S M S_{\mathbb{M}} S M M c \mathbb{M}^{\mathrm{c}} M c M \mathbb{M} M N \mathbb{N} N [21 , 22 ] ).
Theorem 1.2.47. [22 , 23 ] (Frame identity ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N
∑ n ∈ M ∣ ⟨ h , τ n ⟩ ∣ 2 − ∑ n = 1 ∞ ∣ ⟨ S M h , τ ~ n ⟩ ∣ 2 = ∑ n ∈ M c ∣ ⟨ h , τ n ⟩ ∣ 2 − ∑ n = 1 ∞ ∣ ⟨ S M c h , τ ~ n ⟩ ∣ 2 , ∀ h ∈ H . \sum_{n\in\mathbb{M}}|\langle h,\tau_n\rangle|^2-\sum_{n=1}^{\infty}|\langle S_{\mathbb{M}}h,\widetilde{\tau}_n\rangle|^2
=
\sum_{n\in\mathbb{M}^{\mathrm{c}}}|\langle h,\tau_n\rangle|^2-\sum_{n=1}^{\infty}|\langle S_{\mathbb{M}^{\mathrm{c}}}h,\widetilde{\tau}_n\rangle|^2,\quad\forall h\in\mathcal{H}. n ∈ M ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 − n = 1 ∑ ∞ ∣ ⟨ S M h , τ n ⟩ ∣ 2 = n ∈ M c ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 − n = 1 ∑ ∞ ∣ ⟨ S M c h , τ n ⟩ ∣ 2 , ∀ h ∈ H . Theorem 1.2.48. [22 , 23 ] (Parseval frame identity ) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N
∑ n ∈ M ∣ ⟨ h , τ n ⟩ ∣ 2 − ∥ ∑ n ∈ M ⟨ h , τ n ⟩ τ n ∥ 2 = ∑ n ∈ M c ∣ ⟨ h , τ n ⟩ ∣ 2 − ∥ ∑ n ∈ M c ⟨ h , τ n ⟩ τ n ∥ 2 , ∀ h ∈ H . \sum_{n\in\mathbb{M}}|\langle h,\tau_n\rangle|^2-\left\|\sum_{n\in\mathbb{M}}\langle h,\tau_n\rangle\tau_n\right\|^2
=
\sum_{n\in\mathbb{M}^{\mathrm{c}}}|\langle h,\tau_n\rangle|^2-\left\|\sum_{n\in\mathbb{M}^{\mathrm{c}}}\langle h,\tau_n\rangle\tau_n\right\|^2,\quad\forall h\in\mathcal{H}. n ∈ M ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 − n ∈ M ∑ ⟨ h , τ n ⟩ τ n 2 = n ∈ M c ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 − n ∈ M c ∑ ⟨ h , τ n ⟩ τ n 2 , ∀ h ∈ H . Theorem 1.2.48 has applications. It was applied to get the following remarkable lower estimate for Parseval frames.
Theorem 1.2.49. [23 , 115 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H M ⊆ N \mathbb{M} \subseteq\mathbb{N} M ⊆ N
∑ n ∈ M ∣ ⟨ h , τ n ⟩ ∣ 2 + ∥ ∑ n ∈ M c ⟨ h , τ n ⟩ τ n ∥ 2 = ∑ n ∈ M c ∣ ⟨ h , τ n ⟩ ∣ 2 + ∥ ∑ n ∈ M ⟨ h , τ n ⟩ τ n ∥ 2 ≥ 3 4 ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n\in\mathbb{M}} |\langle h,\tau_n\rangle|^2 + \left\lVert\sum_{n\in\mathbb{M}^{c}} \langle h,\tau_n\rangle\tau_n \right\rVert^2 = \sum_{n\in\mathbb{M}^{c}} |\langle h,\tau_n\rangle|^2 + \left\lVert\sum_{n\in\mathbb{M}} \langle h,\tau_n\rangle\tau_n \right\rVert^2 \geq\frac{3}{4}\lVert h\rVert^2,\quad\forall h\in\mathcal{H}. n ∈ M ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 + n ∈ M c ∑ ⟨ h , τ n ⟩ τ n 2 = n ∈ M c ∑ ∣ ⟨ h , τ n ⟩ ∣ 2 + n ∈ M ∑ ⟨ h , τ n ⟩ τ n 2 ≥ 4 3 ∥ h ∥ 2 , ∀ h ∈ H . Further, the bound 3/4 is optimal.
As another application, Theorem 1.2.49 was used in the study of Parseval frames with finite excesses [19 , 24 ] . Like duality, there is another notion called as orthogonality for frames for Hilbert spaces. This was first introduced by Balan [25 ] in his Ph.D. thesis and further studied by Han and Larson [129 ] .
Definition 1.2.50. [25 , 129 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H orthogonal frame for { τ n } n \{\tau_n\}_n { τ n } n
0 = ∑ n = 1 ∞ ⟨ h , ω n ⟩ τ n = ∑ n = 1 ∞ ⟨ h , τ n ⟩ ω n , ∀ h ∈ H . 0 = \sum_{n=1}^{\infty} \langle h,\omega_n\rangle\tau_n = \sum_{n=1}^{\infty} \langle h,\tau_n\rangle\omega_n,\quad\forall h\in\mathcal{H}. 0 = n = 1 ∑ ∞ ⟨ h , ω n ⟩ τ n = n = 1 ∑ ∞ ⟨ h , τ n ⟩ ω n , ∀ h ∈ H . Remarkable property of orthogonal frames is that we can interpolate as well as we can take direct sum of them to get new frames. These are illustrated in the following two results.
Proposition 1.2.51. [128 , 129 ] Let { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H C , D ∈ B ( H ) C,D \in\mathcal{B}(\mathcal{H}) C , D ∈ B ( H ) C ∗ C + D ∗ D = I H C^{*}C + D^{*}D = I_{\mathcal{H}} C ∗ C + D ∗ D = I H { C τ n + D ω n } n \{C\tau_n + D\omega_n\}_n { C τ n + D ω n } n H \mathcal{H} H c , d , e , f c,d,e,f c , d , e , f ∣ c ∣ 2 + ∣ d ∣ 2 = 1 |c|^2 + |d|^2 = 1 ∣ c ∣ 2 + ∣ d ∣ 2 = 1 { c τ n + d ω n } n \{c\tau_n + d\omega_n\}_n { c τ n + d ω n } n
Proposition 1.2.52. [128 , 129 ] If { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H { τ n ⊕ ω n } n \{\tau_n\oplus\omega_n\}_n { τ n ⊕ ω n } n H ⊕ H \mathcal{H}\oplus\mathcal{H} H ⊕ H { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n { τ n ⊕ ω n } n \{\tau_n\oplus\omega_n\}_n { τ n ⊕ ω n } n
Recall that Paley-Wiener theorem 1.2.30 says that sequences which are close to orthonormal bases are Riesz bases. Since a frame will also give a series representation, it is natural to ask whether a sequence close to frame is a frame. This was first derived by Christensen [61 ] which showed that sequences which are quadratically close to frames are again frames.
Theorem 1.2.53. [61 ] (Christensen’s quadratic perturbation ) Let { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H a a a b b b { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H
c : = ∑ n = 1 ∞ ∥ τ n − ω n ∥ 2 < a , c := \sum_{n=1}^{\infty} \lVert\tau_n-\omega_n\rVert^2 < a, c := n = 1 ∑ ∞ ∥ τ n − ω n ∥ 2 < a , then { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H a ( 1 − c a ) 2 a\left(1-\sqrt{\frac{c}{a}}\right)^2 a ( 1 − a c ) 2 b ( 1 + c b ) 2 b\left(1+\sqrt{\frac{c}{b}}\right)^2 b ( 1 + b c ) 2
Three months later, Christensen generalized Theorem 1.2.53.
Theorem 1.2.54. [63 ] (Christensen perturbation ) Let { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H a a a b b b { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H α , γ ≥ 0 \alpha,\gamma\geq0 α , γ ≥ 0 α + γ a < 1 \alpha+\frac{\gamma}{\sqrt{a}}<1 α + a γ < 1
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ 2 ) 1 2 , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … , \left\lVert\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\rVert\leq\alpha\left\lVert\sum_{n=1}^{m}c_n\tau_n\right\rVert+\gamma\left(\sum_{n=1}^{m}|c_n|^2\right)^{\frac{1}{2}},\quad\forall c_1,\ldots,c_m\in\mathbb{K},m=1,2,\ldots, n = 1 ∑ m c n ( τ n − ω n ) ≤ α n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ 2 ) 2 1 , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … , then { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H a ( 1 − ( α + γ a ) ) 2 a\left(1-\left(\alpha+\frac{\gamma}{\sqrt{a}}\right)\right)^2 a ( 1 − ( α + a γ ) ) 2 b ( 1 + ( α + γ b ) ) 2 b\left(1+\left(\alpha+\frac{\gamma}{\sqrt{b}}\right)\right)^2 b ( 1 + ( α + b γ ) ) 2
After two years, Casazza and Christensen further extended Theorem 1.2.54.
Theorem 1.2.55. [48 ] (Casazza-Christensen perturbation ) Let { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H a a a b b b { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H α , β , γ ≥ 0 \alpha,\beta,\gamma\geq0 α , β , γ ≥ 0 max { α + γ a , β } < 1 \max\{\alpha+\frac{\gamma}{\sqrt{a}},\beta\}<1 max { α + a γ , β } < 1
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ 2 ) 1 2 + β ∥ ∑ n = 1 m c n ω n ∥ , \left\lVert\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\rVert\leq\alpha\left\lVert\sum_{n=1}^{m}c_n\tau_n\right\rVert+\gamma\left(\sum_{n=1}^{m}|c_n|^2\right)^{\frac{1}{2}}+\beta\left\lVert\sum_{n=1}^{m}c_n\omega_n\right\rVert, n = 1 ∑ m c n ( τ n − ω n ) ≤ α n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ 2 ) 2 1 + β n = 1 ∑ m c n ω n , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … , \forall c_1,\ldots,c_m\in\mathbb{K},m=1,2,\ldots, ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … , then { ω n } n = 1 ∞ \{\omega_n\}_{n=1}^{\infty} { ω n } n = 1 ∞ H \mathcal{H} H a ( 1 − α + β + γ a 1 + β ) 2 a\left(1-\frac{\alpha+\beta+\frac{\gamma}{\sqrt{a}}}{1+\beta}\right)^2 a ( 1 − 1 + β α + β + a γ ) 2 b ( 1 + α + β + γ b 1 − β ) 2 b\left(1+\frac{\alpha+\beta+\frac{\gamma}{\sqrt{b}}}{1-\beta}\right)^2 b ( 1 + 1 − β α + β + b γ ) 2
We next consider Bessel sequences which is next level of generalization of frames.
Definition 1.2.56. (cf. Christensen [65 ] ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H Bessel sequence for H \mathcal{H} H b > 0 b>0 b > 0
(General Bessel’s inequality )
∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . (6) \sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2\leq b\lVert h\rVert^2,\quad\forall h\in\mathcal{H}. \tag*{(6)} n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . (6) Constant b b b { τ n } n \{\tau_n\}_n { τ n } n
Inequality 1.2.3 is stronger than Inequality 1.2.6. Hence every frame is a Bessel sequence. Using Cauchy-Schwarz inequality, it follows that every finite set of vectors is a Bessel sequence (cf. [65 ] ). Now using Theorem 1.2.34, we get plenty of Bessel sequences which are not frames (in finite dimensions). As an example in infinite dimensions, we claim that { e 2 , e 3 , … } \{e_2,e_3,\ldots\} { e 2 , e 3 , … } ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) { e 2 , e 3 , … } \{e_2,e_3,\ldots\} { e 2 , e 3 , … } a > 0 a>0 a > 0 h = e 1 h=e_1 h = e 1 a ∥ e 1 ∥ 2 ≤ ∑ n = 2 ∞ ∣ ⟨ e 1 , e n ⟩ ∣ 2 = 0 ⇒ a = 0 a\lVert e_1\rVert^2\leq\sum_{n=2}^{\infty}|\langle e_1,e_n\rangle|^2=0\Rightarrow a=0 a ∥ e 1 ∥ 2 ≤ ∑ n = 2 ∞ ∣ ⟨ e 1 , e n ⟩ ∣ 2 = 0 ⇒ a = 0
Example 1.2.57. (i) If { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H S \mathcal{S} S N \mathbb{N} N { τ n } n ∈ S \{\tau_n\}_{n\in\mathcal{S}} { τ n } n ∈ S H \mathcal{H} H ∑ n ∈ S ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 , ∀ h ∈ H \sum_{n\in\mathcal{S}}|\langle h,\tau_n\rangle|^2\leq\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2,\forall h\in\mathcal{H} ∑ n ∈ S ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 , ∀ h ∈ H
(ii) (cf. [65 ] ) Let g ∈ L 2 ( R ) g\in\mathcal{L}^2(\mathbb{R}) g ∈ L 2 ( R ) a , b > 0 a,b>0 a , b > 0 f n , m : R ∋ x ↦ e 2 π i m b x g ( x − n a ) ∈ R , ∀ n , m ∈ Z f_{n,m}:\mathbb{R}\ni x\mapsto e^{2\pi imb x}g(x-na)\in\mathbb{R},\ \forall n,m\in\mathbb{Z} f n , m : R ∋ x ↦ e 2 π imb x g ( x − na ) ∈ R , ∀ n , m ∈ Z { f n , m } n , m ∈ Z \{f_{n,m}\}_{n,m\in\mathbb{Z}} { f n , m } n , m ∈ Z L 2 ( R ) \mathcal{L}^2(\mathbb{R}) L 2 ( R )
(iii) (cf. [65 ] ) If { τ n } n = 1 ∞ \{\tau_n\}_{n=1}^{\infty} { τ n } n = 1 ∞ H \mathcal{H} H { τ n + τ n + 1 } n = 1 ∞ \{\tau_n+\tau_{n+1}\}_{n=1}^{\infty} { τ n + τ n + 1 } n = 1 ∞ H \mathcal{H} H H \mathcal{H} H
Theorem 1.2.58. (cf. [65 ] ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H ℓ 2 ( N ) ∋ { τ n } n ↦ ∑ n = 1 ∞ a n τ n ∈ H \ell^2(\mathbb{N})\ni\{\tau_n\}_n\mapsto\sum_{n=1}^{\infty}a_n\tau_n\in\mathcal{H} ℓ 2 ( N ) ∋ { τ n } n ↦ ∑ n = 1 ∞ a n τ n ∈ H { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H ∋ h ↦ ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H \mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n\in\mathcal{H} H ∋ h ↦ ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H
Since a Bessel sequence need not be a frame, it is natural to ask the following question: Given a Bessel sequence, can we add extra elements to it so that the resulting sequence is a frame? Answer is positive. This result was obtained by Li and Sun (2009).
Theorem 1.2.59. ((Li and Sun (2009))) Every Bessel sequence in a Hilbert space can be expanded to a tight frame. Moreover, we can expand a Bessel sequence in infinitely many ways to tight frames.
Li and Sun (2009) further observed that if a Bessel sequence for a Hilbert space can be expanded finitely to get a tight frame, then the number of elements added can not be small. Precise statement reads as follows.
Theorem 1.2.60. ((Li and Sun (2009))) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H { τ n } n ∪ { ω k } k = 1 N \{\tau_n\}_n\cup\{\omega_k\}_{k=1}^{N} { τ n } n ∪ { ω k } k = 1 N λ \lambda λ H \mathcal{H} H
N ≥ dim ( λ I H − S τ ) ( H ) . (7) N\geq\dim(\lambda I_{\mathcal{H}}-S_{\tau})(\mathcal{H}). \tag*{(7)} N ≥ dim ( λ I H − S τ ) ( H ) . (7) Further, the Inequality (1.2.7) can not be improved.
RIESZ BASES, FRAMES AND BESSEL SEQUENCES FOR BANACH SPACES
Definition of Riesz basis, as given in Definition 1.2.19 requires the notion of inner product. Due to the lack of inner product in a Banach space, Definition 1.2.19 can not be carried over to Banach spaces. However, Theorem 1.2.25 allows to define Riesz basis for Banach spaces as follows.
Definition 1.3.1. (Aldroubi et al. [12 ] ) Let 1 < q < ∞ 1 < q < \infty 1 < q < ∞ X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X
(i) q q q X \mathcal{X} X a , b > 0 a,b > 0 a , b > 0 S S S N \mathbb{N} N
a ( ∑ n ∈ S ∣ c n ∣ q ) 1 q ≤ ∥ ∑ n ∈ S c n τ n ∥ ≤ b ( ∑ n ∈ S ∣ c n ∣ q ) 1 q , ∀ c n ∈ K . (8) a\left(\sum_{n\in S}|c_n|^q\right)^{\frac{1}{q}}\leq\left\|\sum_{n\in S}c_n\tau_n\right\|\leq b\left(\sum_{n\in S}|c_n|^q\right)^{\frac{1}{q}},\qquad\forall c_n\in\mathbb{K}. \tag*{(8)} a ( n ∈ S ∑ ∣ c n ∣ q ) q 1 ≤ n ∈ S ∑ c n τ n ≤ b ( n ∈ S ∑ ∣ c n ∣ q ) q 1 , ∀ c n ∈ K . (8) (ii) q q q X \mathcal{X} X q q q X \mathcal{X} X span ‾ { τ n } n = X \overline{\operatorname{span}}\{\tau_n\}_n=\mathcal{X} span { τ n } n = X
Example 1.3.2. Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) A : ℓ p ( N ) → ℓ p ( N ) A:\ell^p(\mathbb{N})\to\ell^p(\mathbb{N}) A : ℓ p ( N ) → ℓ p ( N ) { A e n } n \{Ae_n\}_n { A e n } n p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
Like Theorem 1.2.22 , we have a similar result for p p p
Theorem 1.3.3. (Christensen and Stoeva [69 ] ) Let { f n } n \{f_n\}_n { f n } n q q q X ∗ \mathcal{X}^* X ∗ p p p q q q p p p { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X
x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X and f = ∑ n = 1 ∞ f ( τ n ) f n , ∀ f ∈ X ∗ . x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\quad\forall x\in\mathcal{X}\quad\text{and}\quad f=\sum_{n=1}^{\infty}f(\tau_n)f_n,\quad\forall f\in\mathcal{X}^*. x = n = 1 ∑ ∞ f n ( x ) τ n , ∀ x ∈ X and f = n = 1 ∑ ∞ f ( τ n ) f n , ∀ f ∈ X ∗ . By realizing that the functional X ∋ h ↦ ⟨ h , τ n ⟩ ∈ K \mathcal{X}\ni h\mapsto\langle h,\tau_n\rangle\in\mathbb{K} X ∋ h ↦ ⟨ h , τ n ⟩ ∈ K Definition 1.2.31 leads to the following in Banach spaces.
Definition 1.3.4. (Aldroubi et al. [12 ] ; Christensen and Stoeva [69 ] ) Let 1 < p < ∞ 1 < p < \infty 1 < p < ∞ X \mathcal{X} X
(i) A collection { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ p p p X \mathcal{X} X a , b > 0 a,b > 0 a , b > 0
a ∥ x ∥ ≤ ( ∑ n = 1 ∞ ∣ f n ( x ) ∣ p ) 1 p ≤ b ∥ x ∥ , ∀ x ∈ X . a\|x\|\leq\left(\sum_{n=1}^{\infty}|f_n(x)|^p\right)^{\frac{1}{p}}\leq b\|x\|,\qquad\forall x\in\mathcal{X}. a ∥ x ∥ ≤ ( n = 1 ∑ ∞ ∣ f n ( x ) ∣ p ) p 1 ≤ b ∥ x ∥ , ∀ x ∈ X . If a a a 0 0 0 { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X
(ii) A collection { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X p p p X ∗ \mathcal{X}^* X ∗ a , b > 0 a,b>0 a , b > 0
a ∥ f ∥ ≤ ( ∑ n = 1 ∞ ∣ f ( τ n ) ∣ p ) 1 p ≤ b ∥ f ∥ , ∀ f ∈ X ∗ . a\lVert f\rVert\le\left(\sum_{n=1}^{\infty}\lvert f(\tau_n)\rvert^p\right)^{\frac{1}{p}} \le b\lVert f\rVert,\quad\forall f\in\mathcal{X}^*. a ∥ f ∥ ≤ ( n = 1 ∑ ∞ ∣ f ( τ n ) ∣ p ) p 1 ≤ b ∥ f ∥ , ∀ f ∈ X ∗ . Example 1.3.5. (i) Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n A : ℓ p ( N ) → ℓ p ( N ) A:\ell^p(\mathbb{N})\to\ell^p(\mathbb{N}) A : ℓ p ( N ) → ℓ p ( N ) { ζ n A } n \{\zeta_nA\}_n { ζ n A } n p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
(ii) ([12 ] ) Let 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ a ∈ R a\in\mathbb{R} a ∈ R
T a : L p ( R ) ∋ f ↦ T a f ∈ L p ( R ) , T a f : R ↦ ( T a f ) ( x ) : = f ( x − a ) ∈ C . T_a:\mathcal{L}^p(\mathbb{R})\ni f\mapsto T_af\in\mathcal{L}^p(\mathbb{R}),\quad T_af:\mathbb{R}\mapsto(T_af)(x):=f(x-a)\in\mathbb{C}. T a : L p ( R ) ∋ f ↦ T a f ∈ L p ( R ) , T a f : R ↦ ( T a f ) ( x ) := f ( x − a ) ∈ C . Define
W : = { f : R → C | sup x ∈ R ∑ k ∈ Z ∣ T k f ( x ) ∣ < ∞ } . W:=\left\{f:\mathbb{R}\to\mathbb{C}\,\middle|\,\sup_{x\in\mathbb{R}}\sum_{k\in\mathbb{Z}}\lvert T_kf(x)\rvert<\infty\right\}. W := { f : R → C x ∈ R sup k ∈ Z ∑ ∣ T k f ( x )∣ < ∞ } . and for 1 < p < ∞ 1<p<\infty 1 < p < ∞ ϕ ∈ W \phi\in W ϕ ∈ W
S p : = { ∑ k ∈ Z c k T k ϕ | { c k } k ∈ Z ∈ ℓ p ( Z ) } . S_p:=\left\{\sum_{k\in\mathbb{Z}}c_kT_k\phi\,\middle|\,\{c_k\}_{k\in\mathbb{Z}}\in\ell^p(\mathbb{Z})\right\}. S p := { k ∈ Z ∑ c k T k ϕ { c k } k ∈ Z ∈ ℓ p ( Z ) } . Then S p S_p S p L p ( R ) \mathcal{L}^p(\mathbb{R}) L p ( R ) { T k ϕ } k ∈ Z \{T_k\phi\}_{k\in\mathbb{Z}} { T k ϕ } k ∈ Z p p p L p ( R ) \mathcal{L}^p(\mathbb{R}) L p ( R )
Like Theorem 1.2.58, we have a similar result for Banach spaces.
Theorem 1.3.6. ([69 ] ) Let X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗
(i) { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X b b b
T : ℓ q ( N ) ∋ { a n } n ↦ ∑ n = 1 ∞ a n f n ∈ X ∗ (9) T:\ell^q(\mathbb{N})\ni\{a_n\}_n\mapsto\sum_{n=1}^{\infty}a_nf_n\in\mathcal{X}^* \tag*{(9)} T : ℓ q ( N ) ∋ { a n } n ↦ n = 1 ∑ ∞ a n f n ∈ X ∗ (9) is a well-defined (hence bounded) linear operator and ∥ T ∥ ≤ b \lVert T\rVert\le b ∥ T ∥ ≤ b q q q p p p
(ii) If X \mathcal{X} X { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X T T T
Rather working on p-frames, one can consider a general notion of frames, which are generalizations of ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
Definition 1.3.7. ([cf. 29 ] A sequence space X d \mathcal{X}_d X d BK-space if it is a Banach space and all the coordinate functionals are continuous, i.e., whenever { x n } n \{x_n\}_n { x n } n X d \mathcal{X}_d X d x ∈ X d x \in\mathcal{X}_d x ∈ X d x n x_n x n x x x
Familiar sequence spaces like ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) c ( N ) c(\mathbb{N}) c ( N ) c 0 ( N ) c_0(\mathbb{N}) c 0 ( N )
Example 1.3.8. The space X d : = { { x n } n = 0 ∞ : x n ∈ K , ∀ n ∈ N ∪ { 0 } } \mathcal{X}_d := \{\{x_n\}_{n=0}^{\infty}: x_n \in\mathbb{K}, \forall n \in\mathbb{N} \cup\{0\}\} X d := {{ x n } n = 0 ∞ : x n ∈ K , ∀ n ∈ N ∪ { 0 }}
d ( { x n } n = 0 ∞ , { y n } n = 0 ∞ ) : = ∑ n = 0 ∞ 1 2 n ∣ x n − y n ∣ 1 + ∣ x n − y n ∣ , ∀ { x n } n = 0 ∞ , { y n } n = 0 ∞ ∈ X d d(\{x_n\}_{n=0}^{\infty},\{y_n\}_{n=0}^{\infty}) := \sum_{n=0}^{\infty} \frac{1}{2^n}\frac{|x_n-y_n|}{1+|x_n-y_n|}, \quad\forall\{x_n\}_{n=0}^{\infty},\{y_n\}_{n=0}^{\infty} \in\mathcal{X}_d d ({ x n } n = 0 ∞ , { y n } n = 0 ∞ ) := n = 0 ∑ ∞ 2 n 1 1 + ∣ x n − y n ∣ ∣ x n − y n ∣ , ∀ { x n } n = 0 ∞ , { y n } n = 0 ∞ ∈ X d is not a BK-space.
Definition 1.3.9. ([45 ] Let X \mathcal{X} X X d \mathcal{X}_d X d { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ X d \mathcal{X}_d X d X \mathcal{X} X
(i) { f n ( x ) } n ∈ X d \{f_n(x)\}_n \in\mathcal{X}_d { f n ( x ) } n ∈ X d x ∈ X x \in\mathcal{X} x ∈ X
(ii) There exist a , b > 0 a,b > 0 a , b > 0 a ∥ x ∥ ≤ ∥ { f n ( x ) } n ∥ ≤ b ∥ x ∥ a\|x\| \le\|\{f_n(x)\}_n\| \le b\|x\| a ∥ x ∥ ≤ ∥ { f n ( x ) } n ∥ ≤ b ∥ x ∥ ∀ x ∈ X \forall x \in\mathcal{X} ∀ x ∈ X
Constants a a a b b b X d \mathcal{X}_d X d
Definition 1.3.10. ([45 ] Let X \mathcal{X} X X d \mathcal{X}_d X d { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X X d \mathcal{X}_d X d X \mathcal{X} X
(i) { f ( τ n ) } n ∈ X d \{f(\tau_n)\}_n \in\mathcal{X}_d { f ( τ n ) } n ∈ X d f ∈ X ∗ f \in\mathcal{X}^* f ∈ X ∗
(ii) There exist a , b > 0 a,b > 0 a , b > 0 a ∥ f ∥ ≤ ∥ { f ( τ n ) } n ∥ ≤ b ∥ f ∥ a\|f\| \le\|\{f(\tau_n)\}_n\| \le b\|f\| a ∥ f ∥ ≤ ∥ { f ( τ n ) } n ∥ ≤ b ∥ f ∥ ∀ f ∈ X ∗ \forall f \in\mathcal{X}^* ∀ f ∈ X ∗
Definition 1.3.11. ([122 ] Let X \mathcal{X} X X d \mathcal{X}_d X d { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ S : X d → X S:\mathcal{X}_d \to\mathcal{X} S : X d → X ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) Banach frame for X \mathcal{X} X
(ii) There exist a , b > 0 a,b>0 a , b > 0 a ∥ x ∥ ≤ ∥ { f n ( x ) } n ∥ ≤ b ∥ x ∥ , ∀ x ∈ X a\lVert x\rVert\leq\lVert\{f_n(x)\}_n\rVert\leq b\lVert x\rVert,\ \forall x\in\mathcal{X} a ∥ x ∥ ≤ ∥{ f n ( x ) } n ∥ ≤ b ∥ x ∥ , ∀ x ∈ X
(iii) S ( { f n ( x ) } n ) = x S(\{f_n(x)\}_n)=x S ({ f n ( x ) } n ) = x x ∈ X x\in\mathcal{X} x ∈ X
Constants a a a b b b lower Banach frame bound and upper Banach frame bound , respectively. The operator S S S reconstruction operator and the operator θ f : X ∋ x ↦ θ f ( x ) : = { f n ( x ) } n ∈ X d \theta_f:\mathcal{X}\ni x\mapsto\theta_f(x):=\{f_n(x)\}_n\in\mathcal{X}_d θ f : X ∋ x ↦ θ f ( x ) := { f n ( x ) } n ∈ X d analysis operator .
Example 1.3.12. (i) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H a a a b b b f ∈ H ∗ f\in\mathcal{H}^* f ∈ H ∗ h f ∈ H h_f\in\mathcal{H} h f ∈ H f ( h ) = ⟨ h , h f ⟩ f(h)=\langle h,h_f\rangle f ( h ) = ⟨ h , h f ⟩ ∀ h ∈ H \forall h\in\mathcal{H} ∀ h ∈ H ∥ f ∥ = ∥ h f ∥ \lVert f\rVert=\lVert h_f\rVert ∥ f ∥ = ∥ h f ∥
a ∥ f ∥ 2 = a ∥ h f ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h f , τ n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ f ( τ n ) ∣ 2 ≤ b ∥ h f ∥ 2 = b ∥ f ∥ 2 . a\lVert f\rVert^2=a\lVert h_f\rVert^2\leq\sum_{n=1}^{\infty}\lvert\langle h_f,\tau_n\rangle\rvert^2=\sum_{n=1}^{\infty}\lvert f(\tau_n)\rvert^2\leq b\lVert h_f\rVert^2=b\lVert f\rVert^2. a ∥ f ∥ 2 = a ∥ h f ∥ 2 ≤ n = 1 ∑ ∞ ∣⟨ h f , τ n ⟩ ∣ 2 = n = 1 ∑ ∞ ∣ f ( τ n ) ∣ 2 ≤ b ∥ h f ∥ 2 = b ∥ f ∥ 2 . Therefore { τ n } n \{\tau_n\}_n { τ n } n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) H \mathcal{H} H
(ii) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H f n ( h ) : = ⟨ h , τ n ⟩ f_n(h):=\langle h,\tau_n\rangle f n ( h ) := ⟨ h , τ n ⟩ ∀ h ∈ H \forall h\in\mathcal{H} ∀ h ∈ H ∀ n \forall n ∀ n S : = S τ − 1 θ τ ∗ S:=S_\tau^{-1}\theta_\tau^* S := S τ − 1 θ τ ∗ ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) H \mathcal{H} H
(iii) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H f n ( h ) : = ⟨ h , S τ − 1 τ n ⟩ f_n(h):=\langle h,S_\tau^{-1}\tau_n\rangle f n ( h ) := ⟨ h , S τ − 1 τ n ⟩ ∀ h ∈ H \forall h\in\mathcal{H} ∀ h ∈ H ∀ n \forall n ∀ n S : = θ τ ∗ S:=\theta_\tau^* S := θ τ ∗ ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) H \mathcal{H} H
(iv) [45 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
X d : = { { ⟨ h , τ n + τ n + 1 ⟩ } n : h ∈ H } = { { a n + a n + 1 } n : { a } n ∈ ℓ 2 ( N ) } . \mathcal{X}_d:=\{\{\langle h,\tau_n+\tau_{n+1}\rangle\}_n:h\in\mathcal{H}\}=\{\{a_n+a_{n+1}\}_n:\{a\}_n\in\ell^2(\mathbb{N})\}. X d := {{⟨ h , τ n + τ n + 1 ⟩ } n : h ∈ H } = {{ a n + a n + 1 } n : { a } n ∈ ℓ 2 ( N )} . equipped with the norm
∥ { a n + a n + 1 } n ∥ : = ( ∑ n = 1 ∞ ∣ a n ∣ 2 ) 1 2 . \lVert\{a_n+a_{n+1}\}_n\rVert:=\left(\sum_{n=1}^{\infty}\lvert a_n\rvert^2\right)^{\frac{1}{2}}. ∥{ a n + a n + 1 } n ∥ := ( n = 1 ∑ ∞ ∣ a n ∣ 2 ) 2 1 . Then X d \mathcal{X}_d X d f n : H ∋ h ↦ ⟨ h , τ n + τ n + 1 ⟩ ∈ K f_n:\mathcal{H}\ni h\mapsto\langle h,\tau_n+\tau_{n+1}\rangle\in\mathbb{K} f n : H ∋ h ↦ ⟨ h , τ n + τ n + 1 ⟩ ∈ K ∀ n \forall n ∀ n T : H ∋ h ↦ { f n ( h ) } n ∈ X d T:\mathcal{H}\ni h\mapsto\{f_n(h)\}_n\in\mathcal{X}_d T : H ∋ h ↦ { f n ( h ) } n ∈ X d S : = T − 1 S:=T^{-1} S := T − 1 ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) H \mathcal{H} H { τ n + τ n + 1 } n \{\tau_n+\tau_{n+1}\}_n { τ n + τ n + 1 } n H \mathcal{H} H
For Hilbert spaces it follows immediately that every Hilbert space has a frame because separable spaces have orthonormal bases and an orthonormal basis is a frame. Using Hahn-Banach theorem, the following result was proved in [53 ] .
Theorem 1.3.13. ([53 ] ) Every separable Banach space admits a Banach frame.
The notion of atomic decomposition is studied along with the notion of frames for Banach spaces. This is defined as follows.
Definition 1.3.14. [122 ] Let X \mathcal{X} X X d \mathcal{X}_d X d { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) atomic decomposition for X \mathcal{X} X
(i) { f n ( x ) } n ∈ X d \{f_n(x)\}_n \in\mathcal{X}_d { f n ( x ) } n ∈ X d x ∈ X x \in\mathcal{X} x ∈ X
(ii) There exist a , b > 0 a,b > 0 a , b > 0 a ∥ x ∥ ≤ ∥ { f n ( x ) } n ∥ ≤ b ∥ x ∥ , ∀ x ∈ X a\lVert x\rVert\le\lVert\{f_n(x)\}_n\rVert\le b\lVert x\rVert,\ \forall x \in\mathcal{X} a ∥ x ∥ ≤ ∥{ f n ( x ) } n ∥ ≤ b ∥ x ∥ , ∀ x ∈ X
(iii) x = ∑ n = 1 ∞ f n ( x ) τ n x = \sum_{n=1}^{\infty} f_n(x)\tau_n x = ∑ n = 1 ∞ f n ( x ) τ n x ∈ X x \in\mathcal{X} x ∈ X
Constants a a a b b b lower atomic bound and upper atomic bound , respectively.
Example 1.3.15. (i) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H f n ( h ) : = ⟨ h , S τ − 1 τ n ⟩ f_n(h) := \langle h,S_\tau^{-1}\tau_n\rangle f n ( h ) := ⟨ h , S τ − 1 τ n ⟩ ∀ h ∈ H , ∀ n \forall h \in\mathcal{H},\ \forall n ∀ h ∈ H , ∀ n ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) H \mathcal{H} H
(ii) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H f n ( h ) : = ⟨ h , τ n ⟩ f_n(h) := \langle h,\tau_n\rangle f n ( h ) := ⟨ h , τ n ⟩ ∀ h ∈ H \forall h \in\mathcal{H} ∀ h ∈ H ω n : = S τ − 1 τ n , ∀ n \omega_n := S_\tau^{-1}\tau_n,\ \forall n ω n := S τ − 1 τ n , ∀ n ( { f n } n , { ω n } n ) (\{f_n\}_n,\{\omega_n\}_n) ({ f n } n , { ω n } n ) H \mathcal{H} H
Now we make a detailed observation that the notions of atomic decompositions and Banach frames are completely different. Definition 1.3.14 of atomic decomposition demands the expression of every element of a Banach space as a convergent series in the same Banach space using a collection of bounded linear functionals on the space and a collection of elements in the same Banach space. On the other hand, Definition 1.3.11 of Banach frame demands the expression of every element of a Banach space using a bounded linear operator from the BK-space to the Banach space and a collection of bounded linear functionals on the same space without bothering about a converging series expansion. Theorem 1.3.13 guarantees the existence of Banach frame for every separable Banach space. However, the following remarkable result gives a lot of conditions on Banach spaces to admit an atomic decomposition.
Theorem 1.3.16. [53 , 140 ] (); Pelczynski 1971) For a Banach space X \mathcal{X} X
(i) X \mathcal{X} X
(ii) X \mathcal{X} X
There is a close relationship between atomic decomposition and Banach frames for certain classes of Banach spaces. These are exhibited in the following theorems.
Theorem 1.3.17. [53 ] Let X \mathcal{X} X X d \mathcal{X}_d X d { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ S : X d → X S:\mathcal{X}_d \to\mathcal{X} S : X d → X { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d
(i) ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) X \mathcal{X} X { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d
(ii) ( { f n } n , { S e n } n ) \left(\{f_n\}_n,\{Se_n\}_n\right) ( { f n } n , { S e n } n ) X \mathcal{X} X
Theorem 1.3.18. [53 , 140 ] (); Pelczynski 1971) Let X \mathcal{X} X X d \mathcal{X}_d X d { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ S : X d → X S:\mathcal{X}_d \to\mathcal{X} S : X d → X { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d
(i) There exists a BK-space X d \mathcal{X}_d X d ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X
(ii) There exists a BK-space Y d \mathcal{Y}_d Y d { e n } n \{e_n\}_n { e n } n S : Y d → X S:\mathcal{Y}_d \to\mathcal{X} S : Y d → X ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) X \mathcal{X} X S S S S e n = τ n Se_n=\tau_n S e n = τ n n ∈ N n \in\mathbb{N} n ∈ N
Next theorem shows that there is a relation between atomic decompositions and projections.
Theorem 1.3.19. [53 ] Let X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ S : X d → X S:\mathcal{X}_d \to\mathcal{X} S : X d → X { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d
(i) There is a BK-space X d \mathcal{X}_d X d ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X
(ii) There is a Banach space Z \mathcal{Z} Z { ω n } n \{\omega_n\}_n { ω n } n X ⊆ Z \mathcal{X} \subseteq\mathcal{Z} X ⊆ Z P : Z → X P:\mathcal{Z} \to\mathcal{X} P : Z → X P ω n = τ n P\omega_n=\tau_n P ω n = τ n ∀ n ∈ N \forall n \in\mathbb{N} ∀ n ∈ N
We mention here before passing that it is known, one can simultaneously construct Banach frames and atomic decompositions for certain classes of Banach spaces such as coorbit spaces Gröchenig [122 ] , α \alpha α [110 ] , decomposition spaces Borup and Nielsen [42 ] , homogeneous spaces Dahlke et al. [77 ] , weighted coorbit spaces Dahlke et al. [76 ] , generalized coorbit spaces Dahlke et al. [75 ] , inhomogeneous function spaces Rauhut and Ullrich (2011) and Bergman spaces Christensen et al. [60 ] .
Using approximation property for Banach spaces, Casazza and Christensen [49 ] proved that X d \mathcal{X}_d X d
Theorem 1.3.20. [45 ] Let X d \mathcal{X}_d X d B K BK B K { f n } n \{f_n\}_n { f n } n X d \mathcal{X}_d X d X \mathcal{X} X θ f : X ∋ x ↦ { f n ( x ) } n ∈ X d \theta_f:\mathcal{X} \ni x \mapsto\{f_n(x)\}_n \in\mathcal{X}_d θ f : X ∋ x ↦ { f n ( x ) } n ∈ X d
(i) θ f ( X ) \theta_f(\mathcal{X}) θ f ( X ) X d \mathcal{X}_d X d
(ii) The operator θ f − 1 : θ f ( X ) → X \theta_f^{-1}:\theta_f(\mathcal{X}) \to\mathcal{X} θ f − 1 : θ f ( X ) → X T f : X d → X T_f:\mathcal{X}_d \to\mathcal{X} T f : X d → X
(iii) There exists a bounded linear operator S : X d → X S:\mathcal{X}_d \to\mathcal{X} S : X d → X ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) X \mathcal{X} X
Also, the condition
(iv) There exists a sequence { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X ∑ n = 1 ∞ a n τ n \sum_{n=1}^{\infty} a_n\tau_n ∑ n = 1 ∞ a n τ n X \mathcal{X} X { a n } n \{a_n\}_n { a n } n X d \mathcal{X}_d X d x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\ \forall x\in\mathcal{X} x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X
implies each of (i)-(iii). If we also assume that the canonical unit vectors { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d
(v) There exists an X d ∗ \mathcal{X}_d^* X d ∗ { τ n } n ⊆ X ⊆ X ∗ ∗ \{\tau_n\}_n\subseteq\mathcal{X}\subseteq\mathcal{X}^{**} { τ n } n ⊆ X ⊆ X ∗∗ X ∗ \mathcal{X}^* X ∗ x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\ \forall x\in\mathcal{X} x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X
If the canonical unit vectors { e n } n \{e_n\}_n { e n } n X d \mathcal{X}_d X d X ∗ \mathcal{X}^* X ∗
(vi) There exists an X d ∗ \mathcal{X}_d^* X d ∗ { τ n } n ⊆ X ⊆ X ∗ ∗ \{\tau_n\}_n\subseteq\mathcal{X}\subseteq\mathcal{X}^{**} { τ n } n ⊆ X ⊆ X ∗∗ X ∗ \mathcal{X}^* X ∗ f = ∑ n = 1 ∞ f ( τ n ) f n , ∀ f ∈ X ∗ f=\sum_{n=1}^{\infty}f(\tau_n)f_n,\ \forall f\in\mathcal{X}^* f = ∑ n = 1 ∞ f ( τ n ) f n , ∀ f ∈ X ∗
In each of the cases (v) and (vi), { τ n } n \{\tau_n\}_n { τ n } n X d ∗ \mathcal{X}_d^* X d ∗ X ∗ \mathcal{X}^* X ∗
We end this section by mentioning a perturbation theorem for Banach frames.
Theorem 1.3.21. [68 ] (Christensen-Heil perturbation ) Let ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) X \mathcal{X} X { g n } n \{g_n\}_n { g n } n X ∗ \mathcal{X}^* X ∗
∥ { ( f n − g n ) ( x ) } n ∥ ≤ α ∥ { f n ( x ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . \left\|\{(f_n-g_n)(x)\}_n\right\| \le\alpha\left\|\{f_n(x)\}_n\right\|+\gamma\|x-y\|,\quad\forall x,y\in\mathcal{X}. ∥ {( f n − g n ) ( x ) } n ∥ ≤ α ∥ { f n ( x ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . Then there exists a reconstruction operator T T T ( { f n } n , T ) \left(\{f_n\}_n,T\right) ( { f n } n , T ) X \mathcal{X} X ∥ S ∥ − 1 − ( α ∥ θ f ∥ + γ ) \|S\|^{-1}-(\alpha\|\theta_f\|+\gamma) ∥ S ∥ − 1 − ( α ∥ θ f ∥ + γ ) ∥ θ f ∥ + ( α ∥ θ f ∥ + γ ) \|\theta_f\|+(\alpha\|\theta_f\|+\gamma) ∥ θ f ∥ + ( α ∥ θ f ∥ + γ )
MULTIPLIERS FOR BANACH SPACES
Let { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N ) { τ n } n , { ω n } n \{\tau_n\}_n,\{\omega_n\}_n { τ n } n , { ω n } n H \mathcal{H} H x , y ∈ H x,y\in\mathcal{H} x , y ∈ H H ∋ h ↦ ⟨ h , y ⟩ x ∈ H \mathcal{H}\ni h\mapsto\langle h,y\rangle x\in\mathcal{H} H ∋ h ↦ ⟨ h , y ⟩ x ∈ H x ⊗ y ‾ x\otimes\overline{y} x ⊗ y
The study of operators of the form
∑ n = 1 ∞ λ n ( τ n ⊗ ω n ‾ ) (10) \sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes\overline{\omega_n}) \tag*{(10)} n = 1 ∑ ∞ λ n ( τ n ⊗ ω n ) (10) began with Schatten (1960), in connection with the study of compact operators. Schatten studied the operator in (1.4.1) whenever { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N ) { τ n } n , { ω n } n \{\tau_n\}_n,\{\omega_n\}_n { τ n } n , { ω n } n H \mathcal{H} H [105 , 33 , 83 , 118 , 71 ] (), Skrettingland 2020)). This was generalized by Balazs [26 ] who replaced orthonormal sequences by Bessel sequences.
Let { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n Y \mathcal{Y} Y τ ⊗ f \tau\otimes f τ ⊗ f τ ⊗ f : X ∋ x ↦ f ( x ) τ ∈ Y \tau\otimes f:\mathcal{X}\ni x\mapsto f(x)\tau\in\mathcal{Y} τ ⊗ f : X ∋ x ↦ f ( x ) τ ∈ Y X \mathcal{X} X X ∗ \mathcal{X}^* X ∗ multipliers ) of the form
M λ , f , τ : = ∑ n = 1 ∞ λ n ( τ n ⊗ f n ) . (11) M_{\lambda,f,\tau}:=\sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes f_n). \tag*{(11)} M λ , f , τ := n = 1 ∑ ∞ λ n ( τ n ⊗ f n ) . (11) Rahimi and Balazs studied the operator in (1.4.2), whenever { τ n } n \{\tau_n\}_n { τ n } n X ∗ \mathcal{X}^* X ∗ { f n } n \{f_n\}_n { f n } n X \mathcal{X} X
Fundamental result obtained by Rahimi and Balazs (2010) is the following. In this section, q q q p p p
Theorem 1.4.1. (Rahimi and Balazs (2010)) Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X b b b { τ n } n \{\tau_n\}_n { τ n } n q q q Y \mathcal{Y} Y d d d { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n \in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N )
T : X ∋ x ↦ ∑ n = 1 ∞ λ n ( τ n ⊗ f n ) x ∈ Y T:\mathcal{X}\ni x\mapsto\sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes f_n)x\in\mathcal{Y} T : X ∋ x ↦ n = 1 ∑ ∞ λ n ( τ n ⊗ f n ) x ∈ Y is a well-defined bounded linear operator and ∥ T ∥ ≤ b d ∥ { λ n } n ∥ ∞ \|T\|\le bd\|\{\lambda_n\}_n\|_\infty ∥ T ∥ ≤ b d ∥ { λ n } n ∥ ∞
By varying only the symbol in a multiplier, we get a bounded linear operator which has the nice property stated in the following theorem.
Proposition 1.4.2. (Rahimi and Balazs (2010)) Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n q q q Y \mathcal{Y} Y { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N )
T : ℓ ∞ ( N ) ∋ { λ n } n ↦ M λ , f , τ ∈ B ( X , Y ) T:\ell^\infty(\mathbb{N})\ni\{\lambda_n\}_n\mapsto M_{\lambda,f,\tau}\in\mathcal{B}(\mathcal{X},\mathcal{Y}) T : ℓ ∞ ( N ) ∋ { λ n } n ↦ M λ , f , τ ∈ B ( X , Y ) is a well-defined injective bounded linear operator.
From the spectral theory of compact operators in Hilbert spaces, it easily follows that the symbol of compact operator converges to zero. Following is a general result for Banach spaces.
Proposition 1.4.3. (Rahimi and Balazs (2010)) Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X b b b { τ n } n \{\tau_n\}_n { τ n } n q q q Y \mathcal{Y} Y d d d { λ n } n ∈ c 0 ( N ) \{\lambda_n\}_n\in c_0(\mathbb{N}) { λ n } n ∈ c 0 ( N ) M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ
Following theorem shows that multipliers behave nicely with respect to change in its parameters. These are known as continuity properties of multipliers in the literature.
Theorem 1.4.4. (Rahimi and Balazs (2010)) Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X b b b { τ n } n \{\tau_n\}_n { τ n } n q q q Y \mathcal{Y} Y d d d { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N ) k ∈ N k\in\mathbb{N} k ∈ N λ ( k ) = { λ 1 ( k ) , λ 2 ( k ) , … } \lambda^{(k)}=\{\lambda_1^{(k)},\lambda_2^{(k)},\ldots\} λ ( k ) = { λ 1 ( k ) , λ 2 ( k ) , … } λ = { λ 1 , λ 2 , … } \lambda=\{\lambda_1,\lambda_2,\ldots\} λ = { λ 1 , λ 2 , … } τ ( k ) = { τ 1 ( k ) , τ 2 ( k ) , … } \tau^{(k)}=\{\tau_1^{(k)},\tau_2^{(k)},\ldots\} τ ( k ) = { τ 1 ( k ) , τ 2 ( k ) , … } τ n k ∈ X \tau_n^k\in\mathcal{X} τ n k ∈ X τ = { τ 1 , τ 2 , … } \tau=\{\tau_1,\tau_2,\ldots\} τ = { τ 1 , τ 2 , … } k k k λ ( k ) ∈ ℓ ∞ ( N ) \lambda^{(k)}\in\ell^\infty(\mathbb{N}) λ ( k ) ∈ ℓ ∞ ( N ) τ ( k ) \tau^{(k)} τ ( k ) q q q Y \mathcal{Y} Y
(i) If λ ( k ) → λ \lambda^{(k)} \to\lambda λ ( k ) → λ k → ∞ k \to\infty k → ∞ p p p
∥ M λ ( k ) , f , τ − M λ , f , τ ∥ → 0 as k → ∞ . \|M_{\lambda^{(k)},f,\tau}-M_{\lambda,f,\tau}\| \to0 \quad\text{as } k \to\infty. ∥ M λ ( k ) , f , τ − M λ , f , τ ∥ → 0 as k → ∞. (ii) If { λ n } n ∈ ℓ p ( N ) \{\lambda_n\}_n \in\ell^p(\mathbb{N}) { λ n } n ∈ ℓ p ( N ) ∑ n = 1 ∞ ∥ τ n ( k ) − τ n ∥ q → 0 \sum_{n=1}^{\infty}\|\tau_n^{(k)}-\tau_n\|^q \to0 ∑ n = 1 ∞ ∥ τ n ( k ) − τ n ∥ q → 0 k → ∞ k \to\infty k → ∞
∥ M λ , f , τ ( k ) − M λ , f , τ ∥ → 0 as k → ∞ . \|M_{\lambda,f,\tau^{(k)}}-M_{\lambda,f,\tau}\| \to0 \quad\text{as } k \to\infty. ∥ M λ , f , τ ( k ) − M λ , f , τ ∥ → 0 as k → ∞. LIPSCHITZ OPERATORS AND LIPSCHITZ COMPACT OPERATORS
We first recall the definition of Lipschitz function.
Definition 1.5.1. (cf. Weaver (2018)) Let M , N \mathcal{M}, \mathcal{N} M , N f : M → N f:\mathcal{M}\to\mathcal{N} f : M → N Lipschitz if there exists b > 0 b>0 b > 0
d ( f ( x ) , f ( y ) ) ≤ b d ( x , y ) , ∀ x , y ∈ M . d(f(x),f(y)) \leq b d(x,y), \quad\forall x,y \in\mathcal{M}. d ( f ( x ) , f ( y )) ≤ b d ( x , y ) , ∀ x , y ∈ M . One of the most important results in the study of Lipschitz functions is the following (which is similar to the Banach-Mazur theorem for Banach spaces which shows that every separable Banach space embeds isometrically in the Banach space of continuous functions on [ 0 , 1 ] [0,1] [ 0 , 1 ] [6 ] )).
Theorem 1.5.2. (cf. Aharoni [4 ] ; Kalton and Lancien [147 ] ) (Aharoni’s theorem ) If M \mathcal{M} M f : M → c 0 ( N ) f:\mathcal{M}\to c_0(\mathbb{N}) f : M → c 0 ( N ) b > 0 b>0 b > 0
d ( x , y ) ≤ ∥ f ( x ) − f ( y ) ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . d(x,y) \leq\|f(x)-f(y)\| \leq b d(x,y), \quad\forall x,y \in\mathcal{M}. d ( x , y ) ≤ ∥ f ( x ) − f ( y ) ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . Definition 1.5.3. (cf. Weaver (2018)) A metric space M \mathcal{M} M 0 0 0 pointed metric space . In this case, we write ( M , 0 ) (\mathcal{M},0) ( M , 0 )
Note that every metric space is a pointed metric space by fixing any point of the space.
Just like norm of linear operator, a reasonable measure of a Lipschitz function can be defined. This is exhibited in the next definition.
Definition 1.5.4. (cf. Weaver (2018)) Let X \mathcal{X} X
(i) Let M \mathcal{M} M Lip ( M , X ) \operatorname{Lip}(\mathcal{M},\mathcal{X}) Lip ( M , X ) Lip ( M , X ) : = { f : f : M → X is Lipschitz } \operatorname{Lip}(\mathcal{M},\mathcal{X}):=\{f:f:\mathcal{M}\to\mathcal{X}\text{ is Lipschitz}\} Lip ( M , X ) := { f : f : M → X is Lipschitz } f ∈ Lip ( M , X ) f\in\operatorname{Lip}(\mathcal{M},\mathcal{X}) f ∈ Lip ( M , X ) Lipschitz number is defined as
Lip ( f ) : = sup x , y ∈ M , x ≠ y ∥ f ( x ) − f ( y ) ∥ d ( x , y ) . \operatorname{Lip}(f):=\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}}\frac{\lVert f(x)-f(y)\rVert}{d(x,y)}. Lip ( f ) := x , y ∈ M , x = y sup d ( x , y ) ∥ f ( x ) − f ( y )∥ . (ii) Let ( M , 0 ) (\mathcal{M},0) ( M , 0 ) Lip 0 ( M , X ) \operatorname{Lip}_0(\mathcal{M},\mathcal{X}) Lip 0 ( M , X ) Lip 0 ( M , X ) : = { f : f : M → X is Lipschitz and f ( 0 ) = 0 } \operatorname{Lip}_0(\mathcal{M},\mathcal{X}):=\{f:f:\mathcal{M}\to\mathcal{X}\text{ is Lipschitz and }f(0)=0\} Lip 0 ( M , X ) := { f : f : M → X is Lipschitz and f ( 0 ) = 0 } f ∈ Lip 0 ( M , X ) f\in\operatorname{Lip}_0(\mathcal{M},\mathcal{X}) f ∈ Lip 0 ( M , X ) Lipschitz norm is defined as
∥ f ∥ Lip 0 : = sup x , y ∈ M , x ≠ y ∥ f ( x ) − f ( y ) ∥ d ( x , y ) . \lVert f\rVert_{\operatorname{Lip}_0}:=\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}}\frac{\lVert f(x)-f(y)\rVert}{d(x,y)}. ∥ f ∥ Lip 0 := x , y ∈ M , x = y sup d ( x , y ) ∥ f ( x ) − f ( y )∥ . It is well-known that given two Banach spaces X \mathcal{X} X Y \mathcal{Y} Y X \mathcal{X} X Y \mathcal{Y} Y
Theorem 1.5.5. ((cf. Weaver (2018))) Let X \mathcal{X} X
(i) If M \mathcal{M} M Lip ( M , X ) \operatorname{Lip}(\mathcal{M},\mathcal{X}) Lip ( M , X ) Lip ( ⋅ ) \operatorname{Lip}(\cdot) Lip ( ⋅ )
(ii) If ( M , 0 ) (\mathcal{M},0) ( M , 0 ) Lip 0 ( M , X ) \operatorname{Lip}_0(\mathcal{M},\mathcal{X}) Lip 0 ( M , X ) ∥ ⋅ ∥ Lip 0 \lVert\cdot\rVert_{\operatorname{Lip}_0} ∥ ⋅ ∥ Lip 0 Lip 0 ( X ) : = Lip 0 ( X , X ) \operatorname{Lip}_0(\mathcal{X}):=\operatorname{Lip}_0(\mathcal{X},\mathcal{X}) Lip 0 ( X ) := Lip 0 ( X , X ) T ∈ Lip 0 ( X ) T\in\operatorname{Lip}_0(\mathcal{X}) T ∈ Lip 0 ( X ) ∥ T − I X ∥ Lip 0 < 1 \lVert T-I_{\mathcal{X}}\rVert_{\operatorname{Lip}_0}<1 ∥ T − I X ∥ Lip 0 < 1 T T T T − 1 ∈ Lip 0 ( X ) T^{-1}\in\operatorname{Lip}_0(\mathcal{X}) T − 1 ∈ Lip 0 ( X )
In the study of Lipschitz functions it is natural to shift from metric space to the setting of Banach spaces and use functional analysis tools on Banach spaces. This is achieved through the following theorem.
Theorem 1.5.6. ([cf. 16 , 146 ] (); Weaver 2018)) Let ( M , 0 ) (\mathcal{M},0) ( M , 0 ) F ( M ) \mathcal{F}(\mathcal{M}) F ( M ) e : M → F ( M ) e:\mathcal{M}\to\mathcal{F}(\mathcal{M}) e : M → F ( M ) X \mathcal{X} X f ∈ Lip 0 ( M , X ) f\in\operatorname{Lip}_0(\mathcal{M},\mathcal{X}) f ∈ Lip 0 ( M , X ) T f : F ( M ) → X T_f:\mathcal{F}(\mathcal{M})\to\mathcal{X} T f : F ( M ) → X T f e = f T_f e=f T f e = f
M → f X e ↓ T f ↑ F ( M ) → X \begin{CD}
\mathcal{M} @>{f}>> \mathcal{X} \\
@V{e}VV @A{T_f}AA \\
\mathcal{F}(\mathcal{M}) @>>> \mathcal{X}
\end{CD} M e ↓ ⏐ F ( M ) f X T f ⏐ ↑ X Further, ∥ T f ∥ = ∥ f ∥ Lip 0 \lVert T_f\rVert=\lVert f\rVert_{\operatorname{Lip}_0} ∥ T f ∥ = ∥ f ∥ Lip 0 ( F ( M ) , e ) (\mathcal{F}(\mathcal{M}),e) ( F ( M ) , e ) Lip 0 ( M , X ) ∋ f ↦ T f ∈ B ( F ( M ) , X ) \operatorname{Lip}_0(\mathcal{M},\mathcal{X})\ni f\mapsto T_f\in\mathcal{B}(\mathcal{F}(\mathcal{M}),\mathcal{X}) Lip 0 ( M , X ) ∋ f ↦ T f ∈ B ( F ( M ) , X )
The space F ( M ) \mathcal{F}(\mathcal{M}) F ( M ) Arens-Eells space or Lipschitz-free Banach space (Godefroy [120 ] ). Theorem 1.5.6 tells that in order to ‘find’ the space Lip 0 ( M , X ) \operatorname{Lip}_0(\mathcal{M},\mathcal{X}) Lip 0 ( M , X ) F ( M ) \mathcal{F}(\mathcal{M}) F ( M ) B ( F ( M ) , X ) \mathcal{B}(\mathcal{F}(\mathcal{M}),\mathcal{X}) B ( F ( M ) , X ) Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) F ( M ) ∗ \mathcal{F}(\mathcal{M})^* F ( M ) ∗ F ( M ) \mathcal{F}(\mathcal{M}) F ( M ) predual of metric space M \mathcal{M} M T f T_f T f linearization of f f f
Example 1.5.7. (Weaver (2018); ), Dubei et al. [88 ] )
If R \mathbb{R} R F ( R ) ≅ L 1 ( R ) \mathcal{F}(\mathbb{R})\cong\mathcal{L}^1(\mathbb{R}) F ( R ) ≅ L 1 ( R )
If M \mathcal{M} M F ( M ) ≅ L 1 ( [ 0 , 1 ] ) \mathcal{F}(\mathcal{M})\cong\mathcal{L}^1([0,1]) F ( M ) ≅ L 1 ([ 0 , 1 ])
If M \mathcal{M} M d ( x , y ) : = 2 d(x,y):=2 d ( x , y ) := 2 x , y ∈ M x,y\in\mathcal{M} x , y ∈ M x ≠ y x\ne y x = y d ( x , x ) : = 0 d(x,x):=0 d ( x , x ) := 0 ∀ x ∈ M \forall x\in\mathcal{M} ∀ x ∈ M F ( M ) ≅ ℓ 1 ( M ) \mathcal{F}(\mathcal{M})\cong\ell^1(\mathcal{M}) F ( M ) ≅ ℓ 1 ( M )
If N \mathbb{N} N F ( N ) ≅ ℓ 1 ( N ) \mathcal{F}(\mathbb{N})\cong\ell^1(\mathbb{N}) F ( N ) ≅ ℓ 1 ( N )
F ( ℓ 1 ( N ) ) ≅ L 1 ( R ) \mathcal{F}(\ell^1(\mathbb{N}))\cong\mathcal{L}^1(\mathbb{R}) F ( ℓ 1 ( N )) ≅ L 1 ( R )
In the theory of bounded linear operators between Banach spaces, an operator is said to be compact if the image of the unit ball under the operator is precompact (Fabian et al. [95 ] ). Linearity of the operator now gives various characterizations of compactness and plays an important role in rich theories such as theory of integral equations, spectral theory, theory of Fredholm operators, operator algebra (C*-algebra), K-theory, Calkin algebra, (operator) ideal theory, approximation properties of Banach spaces, Schauder basis theory. Lack of linearity is a hurdle when one tries to define compactness of nonlinear maps. This hurdle was successfully crossed in the paper (Jiménez-Vargas et al. [139 ] ) which began the study of Lipschitz compact operators.
Definition 1.5.8. (Jiménez-Vargas et al. [139 ] ) If M \mathcal{M} M X \mathcal{X} X Lipschitz image of a Lipschitz map (also called as Lipschitz operator ) f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X
{ f ( x ) − f ( y ) d ( x , y ) : x , y ∈ M , x ≠ y } . (12) \left\{\frac{f(x)-f(y)}{d(x,y)}:x,y\in\mathcal{M},x\ne y\right\}. \tag*{(12)} { d ( x , y ) f ( x ) − f ( y ) : x , y ∈ M , x = y } . (12) We observe that whenever an operator is linear, the set in (1.5.1) is simply the image of the unit sphere.
Definition 1.5.9. (Jiménez-Vargas et al. [139 ] ) If ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 Lipschitz compact if its Lipschitz image is relatively compact in X \mathcal{X} X X \mathcal{X} X
As showed in (Jiménez-Vargas et al. [139 ] ), there is a large collection of Lipschitz compact operators. To state this, first we need a definition.
Definition 1.5.10. (Chen and Zheng [59 ] ) Let ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 strongly Lipschitz p p p (1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ A ∈ B ( ℓ p ( N ) , X ) A\in\mathcal{B}(\ell^{p}(\mathbb{N}),\mathcal{X}) A ∈ B ( ℓ p ( N ) , X ) g ∈ Lip 0 ( M , ℓ ∞ ( N ) ) g\in\operatorname{Lip}_{0}(\mathcal{M},\ell^{\infty}(\mathbb{N})) g ∈ Lip 0 ( M , ℓ ∞ ( N )) M λ ∈ B ( ℓ ∞ ( N ) , ℓ p ( N ) ) M_{\lambda}\in\mathcal{B}(\ell^{\infty}(\mathbb{N}),\ell^{p}(\mathbb{N})) M λ ∈ B ( ℓ ∞ ( N ) , ℓ p ( N )) λ ∈ ℓ p ( N ) \lambda\in\ell^{p}(\mathbb{N}) λ ∈ ℓ p ( N ) f = A M λ g f=AM_{\lambda}g f = A M λ g
\begin{CD}
\mathcal{M} @>f>> \mathcal{X}\\
@VgVV @AA V\\
\ell^{\infty}(\mathbb{N}) @>M_{\lambda}>> \ell^{p}(\mathbb{N})
\end{CD}
Proposition 1.5.11. (Jiménez-Vargas et al. [139 ] ) Every strongly Lipschitz p p p
Since the image of a linear operator is a subspace, the natural definition of finite rank operator is that image is a finite dimensional subspace. The image of Lipschitz map may not be a subspace. Thus care has to be taken while defining rank of such maps.
Definition 1.5.12. (Jiménez-Vargas et al. [139 ] ) If ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 Lipschitz finite dimensional rank if the linear hull of its Lipschitz image is a finite dimensional subspace of X \mathcal{X} X
Definition 1.5.13. [139 ] If M \mathcal{M} M X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X finite dimensional rank if the linear hull of its image is a finite dimensional subspace of X \mathcal{X} X
Next theorem shows that for pointed metric spaces, Definitions 1.5.12 and 1.5.13 are equivalent.
Theorem 1.5.14. [1 , 139 ] Let ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0
(i) f f f
(ii) f f f
(iii) There exist f 1 , … , f n f_1,\ldots,f_n f 1 , … , f n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) τ 1 , … , τ n \tau_1,\ldots,\tau_n τ 1 , … , τ n X \mathcal{X} X
f ( x ) = ∑ k = 1 n f k ( x ) τ k , ∀ x ∈ M . f(x)=\sum_{k=1}^{n}f_k(x)\tau_k,\qquad\forall x\in\mathcal{M}. f ( x ) = k = 1 ∑ n f k ( x ) τ k , ∀ x ∈ M . In Hilbert spaces (and not in Banach spaces), every compact operator is approximable by finite rank operators in the operator norm [cf. 95 ] . Following is the definition of approximable operator for Lipschitz maps.
Definition 1.5.15. [139 ] If ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X f : M → X f:\mathcal{M}\to\mathcal{X} f : M → X f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 Lipschitz approximable if it is the limit in the Lipschitz norm of a sequence of Lipschitz finite rank operators from M \mathcal{M} M X \mathcal{X} X
Theorem 1.5.16. [139 ] Every Lipschitz approximable operator from pointed metric space ( M , 0 ) (\mathcal{M},0) ( M , 0 ) X \mathcal{X} X
OPERATOR-VALUED ORTHONORMAL BASES, RIESZ BASES, FRAMES AND BESSEL SEQUENCES IN HILBERT SPACES
Through a decade long research, the frame theory for Hilbert spaces was extended to larger extent by Kaftal, Larson and Zhang with the introduction of the notion of operator-valued frame (OVF) in 2009. In the theory of operator-valued frames, the sequence { L n } n \{L_n\}_n { L n } n
Definition 1.6.1. (Kaftal et al. [144 ] ) Given n ∈ N n \in\mathbb{N} n ∈ N
L n : H 0 ∋ h ↦ L n h : = e n ⊗ h ∈ ℓ 2 ( N ) ⊗ H 0 , L_n:\mathcal{H}_0 \ni h \mapsto L_nh := e_n \otimes h \in\ell^2(\mathbb{N})\otimes\mathcal{H}_0, L n : H 0 ∋ h ↦ L n h := e n ⊗ h ∈ ℓ 2 ( N ) ⊗ H 0 , where { e n } n \{e_n\}_n { e n } n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N )
Following proposition shows properties of the operator L n L_n L n
Proposition 1.6.2. (Kaftal et al. [144 ] ) The operators L n L_n L n Definition 1.6.1 satisfy the following.
(i) Each L n L_n L n H 0 \mathcal{H}_0 H 0 ℓ 2 ( N ) ⊗ H 0 \ell^2(\mathbb{N})\otimes\mathcal{H}_0 ℓ 2 ( N ) ⊗ H 0 n , m ∈ N n,m\in\mathbb{N} n , m ∈ N
L n ∗ L m = { I H 0 if n = m 0 if n ≠ m and ∑ n = 1 ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . (13) L_n^*L_m =
\begin{cases}
I_{\mathcal{H}_0} & \text{if } n=m\\
0 & \text{if } n\ne m
\end{cases}
\quad\text{and}\quad
\sum_{n=1}^{\infty}L_nL_n^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}. \tag*{(13)} L n ∗ L m = { I H 0 0 if n = m if n = m and n = 1 ∑ ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . (13) where the convergence is in the strong-operator topology.
(ii) L m ∗ ( { a n } n ⊗ y ) = a m y L_m^*(\{a_n\}_n\otimes y)=a_my L m ∗ ({ a n } n ⊗ y ) = a m y ∀ { a n } n ∈ ℓ 2 ( N ) , ∀ y ∈ H 0 \forall\{a_n\}_n\in\ell^2(\mathbb{N}),\forall y\in\mathcal{H}_0 ∀ { a n } n ∈ ℓ 2 ( N ) , ∀ y ∈ H 0 m ∈ N m\in\mathbb{N} m ∈ N
Orthonormal basis for Hilbert spaces are defined in Definition 1.2.8 . Considering Definition 1.2.8 and Parseval identity ((iv) in Theorem 1.2.10 ) for orthonormal basis, Sun (2006) defined the notion of orthonormal basis for operators.
Definition 1.6.3. (Sun (2006)) A collection { F n } n \{F_n\}_n { F n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
⟨ F n ∗ y , F k ∗ z ⟩ = δ n , k ⟨ y , z ⟩ , ∀ y , z ∈ H 0 , ∀ n , k ∈ N \langle F_n^*y,F_k^*z\rangle=\delta_{n,k}\langle y,z\rangle,\quad\forall y,z\in\mathcal{H}_0,\ \forall n,k\in\mathbb{N} ⟨ F n ∗ y , F k ∗ z ⟩ = δ n , k ⟨ y , z ⟩ , ∀ y , z ∈ H 0 , ∀ n , k ∈ N and
∑ n = 1 ∞ ∥ F n h ∥ 2 = ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n=1}^{\infty}\lVert F_nh\rVert^2=\lVert h\rVert^2,\quad\forall h\in\mathcal{H}. n = 1 ∑ ∞ ∥ F n h ∥ 2 = ∥ h ∥ 2 , ∀ h ∈ H . We observe ⟨ F n ∗ y , F k ∗ z ⟩ = δ n , k ⟨ y , z ⟩ , ∀ y , z ∈ H 0 , ∀ n , k ∈ N \langle F_n^*y,F_k^*z\rangle=\delta_{n,k}\langle y,z\rangle,\forall y,z\in\mathcal{H}_0,\ \forall n,k\in\mathbb{N} ⟨ F n ∗ y , F k ∗ z ⟩ = δ n , k ⟨ y , z ⟩ , ∀ y , z ∈ H 0 , ∀ n , k ∈ N F n F k ∗ = δ n , k I H 0 , ∀ n , k ∈ N F_nF_k^*=\delta_{n,k}I_{\mathcal{H}_0},\ \forall n,k\in\mathbb{N} F n F k ∗ = δ n , k I H 0 , ∀ n , k ∈ N { F n } n \{F_n\}_n { F n } n ∥ F n ∥ 2 = ∥ F n F n ∗ ∥ = 1 , ∀ n ∈ N \lVert F_n\rVert^2=\lVert F_nF_n^*\rVert=1,\forall n\in\mathbb{N} ∥ F n ∥ 2 = ∥ F n F n ∗ ∥ = 1 , ∀ n ∈ N ∑ n = 1 ∞ F n ∗ F n = I H \sum_{n=1}^{\infty}F_n^*F_n=I_{\mathcal{H}} ∑ n = 1 ∞ F n ∗ F n = I H Proposition 1.6.2 we have
( ∑ n = 1 ∞ A n ∗ L n ∗ ) ( ∑ k = 1 ∞ L k A k ) = I H . \left(\sum_{n=1}^{\infty}A_n^*L_n^*\right)\left(\sum_{k=1}^{\infty}L_kA_k\right)=I_{\mathcal{H}}. ( n = 1 ∑ ∞ A n ∗ L n ∗ ) ( k = 1 ∑ ∞ L k A k ) = I H . Consider the case H 0 = K \mathcal{H}_0=\mathbb{K} H 0 = K n ∈ N n\in\mathbb{N} n ∈ N [158 ] ), there exists a unique τ n ∈ H \tau_n\in\mathcal{H} τ n ∈ H F n h = ⟨ h , τ n ⟩ , ∀ h ∈ H F_nh=\langle h,\tau_n\rangle,\forall h\in\mathcal{H} F n h = ⟨ h , τ n ⟩ , ∀ h ∈ H Definition 1.6.3 tells $\langle F_n^*y,F_k^*z\rangle=\overline{y}z\langle\tau_n,\tau_k\rangle=\overline{y}z\delta_{j,k},\forall j,k\in\mathbb{N},\forall y,z\in K \mathbb{K} K { τ n } n \{\tau_n\}_n { τ n } n
∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 = ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n=1}^{\infty} |\langle h,\tau_n\rangle|^2 = \|h\|^2,\quad\forall h \in\mathcal{H}. n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 = ∥ h ∥ 2 , ∀ h ∈ H . Theorem 1.2.10 now tells that { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
Example 1.6.4. (i) (Sun (2006)) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H F n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K F_n:\mathcal{H}\ni h\mapsto\langle h,\tau_n\rangle\in\mathbb{K} F n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K n ∈ N n\in\mathbb{N} n ∈ N { F n } n \{F_n\}_n { F n } n B ( H , K ) \mathcal{B}(\mathcal{H},\mathbb{K}) B ( H , K )
(ii) If U : H → H 0 U:\mathcal{H}\to\mathcal{H}_0 U : H → H 0 { U } \{U\} { U } B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(iii) Let n ≥ 2 n\ge2 n ≥ 2 A 1 , … , A n A_1,\ldots,A_n A 1 , … , A n n n n ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) A 1 A 1 ∗ + ⋯ + A n A n ∗ = I ℓ 2 ( N ) A_1A_1^*+\cdots+A_nA_n^*=I_{\ell^2(\mathbb{N})} A 1 A 1 ∗ + ⋯ + A n A n ∗ = I ℓ 2 ( N ) [73 ] )). We then have
( A j A j ∗ ) 2 = A j ( A j ∗ A j ) A j ∗ = A j I H A j ∗ = A j A j ∗ . (A_jA_j^*)^2=A_j(A_j^*A_j)A_j^*=A_jI_{\mathcal{H}}A_j^*=A_jA_j^*. ( A j A j ∗ ) 2 = A j ( A j ∗ A j ) A j ∗ = A j I H A j ∗ = A j A j ∗ . Hence A j A j ∗ A_jA_j^* A j A j ∗ A j ∗ A k = 0 A_j^*A_k=0 A j ∗ A k = 0 ∀ j ≠ k \forall j\ne k ∀ j = k { A j ∗ } j = 1 n \{A_j^*\}_{j=1}^n { A j ∗ } j = 1 n B ( ℓ 2 ( N ) ) \mathcal{B}(\ell^2(\mathbb{N})) B ( ℓ 2 ( N ))
(iv) Equation (13) says that { L n ∗ } n \{L_n^*\}_n { L n ∗ } n B ( ℓ 2 ( N ) ⊗ H 0 , H 0 ) \mathcal{B}(\ell^2(\mathbb{N})\otimes\mathcal{H}_0,\mathcal{H}_0) B ( ℓ 2 ( N ) ⊗ H 0 , H 0 )
Using Theorem 1.2.25, Sun (2006) defined the notion of Riesz basis for operators.
Definition 1.6.5. (Sun (2006)) A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(i) { h ∈ H : A n h = 0 , ∀ n ∈ N } = { 0 } \{h\in\mathcal{H}:A_nh=0,\forall n\in\mathbb{N}\}=\{0\} { h ∈ H : A n h = 0 , ∀ n ∈ N } = { 0 }
(ii) There exist a , b > 0 a,b>0 a , b > 0 S S S N \mathbb{N} N
a ∑ n ∈ S ∥ h n ∥ 2 ≤ ∥ ∑ n ∈ S A n ∗ h n ∥ 2 ≤ b ∑ n ∈ S ∥ h n ∥ 2 , ∀ h n ∈ H 0 . a\sum_{n\in S}\|h_n\|^2\le\left\|\sum_{n\in S}A_n^*h_n\right\|^2\le b\sum_{n\in S}\|h_n\|^2,\quad\forall h_n\in\mathcal{H}_0. a n ∈ S ∑ ∥ h n ∥ 2 ≤ n ∈ S ∑ A n ∗ h n 2 ≤ b n ∈ S ∑ ∥ h n ∥ 2 , ∀ h n ∈ H 0 . Example 1.6.6. (i) (Sun (2006)) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K A_n:\mathcal{H}\ni h\mapsto\langle h,\tau_n\rangle\in\mathbb{K} A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K n ∈ N n\in\mathbb{N} n ∈ N A n ∗ y = y τ n , ∀ y ∈ K A_n^*y=y\tau_n,\forall y\in\mathbb{K} A n ∗ y = y τ n , ∀ y ∈ K Theorem 1.2.25 , { A n } n \{A_n\}_n { A n } n B ( H , K ) \mathcal{B}(\mathcal{H},\mathbb{K}) B ( H , K )
(ii) If U : H → H 0 U:\mathcal{H}\to\mathcal{H}_0 U : H → H 0 { U } \{U\} { U } B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(iii) Let A 1 , … , A n A_1,\ldots,A_n A 1 , … , A n A , B ∈ B ( H ) A,B\in\mathcal{B}(\mathcal{H}) A , B ∈ B ( H ) { A A j ∗ B } j = 1 n \{AA_j^*B\}_{j=1}^n { A A j ∗ B } j = 1 n B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) A A j ∗ B h = AA_j^*Bh= A A j ∗ B h =
0 0 0 ∀ j \forall j ∀ j A j ∗ B h = 0 A_j^*Bh=0 A j ∗ B h = 0 ∀ j \forall j ∀ j
h = B − 1 B h = B − 1 ( ∑ k = 1 n A k A k ∗ B h ) = 0 h=B^{-1}Bh=B^{-1}\left(\sum_{k=1}^{n}A_kA_k^*Bh\right)=0 h = B − 1 B h = B − 1 ( k = 1 ∑ n A k A k ∗ B h ) = 0 and every finite subset S S S N \mathbb{N} N
( ∥ B − 1 ∥ ∥ A − 1 ∥ ) − 1 ∑ j ∈ S ∥ h j ∥ 2 ≤ ∥ ∑ j ∈ S B ∗ A j A j ∗ h j ∥ 2 ≤ ∥ B ∥ ∥ A ∥ ∑ j ∈ S ∥ h j ∥ 2 , \left(\left\|B^{-1}\right\|\left\|A^{-1}\right\|\right)^{-1}\sum_{j\in S}\|h_j\|^2\leq\left\|\sum_{j\in S}B^*A_jA_j^*h_j\right\|^2\leq\|B\|\|A\|\sum_{j\in S}\|h_j\|^2, ( B − 1 A − 1 ) − 1 j ∈ S ∑ ∥ h j ∥ 2 ≤ j ∈ S ∑ B ∗ A j A j ∗ h j 2 ≤ ∥ B ∥∥ A ∥ j ∈ S ∑ ∥ h j ∥ 2 , for all h j ∈ H h_j\in\mathcal{H} h j ∈ H
Sun (2006) derived the following result which tells that we can define the notion of operator-valued Riesz basis in a manner similar to Definition 1.2.19.
Theorem 1.6.7. (Sun (2006)) A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { F n } n \{F_n\}_n { F n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) T ∈ B ( H ) \mathrm{T}\in\mathcal{B}(\mathcal{H}) T ∈ B ( H ) A n = F n U , ∀ n A_n=F_nU,\forall n A n = F n U , ∀ n
Historically, many generalizations of frames for Hilbert spaces are proposed such as frames for subspaces (Casazza and Kutyniok [55 ] ), fusion frames (Casazza et al. [57 ] ), outer frames (Aldroubi et al. [10 ] ), oblique frames (Christensen and Eldar [66 ] ), pseudo frames (Li and Ogawa (2004)), quasi-projectors (Fornasier [109 ] ). It was in 2006, when Sun gave the definition of G-frame which unified all these notions of frames for Hilbert spaces.
Definition 1.6.8. (Sun (2006)) A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) a , b > 0 a,b>0 a , b > 0
a ∥ h ∥ 2 ≤ ∑ n = 1 ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . a\|h\|^2\leq\sum_{n=1}^{\infty}\|A_nh\|^2\leq b\|h\|^2,\qquad\forall h\in\mathcal{H}. a ∥ h ∥ 2 ≤ n = 1 ∑ ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . Basic idea for the notion of OVF is the following. Definition 1.2.31 can be written in an equivalent form as
the map S τ : H ∋ h ↦ ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H is a well-defined bounded positive invertible operator. (14) \text{the map }S_\tau:\mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n\in\mathcal{H}\text{ is a well-defined bounded positive invertible operator.} \tag*{(14)} the map S τ : H ∋ h ↦ n = 1 ∑ ∞ ⟨ h , τ n ⟩ τ n ∈ H is a well-defined bounded positive invertible operator. (14) If we now define A n : H ∋ h ↦ ⟨ h , x n ⟩ ∈ K A_n:\mathcal{H}\ni h\mapsto\langle h,x_n\rangle\in\mathcal{K} A n : H ∋ h ↦ ⟨ h , x n ⟩ ∈ K n ∈ N n\in\mathbb{N} n ∈ N
∑ n = 1 ∞ A n ∗ A n converges in the strong-operator topology on B ( H ) to a bounded positive invertible operator. (15) \sum_{n=1}^{\infty} A_n^*A_n \text{ converges in the strong-operator topology on } \mathcal{B}(\mathcal{H}) \text{ to a bounded positive invertible operator.} \tag*{(15)} n = 1 ∑ ∞ A n ∗ A n converges in the strong-operator topology on B ( H ) to a bounded positive invertible operator. (15) Now Statement (1.6.3) leads to
Definition 1.6.9. ([144 ] ) A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
S A : = ∑ n = 1 ∞ A n ∗ A n S_A := \sum_{n=1}^{\infty} A_n^*A_n S A := n = 1 ∑ ∞ A n ∗ A n converges in the strong-operator topology on B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) a a a b b b
Before proceeding, we first show that Definitions 1.6.8 and 1.6.9 are equivalent. For this, we first need a result.
Theorem 1.6.10. ((Sun (2006))) If { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) S A : H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h ∈ H S_A:\mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}A_n^*A_nh\in\mathcal{H} S A : H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h ∈ H
Following theorem will reflect the equivalence of notions of OVF and G-frame.
Theorem 1.6.11. A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) S A S_A S A A n ∗ A n ≥ 0 , ∀ n ∈ N A_n^*A_n\ge0,\forall n\in\mathbb{N} A n ∗ A n ≥ 0 , ∀ n ∈ N S A S_A S A { A n } n \{A_n\}_n { A n } n
( ⇐ ) (\Leftarrow) ( ⇐ ) ∑ n = 1 ∞ A n ∗ A n \sum_{n=1}^{\infty}A_n^*A_n ∑ n = 1 ∞ A n ∗ A n a , b > 0 a,b>0 a , b > 0 a I H ≤ ∑ n = 1 ∞ A n ∗ A n ≤ b I H aI_{\mathcal{H}}\le\sum_{n=1}^{\infty}A_n^*A_n\le bI_{\mathcal{H}} a I H ≤ ∑ n = 1 ∞ A n ∗ A n ≤ b I H a ∥ h ∥ 2 ≤ ⟨ ∑ n = 1 ∞ A n ∗ A n h , h ⟩ = ∑ n = 1 ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H a\lVert h\rVert^2\le\left\langle\sum_{n=1}^{\infty}A_n^*A_nh,h\right\rangle=\sum_{n=1}^{\infty}\lVert A_nh\rVert^2\le b\lVert h\rVert^2,\ \forall h\in\mathcal{H} a ∥ h ∥ 2 ≤ ⟨ ∑ n = 1 ∞ A n ∗ A n h , h ⟩ = ∑ n = 1 ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H { A n } n \{A_n\}_n { A n } n
Example 1.6.13. (i) (Sun (2006)) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K A_n:\mathcal{H}\ni h\mapsto\langle h,\tau_n\rangle\in\mathbb{K} A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K n ∈ N n\in\mathbb{N} n ∈ N A n ∗ y = y τ n A_n^*y=y\tau_n A n ∗ y = y τ n ∀ y ∈ K \forall y\in\mathbb{K} ∀ y ∈ K Theorem 1.2.36 , the map H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H \mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}A_n^*A_nh=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n\in\mathcal{H} H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H { A n } n \{A_n\}_n { A n } n B ( H , K ) \mathcal{B}(\mathcal{H},\mathbb{K}) B ( H , K )
(ii) If A : H → H 0 A:\mathcal{H}\to\mathcal{H}_0 A : H → H 0 { A } \{A\} { A } B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(iii) Let A 1 , … , A n A_1,\ldots,A_n A 1 , … , A n A , B ∈ B ( H ) A,B\in\mathcal{B}(\mathcal{H}) A , B ∈ B ( H ) { A A j ∗ B } j = 1 n \{AA_j^*B\}_{j=1}^n { A A j ∗ B } j = 1 n B ( H ) \mathcal{B}(\mathcal{H}) B ( H )
The fundamental tool used in the study of OVF is the factorization of frame operator S A S_A S A
Theorem 1.6.14. (Kaftal et al. [144 ] , Sun (2006)) Let { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(i) span ‾ ⋃ n = 1 ∞ A n ∗ ( H 0 ) = H \overline{\operatorname{span}}\bigcup_{n=1}^{\infty}A_n^*(\mathcal{H}_0)=\mathcal{H} span ⋃ n = 1 ∞ A n ∗ ( H 0 ) = H
(ii) The analysis operator
θ A : H ∋ h ↦ θ A h : = ∑ n = 1 ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 \theta_A:\mathcal{H}\ni h\mapsto\theta_Ah:=\sum_{n=1}^{\infty}L_nA_nh\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 θ A : H ∋ h ↦ θ A h := n = 1 ∑ ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 is a well-defined bounded linear operator. Further, a ∥ h ∥ ≤ ∥ θ A h ∥ ≤ b ∥ h ∥ \sqrt{a}\lVert h\rVert\leq\lVert\theta_Ah\rVert\leq\sqrt{b}\lVert h\rVert a ∥ h ∥ ≤ ∥ θ A h ∥ ≤ b ∥ h ∥ ∀ h ∈ H \forall h\in\mathcal{H} ∀ h ∈ H θ A \theta_A θ A
(iii) We have
a ∥ h ∥ 2 ≤ ⟨ S A h , h ⟩ ≤ b ∥ h ∥ 2 , ∀ h ∈ H and a ∥ h ∥ ≤ ∥ S A h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . a\lVert h\rVert^2\leq\langle S_Ah,h\rangle\leq b\lVert h\rVert^2,\ \forall h\in\mathcal{H}\ \text{and}\ a\lVert h\rVert\leq\lVert S_Ah\rVert\leq b\lVert h\rVert,\ \forall h\in\mathcal{H}. a ∥ h ∥ 2 ≤ ⟨ S A h , h ⟩ ≤ b ∥ h ∥ 2 , ∀ h ∈ H and a ∥ h ∥ ≤ ∥ S A h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . (iv) h = ∑ n = 1 ∞ ( A n S A − 1 ) ∗ A n h = ∑ n = 1 ∞ A n ∗ ( A n S A − 1 ) h , ∀ h ∈ H h=\sum_{n=1}^{\infty}(A_nS_A^{-1})^*A_nh=\sum_{n=1}^{\infty}A_n^*(A_nS_A^{-1})h,\ \forall h\in\mathcal{H} h = ∑ n = 1 ∞ ( A n S A − 1 ) ∗ A n h = ∑ n = 1 ∞ A n ∗ ( A n S A − 1 ) h , ∀ h ∈ H
(v) θ A ∗ z = ∑ n = 1 ∞ A n ∗ L n z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 \theta_A^*z=\sum_{n=1}^{\infty}A_n^*L_nz,\ \forall z\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 θ A ∗ z = ∑ n = 1 ∞ A n ∗ L n z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0
(vi) Frame operator factors as S A = θ A ∗ θ A S_A=\theta_A^*\theta_A S A = θ A ∗ θ A
(vii) θ A ∗ \theta_A^* θ A ∗
(viii) ∥ S A − 1 ∥ − 1 \lVert S_A^{-1}\rVert^{-1} ∥ S A − 1 ∥ − 1 ∥ S A ∥ = ∥ θ A ∥ 2 \lVert S_A\rVert=\lVert\theta_A\rVert^2 ∥ S A ∥ = ∥ θ A ∥ 2
(ix) P A : = θ A S A − 1 θ A ∗ : ℓ 2 ( N ) ⊗ H 0 → ℓ 2 ( N ) ⊗ H 0 P_A:=\theta_AS_A^{-1}\theta_A^*:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 P A := θ A S A − 1 θ A ∗ : ℓ 2 ( N ) ⊗ H 0 → ℓ 2 ( N ) ⊗ H 0 θ A ( H ) \theta_A(\mathcal{H}) θ A ( H )
(xi) { A n S A − 1 } n \{A_nS_A^{-1}\}_n { A n S A − 1 } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) b − 1 b^{-1} b − 1 a − 1 a^{-1} a − 1
(xii) { A n S A − 1 / 2 } n \{A_nS_A^{-1/2}\}_n { A n S A − 1/2 } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
∑ n = 1 ∞ ∥ y n ∥ 2 = ∑ n = 1 ∞ ∥ A n S A − 1 h ∥ 2 + ∑ n = 1 ∞ ∥ y n − A n S A − 1 h ∥ 2 . \sum_{n=1}^{\infty}\lVert y_n\rVert^2=\sum_{n=1}^{\infty}\lVert A_nS_A^{-1}h\rVert^2+\sum_{n=1}^{\infty}\lVert y_n-A_nS_A^{-1}h\rVert^2. n = 1 ∑ ∞ ∥ y n ∥ 2 = n = 1 ∑ ∞ ∥ A n S A − 1 h ∥ 2 + n = 1 ∑ ∞ ∥ y n − A n S A − 1 h ∥ 2 . Similar to the notion of duality, orthogonality and similarity for frames in Hilbert spaces, there are similar notions for operator-valued frames. We now recall these notions and mention some results.
Definition 1.6.15. (Kaftal et al. [144 ] ) An OVF { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) dual for an OVF { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ∑ n = 1 ∞ B n ∗ A n = I H \sum_{n=1}^{\infty}B_n^*A_n=I_{\mathcal{H}} ∑ n = 1 ∞ B n ∗ A n = I H
Definition 1.6.16. (Kaftal et al. [144 ] ) An OVF { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) orthogonal to an OVF { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ∑ n = 1 ∞ B n ∗ A n = 0 \sum_{n=1}^{\infty}B_n^*A_n=0 ∑ n = 1 ∞ B n ∗ A n = 0
Proposition 1.6.17. (Kaftal et al. [144 ] ) Let { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) C , D , E , F ∈ B ( H ) C,D,E,F\in\mathcal{B}(\mathcal{H}) C , D , E , F ∈ B ( H ) C ∗ C + D ∗ D = I H C^*C+D^*D=I_{\mathcal{H}} C ∗ C + D ∗ D = I H { A n C + B n D } n \{A_nC+B_nD\}_n { A n C + B n D } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) c , d c,d c , d ∣ c ∣ 2 + ∣ d ∣ 2 = 1 |c|^2+|d|^2=1 ∣ c ∣ 2 + ∣ d ∣ 2 = 1 { c A n + d B n } n \{cA_n+dB_n\}_n { c A n + d B n } n
Proposition 1.6.18. (Kaftal et al. [144 ] ) If { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { A n ⊕ B n } n \{A_n\oplus B_n\}_n { A n ⊕ B n } n B ( H ⊕ H , H 0 ) \mathcal{B}(\mathcal{H}\oplus\mathcal{H},\mathcal{H}_0) B ( H ⊕ H , H 0 ) { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n { A n ⊕ B n } n \{A_n\oplus B_n\}_n { A n ⊕ B n } n
Definition 1.6.19. (Kaftal et al. [144 ] ) An OVF { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) similar or equivalent to an OVF { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) R A , B ∈ B ( H ) R_{A,B}\in\mathcal{B}(\mathcal{H}) R A , B ∈ B ( H ) B n = A n R A , B , ∀ n ∈ N B_n=A_nR_{A,B},\forall n\in\mathbb{N} B n = A n R A , B , ∀ n ∈ N
Similar frames share some nice properties that knowing analysis, synthesis and frame operators of one give that of another.
Lemma 1.6.20. (Kaftal et al. [144 ] ) Let { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n B n = A n R A , B , ∀ n ∈ N B_n=A_nR_{A,B},\forall n\in\mathbb{N} B n = A n R A , B , ∀ n ∈ N R A , B ∈ B ( H ) R_{A,B}\in\mathcal{B}(\mathcal{H}) R A , B ∈ B ( H )
(i) θ B = θ A R A , B \theta_B=\theta_AR_{A,B} θ B = θ A R A , B
(ii) S B = R A , B ∗ S A R A , B S_B = R_{A,B}^{*}S_A R_{A,B} S B = R A , B ∗ S A R A , B
(iii) P B = P A P_B = P_A P B = P A
There is a complete classification of similarity using operators.
Theorem 1.6.21. (Kaftal et al. [144 ] ) For two OVFs { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n
(i) B n = A n R A , B , ∀ n ∈ N B_n = A_nR_{A,B},\forall n \in\mathbb{N} B n = A n R A , B , ∀ n ∈ N R A , B ∈ B ( H ) R_{A,B} \in\mathcal{B}(\mathcal{H}) R A , B ∈ B ( H )
(ii) θ B = θ A R A , B \theta_B = \theta_A R_{A,B} θ B = θ A R A , B R A , B ∈ B ( H ) R_{A,B} \in\mathcal{B}(\mathcal{H}) R A , B ∈ B ( H )
(iii) P B = P A P_B = P_A P B = P A
If one of the above conditions is satisfied, then invertible operators in (i) and (ii) are unique and are given by R A , B = S A − 1 θ A ∗ θ B R_{A,B}=S_A^{-1}\theta_A^{*}\theta_B R A , B = S A − 1 θ A ∗ θ B { A n } n \{A_n\}_n { A n } n { B n } n \{B_n\}_n { B n } n R A , B R_{A,B} R A , B
In the study of frames, rather indexing with natural numbers or other indexing sets, it is more useful in some cases to study frames indexed by groups and generated by a single operator.
Let G G G { χ g } g ∈ G \{\chi_g\}_{g\in G} { χ g } g ∈ G ℓ 2 ( G ) \ell^2(G) ℓ 2 ( G ) λ \lambda λ G G G λ g χ q ( r ) = χ q ( g − 1 r ) \lambda_g\chi_q(r)=\chi_q(g^{-1}r) λ g χ q ( r ) = χ q ( g − 1 r ) ∀ g , q , r ∈ G \forall g,q,r\in G ∀ g , q , r ∈ G ρ \rho ρ G G G ρ g χ q ( r ) = χ q ( r g ) \rho_g\chi_q(r)=\chi_q(rg) ρ g χ q ( r ) = χ q ( r g ) ∀ g , q , r ∈ G \forall g,q,r\in G ∀ g , q , r ∈ G L ( G ) \mathcal{L}(G) L ( G ) { λ g } g ∈ G \{\lambda_g\}_{g\in G} { λ g } g ∈ G B ( ℓ 2 ( G ) ) \mathcal{B}(\ell^2(G)) B ( ℓ 2 ( G )) R ( G ) \mathcal{R}(G) R ( G ) { ρ g } g ∈ G \{\rho_g\}_{g\in G} { ρ g } g ∈ G B ( ℓ 2 ( G ) ) \mathcal{B}(\ell^2(G)) B ( ℓ 2 ( G )) L ( G ) ′ = R ( G ) \mathcal{L}(G)^{\prime}=\mathcal{R}(G) L ( G ) ′ = R ( G ) R ( G ) ′ = L ( G ) \mathcal{R}(G)^{\prime}=\mathcal{L}(G) R ( G ) ′ = L ( G ) [70 ] ), where A ′ \mathcal{A}^{\prime} A ′ A ⊆ B ( H ) \mathcal{A}\subseteq\mathcal{B}(\mathcal{H}) A ⊆ B ( H )
Definition 1.6.22. (Kaftal et al. [144 ] ) Let π \pi π G G G H \mathcal{H} H A A A B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) operator frame generator (resp. a Parseval frame generator) w.r.t. an operator Ψ \Psi Ψ B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { A g : = A π g − 1 } g ∈ G \{A_g:=A\pi_{g^{-1}}\}_{g\in G} { A g := A π g − 1 } g ∈ G B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) A A A π \pi π
Frames generated by groups have the remarkable property that the frame operator belongs to the commutant of π ( G ) \pi(G) π ( G )
Proposition 1.6.23. (Kaftal et al. [144 ] ) Let A A A B B B B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) π \pi π G G G H \mathcal{H} H
(i) θ A π g = ( λ g ⊗ I H 0 ) θ A , θ B π g = ( λ g ⊗ I H 0 ) θ B , ∀ g ∈ G \theta_A\pi_g=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A,\theta_B\pi_g=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_B,\forall g\in G θ A π g = ( λ g ⊗ I H 0 ) θ A , θ B π g = ( λ g ⊗ I H 0 ) θ B , ∀ g ∈ G
(ii) θ A ∗ θ B \theta_A^*\theta_B θ A ∗ θ B π ( G ) ′ \pi(G)' π ( G ) ′ π ( G ) ′ ′ \pi(G)'' π ( G ) ′′ S A ∈ π ( G ) ′ S_A\in\pi(G)' S A ∈ π ( G ) ′
(iii) θ A T θ A ∗ , θ A T θ B ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ T ∈ π ( G ) ′ \theta_A T\theta_A^*,\theta_A T\theta_B^*\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0),\forall T\in\pi(G)' θ A T θ A ∗ , θ A T θ B ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ T ∈ π ( G ) ′ P A ∈ R ( G ) ⊗ B ( H 0 ) P_A\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0) P A ∈ R ( G ) ⊗ B ( H 0 )
Following theorem gives a characterization of frames generated by unitary representation without using representation.
Theorem 1.6.24. [144 ] Let G G G e e e G G G { A g } g ∈ G \{A_g\}_{g\in G} { A g } g ∈ G B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) π \pi π G G G H \mathcal{H} H
A g = A e π g − 1 , ∀ g ∈ G A_g=A_e\pi_{g^{-1}},\qquad\forall g\in G A g = A e π g − 1 , ∀ g ∈ G if and only if
A g A g ∗ = A p A q ∗ , ∀ g , p , q ∈ G . A_gA_g^*=A_pA_q^*,\qquad\forall g,p,q\in G. A g A g ∗ = A p A q ∗ , ∀ g , p , q ∈ G . One of the most important results obtained by Kaftal et al. [144 ] is the connectedness of the set of all generators of operator-valued frames generated by groups.
Theorem 1.6.25. [144 ] Let π \pi π G G G H \mathcal{H} H
∅ ≠ F G : = { A ∈ B ( H , H 0 ) : { A π g − 1 } g ∈ G is an operator-valued frame in B ( H , H 0 ) } . \varnothing\ne\mathcal{F}_G:=\{A\in\mathcal{B}(\mathcal{H},\mathcal{H}_0):\{A\pi_{g^{-1}}\}_{g\in G}\text{ is an operator-valued frame in }\mathcal{B}(\mathcal{H},\mathcal{H}_0)\}. ∅ = F G := { A ∈ B ( H , H 0 ) : { A π g − 1 } g ∈ G is an operator-valued frame in B ( H , H 0 )} . (i) If dim H 0 < ∞ \dim\mathcal{H}_0<\infty dim H 0 < ∞ F G \mathcal{F}_G F G path-connected in the operator-norm topology on B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(ii) If dim H 0 = ∞ \dim\mathcal{H}_0=\infty dim H 0 = ∞ F G \mathcal{F}_G F G path-connected in the operator-norm topology on B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) R ( G ) \mathcal{R}(G) R ( G ) G G G diffuse (i.e., R ( G ) \mathcal{R}(G) R ( G )
Stability result for OVFs is due to Sun (2007).
Theorem 1.6.26. (Sun (2007)) Let { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) a a a b b b { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 max { α + γ a , β } < 1 \max\{\alpha+\frac{\gamma}{\sqrt{a}},\beta\}<1 max { α + a γ , β } < 1 m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , …
∥ ∑ n = 1 m ( A n ∗ − B n ∗ ) y n ∥ ≤ α ∥ ∑ n = 1 m A n ∗ y n ∥ + β ∥ ∑ n = 1 m B n ∗ y n ∥ + γ ( ∑ n = 1 m ∥ y n ∥ 2 ) 1 2 , ∀ y n ∈ H 0 . \left\|\sum_{n=1}^{m}(A_n^*-B_n^*)y_n\right\|
\leq\alpha\left\|\sum_{n=1}^{m}A_n^*y_n\right\|
+\beta\left\|\sum_{n=1}^{m}B_n^*y_n\right\|
+\gamma\left(\sum_{n=1}^{m}\|y_n\|^2\right)^{\frac{1}{2}},
\qquad\forall y_n\in\mathcal{H}_0. n = 1 ∑ m ( A n ∗ − B n ∗ ) y n ≤ α n = 1 ∑ m A n ∗ y n + β n = 1 ∑ m B n ∗ y n + γ ( n = 1 ∑ m ∥ y n ∥ 2 ) 2 1 , ∀ y n ∈ H 0 . Then { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
a ( 1 − α + β + γ a 1 + β ) 2 and b ( 1 + α + β + γ b 1 − β ) 2 . a\left(1-\frac{\alpha+\beta+\frac{\gamma}{\sqrt{a}}}{1+\beta}\right)^2 \quad\text{and}\quad b\left(1+\frac{\alpha+\beta+\frac{\gamma}{\sqrt{b}}}{1-\beta}\right)^2. a ( 1 − 1 + β α + β + a γ ) 2 and b ( 1 + 1 − β α + β + b γ ) 2 . Like Bessel sequences for Hilbert spaces, there is a similar notion for operators.
Definition 1.6.27. ((Sun (2006))) A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) b > 0 b>0 b > 0
(Operator-valued Bessel’s inequality)
∑ n = 1 ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . \sum_{n=1}^{\infty}\left\|A_nh\right\|^2\leq b\left\|h\right\|^2,\quad\forall h\in\mathcal{H}. n = 1 ∑ ∞ ∥ A n h ∥ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . Constant b b b { A n } n \{A_n\}_n { A n } n
Similar to Theorem 1.6.11 we have the following result.
Theorem 1.6.28. A collection { A n } n \{A_n\}_n { A n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ∑ n A n ∗ A n \sum_n A_n^*A_n ∑ n A n ∗ A n B ( H ) \mathcal{B}(\mathcal{H}) B ( H )
Following are some examples of operator-valued Bessel sequences.
Example 1.6.29. (i) ((Sun (2006))) Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K A_n:\mathcal{H}\ni h\mapsto\langle h,\tau_n\rangle\in\mathbb{K} A n : H ∋ h ↦ ⟨ h , τ n ⟩ ∈ K n ∈ N n\in\mathbb{N} n ∈ N A n ∗ y = y τ n , ∀ y ∈ K A_n^*y=y\tau_n,\forall y\in\mathbb{K} A n ∗ y = y τ n , ∀ y ∈ K Theorem 1.2.58 , the map H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H \mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}A_n^*A_nh=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n\in\mathcal{H} H ∋ h ↦ ∑ n = 1 ∞ A n ∗ A n h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H { A n } n \{A_n\}_n { A n } n B ( H , K ) \mathcal{B}(\mathcal{H},\mathbb{K}) B ( H , K )
(ii) From operator-norm inequality, we see that any finite collection of operators is an operator-valued Bessel sequence.
FRAMES FOR METRIC SPACES
BASIC PROPERTIES
In this chapter, we define frames for metric spaces and derive several fundamental properties.
Definition 2.1.1. (p -frame for metric space) Let M \mathcal{M} M p ∈ [ 1 , ∞ ) p \in[1,\infty) p ∈ [ 1 , ∞ ) { f n } n \{f_n\}_n { f n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K ) metric p -frame or Lipschitz p -frame for M \mathcal{M} M a , b > 0 a,b > 0 a , b > 0
a d ( x , y ) ≤ ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ b d ( x , y ) , ∀ x , y ∈ M . a d(x,y) \le\left(\sum_{n=1}^{\infty} \lvert f_n(x)-f_n(y)\rvert^p\right)^{\frac{1}{p}} \le b d(x,y), \qquad\forall x,y \in\mathcal{M}. a d ( x , y ) ≤ ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ b d ( x , y ) , ∀ x , y ∈ M . If we do not demand the first inequality, then we say { f n } n \{f_n\}_n { f n } n p -Bessel sequence for M \mathcal{M} M
We now see that whenever M \mathcal{M} M f n f_n f n
Example 2.1.2. Let { f n } n \{f_n\}_n { f n } n p -frame for a Banach space X \mathcal{X} X A : X → X A:\mathcal{X}\to\mathcal{X} A : X → X { f n A } n \{f_nA\}_n { f n A } n p -frame for X \mathcal{X} X
Example 2.1.3. Let 1 < a < ∞ 1 < a < \infty 1 < a < ∞ M : = [ a , ∞ ) \mathcal{M}:=[a,\infty) M := [ a , ∞ ) f n : M → R f_n:\mathcal{M}\to\mathbb{R} f n : M → R
f 0 ( x ) : = 1 , ∀ x ∈ M f_0(x):=1,\qquad\forall x\in\mathcal{M} f 0 ( x ) := 1 , ∀ x ∈ M f n ( x ) : = ( log x ) n n ! , ∀ x ∈ M , ∀ n ≥ 1. f_n(x):=\frac{(\log x)^n}{n!},\qquad\forall x\in\mathcal{M},\ \forall n\ge1. f n ( x ) := n ! ( log x ) n , ∀ x ∈ M , ∀ n ≥ 1. Then f n ′ ( x ) = ( log x ) n − 1 ( n − 1 ) ! x , ∀ x ∈ M , ∀ n ≥ 1 f_n'(x)=\frac{(\log x)^{n-1}}{(n-1)!x},\ \forall x\in\mathcal{M},\ \forall n\ge1 f n ′ ( x ) = ( n − 1 )! x ( l o g x ) n − 1 , ∀ x ∈ M , ∀ n ≥ 1 f n ′ f_n' f n ′ M \mathcal{M} M ∀ n ≥ 1 \forall n\ge1 ∀ n ≥ 1 f n f_n f n M \mathcal{M} M ∀ n ≥ 1 \forall n \ge1 ∀ n ≥ 1 x , y ∈ M x,y \in\mathcal{M} x , y ∈ M x < y x < y x < y
∑ n = 0 ∞ ∣ f n ( x ) − f n ( y ) ∣ = ∑ n = 0 ∞ ∣ ( log x ) n n ! − ( log y ) n n ! ∣ = ∑ n = 0 ∞ ( log y ) n n ! − ∑ n = 0 ∞ ( log x ) n n ! = e log y − e log x = y − x = ∣ x − y ∣ . \begin{aligned}
\sum_{n=0}^{\infty} \left|f_n(x)-f_n(y)\right|=\sum_{n=0}^{\infty}\left|\frac{(\log x)^n}{n!}-\frac{(\log y)^n}{n!}\right|=\sum_{n=0}^{\infty}\frac{(\log y)^n}{n!}-\sum_{n=0}^{\infty}\frac{(\log x)^n}{n!} \\
&=e^{\log y}-e^{\log x}=y-x=|x-y|.
\end{aligned} n = 0 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ = n = 0 ∑ ∞ n ! ( log x ) n − n ! ( log y ) n = n = 0 ∑ ∞ n ! ( log y ) n − n = 0 ∑ ∞ n ! ( log x ) n = e l o g y − e l o g x = y − x = ∣ x − y ∣. Hence { f n } n \{f_n\}_n { f n } n 1 1 1 M \mathcal{M} M
Example 2.1.4. Let 1 < a < b < ∞ 1<a<b<\infty 1 < a < b < ∞ M : = [ 1 1 − a , 1 1 − b ] \mathcal{M}:=\left[\frac{1}{1-a},\frac{1}{1-b}\right] M := [ 1 − a 1 , 1 − b 1 ] f n : M → R f_n:\mathcal{M}\to\mathbb{R} f n : M → R
f n ( x ) : = ( 1 − 1 x ) n , ∀ x ∈ M , ∀ n ≥ 0. f_n(x):=\left(1-\frac{1}{x}\right)^n,\qquad\forall x\in\mathcal{M},\ \forall n\ge0. f n ( x ) := ( 1 − x 1 ) n , ∀ x ∈ M , ∀ n ≥ 0. Then f n ′ ( x ) = n − x 2 ( 1 − 1 x ) n − 1 , ∀ x ∈ M , ∀ n ≥ 1 f_n'(x)=\frac{n}{-x^2}\left(1-\frac{1}{x}\right)^{n-1},\ \forall x\in\mathcal{M},\ \forall n\ge1 f n ′ ( x ) = − x 2 n ( 1 − x 1 ) n − 1 , ∀ x ∈ M , ∀ n ≥ 1 f n f_n f n n ≥ 1 n\ge1 n ≥ 1 { f n } n \{f_n\}_n { f n } n 1 1 1 M \mathcal{M} M x , y ∈ M x,y\in\mathcal{M} x , y ∈ M x < y x<y x < y
∑ n = 0 ∞ ∣ f n ( x ) − f n ( y ) ∣ = ∑ n = 0 ∞ ∣ ( 1 − 1 x ) n − ( 1 − 1 y ) n ∣ = ∑ n = 0 ∞ ( 1 − 1 y ) n − ∑ n = 0 ∞ ( 1 − 1 x ) n = y − x = ∣ x − y ∣ . \begin{aligned}
\sum_{n=0}^{\infty}|f_n(x)-f_n(y)|=\sum_{n=0}^{\infty}\left|\left(1-\frac{1}{x}\right)^n-\left(1-\frac{1}{y}\right)^n\right|=\sum_{n=0}^{\infty}\left(1-\frac{1}{y}\right)^n-\sum_{n=0}^{\infty}\left(1-\frac{1}{x}\right)^n \\
&=y-x=|x-y|.
\end{aligned} n = 0 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ = n = 0 ∑ ∞ ( 1 − x 1 ) n − ( 1 − y 1 ) n = n = 0 ∑ ∞ ( 1 − y 1 ) n − n = 0 ∑ ∞ ( 1 − x 1 ) n = y − x = ∣ x − y ∣. Example 2.1.5. Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X ϕ : K → K \phi:\mathbb{K}\to\mathbb{K} ϕ : K → K g n ( x ) : = ϕ ( f n ( x ) ) , ∀ x ∈ X , ∀ n ∈ N g_n(x):=\phi(f_n(x)),\ \forall x\in\mathcal{X},\ \forall n\in\mathbb{N} g n ( x ) := ϕ ( f n ( x )) , ∀ x ∈ X , ∀ n ∈ N { g n } n \{g_n\}_n { g n } n p p p X \mathcal{X} X
By looking at Theorem 1.3.6 we can ask whether there is a result similar for metric p p p
Theorem 2.1.6. Let ( M , 0 ) (\mathcal{M},0) ( M , 0 ) { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) { f n } n \{f_n\}_n { f n } n p p p M \mathcal{M} M b b b
T : ℓ q ( N ) ∋ { a n } n ↦ T { a n } n ∈ Lip 0 ( M × M , K ) , (16) T:\ell^q(\mathbb{N})\ni\{a_n\}_n\mapsto T\{a_n\}_n\in\operatorname{Lip}_0(\mathcal{M}\times\mathcal{M},\mathbb{K}), \tag*{(16)} T : ℓ q ( N ) ∋ { a n } n ↦ T { a n } n ∈ Lip 0 ( M × M , K ) , (16) T { a n } n : M × M ∋ ( x , y ) ↦ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( y ) ) ∈ K T\{a_n\}_n:\mathcal{M}\times\mathcal{M}\ni(x,y)\mapsto\sum_{n=1}^{\infty}a_n(f_n(x)-f_n(y))\in\mathbb{K} T { a n } n : M × M ∋ ( x , y ) ↦ n = 1 ∑ ∞ a n ( f n ( x ) − f n ( y )) ∈ K is a well-defined (hence bounded) linear operator and ∥ T ∥ ≤ b \|T\|\le b ∥ T ∥ ≤ b q q q p p p
Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) { a n } n ∈ ℓ q ( N ) \{a_n\}_n\in\ell^q(\mathbb{N}) { a n } n ∈ ℓ q ( N ) n , m ∈ N n,m\in\mathbb{N} n , m ∈ N n < m n<m n < m x , y ∈ M x,y \in\mathcal{M} x , y ∈ M
∣ ∑ k = n m a k ( f k ( x ) − f k ( y ) ) ∣ ≤ ( ∑ k = n m ∣ a k ∣ q ) 1 q ( ∑ k = n m ∣ f k ( x ) − f k ( y ) ∣ p ) 1 p ≤ b ( ∑ k = n m ∣ a k ∣ q ) 1 q d ( x , y ) . \begin{aligned}
\left|\sum_{k=n}^{m}a_k(f_k(x)-f_k(y))\right|
\leq\left(\sum_{k=n}^{m}|a_k|^q\right)^{\frac{1}{q}}\left(\sum_{k=n}^{m}|f_k(x)-f_k(y)|^p\right)^{\frac{1}{p}} \\
&\leq b\left(\sum_{k=n}^{m}|a_k|^q\right)^{\frac{1}{q}}d(x,y).
\end{aligned} k = n ∑ m a k ( f k ( x ) − f k ( y )) ≤ ( k = n ∑ m ∣ a k ∣ q ) q 1 ( k = n ∑ m ∣ f k ( x ) − f k ( y ) ∣ p ) p 1 ≤ b ( k = n ∑ m ∣ a k ∣ q ) q 1 d ( x , y ) . Therefore the series in (2.1.1) converges. We next show that the map T { a n } n T\{a_n\}_n T { a n } n
∥ T { a n } n ∥ Lip 0 = sup ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) ≠ ( u , v ) ∣ T { a n } n ( x , y ) − T { a n } n ( u , v ) ∣ d ( x , u ) + d ( y , v ) = sup ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) ≠ ( u , v ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( u ) ) − ∑ n = 1 ∞ a n ( f n ( y ) − f n ( v ) ) ∣ d ( x , u ) + d ( y , v ) ≤ sup ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) ≠ ( u , v ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( u ) ) ∣ + ∣ ∑ n = 1 ∞ a n ( f n ( y ) − f n ( v ) ) ∣ d ( x , u ) + d ( y , v ) ≤ b sup ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) ≠ ( u , v ) ( ∑ n = 1 ∞ ∣ a n ∣ q ) 1 q d ( x , u ) + ( ∑ n = 1 ∞ ∣ a n ∣ q ) 1 q d ( y , v ) d ( x , u ) + d ( y , v ) = b ( ∑ n = 1 ∞ ∣ a n ∣ q ) 1 q . \begin{aligned}
\|T\{a_n\}_n\|_{\operatorname{Lip}_0}
=
\sup_{\substack{(x,y),(u,v)\in\mathcal{M}\times\mathcal{M},\\(x,y)\ne(u,v)}}
\frac{|T\{a_n\}_n(x,y)-T\{a_n\}_n(u,v)|}{d(x,u)+d(y,v)} \\
&=
\sup_{\substack{(x,y),(u,v)\in\mathcal{M}\times\mathcal{M},\\(x,y)\ne(u,v)}}
\frac{\left|\sum_{n=1}^{\infty}a_n(f_n(x)-f_n(u))-\sum_{n=1}^{\infty}a_n(f_n(y)-f_n(v))\right|}{d(x,u)+d(y,v)} \\
&\leq
\sup_{\substack{(x,y),(u,v)\in\mathcal{M}\times\mathcal{M},\\(x,y)\ne(u,v)}}
\frac{\left|\sum_{n=1}^{\infty}a_n(f_n(x)-f_n(u))\right|+\left|\sum_{n=1}^{\infty}a_n(f_n(y)-f_n(v))\right|}{d(x,u)+d(y,v)} \\
&\leq b
\sup_{\substack{(x,y),(u,v)\in\mathcal{M}\times\mathcal{M},\\(x,y)\ne(u,v)}}
\frac{\left(\sum_{n=1}^{\infty}|a_n|^q\right)^{\frac{1}{q}}d(x,u)+\left(\sum_{n=1}^{\infty}|a_n|^q\right)^{\frac{1}{q}}d(y,v)}{d(x,u)+d(y,v)} \\
&= b\left(\sum_{n=1}^{\infty}|a_n|^q\right)^{\frac{1}{q}}.
\end{aligned} ∥ T { a n } n ∥ Lip 0 = ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) = ( u , v ) sup d ( x , u ) + d ( y , v ) ∣ T { a n } n ( x , y ) − T { a n } n ( u , v ) ∣ = ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) = ( u , v ) sup d ( x , u ) + d ( y , v ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( u )) − ∑ n = 1 ∞ a n ( f n ( y ) − f n ( v )) ∣ ≤ ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) = ( u , v ) sup d ( x , u ) + d ( y , v ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( u )) ∣ + ∣ ∑ n = 1 ∞ a n ( f n ( y ) − f n ( v )) ∣ ≤ b ( x , y ) , ( u , v ) ∈ M × M , ( x , y ) = ( u , v ) sup d ( x , u ) + d ( y , v ) ( ∑ n = 1 ∞ ∣ a n ∣ q ) q 1 d ( x , u ) + ( ∑ n = 1 ∞ ∣ a n ∣ q ) q 1 d ( y , v ) = b ( n = 1 ∑ ∞ ∣ a n ∣ q ) q 1 . Hence T T T T T T T T T b b b
( ⇐ ) (\Leftarrow) ( ⇐ ) T T T x , y ∈ M x,y \in\mathcal{M} x , y ∈ M
Φ x , y : ℓ q ( N ) ∋ { a n } n ↦ Φ x , y { a n } n : = ∑ n = 1 ∞ a n ( f n ( x ) − f n ( y ) ) ∈ K \Phi_{x,y}:\ell^q(\mathbb{N})\ni\{a_n\}_n\mapsto\Phi_{x,y}\{a_n\}_n:=\sum_{n=1}^{\infty}a_n(f_n(x)-f_n(y))\in\mathbb{K} Φ x , y : ℓ q ( N ) ∋ { a n } n ↦ Φ x , y { a n } n := n = 1 ∑ ∞ a n ( f n ( x ) − f n ( y )) ∈ K which is a bounded linear functional. Hence { f n ( x ) − f n ( y ) } n ∈ ℓ p ( N ) \{f_n(x)-f_n(y)\}_n\in\ell^p(\mathbb{N}) { f n ( x ) − f n ( y ) } n ∈ ℓ p ( N ) { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
∥ Φ x , y ∥ = ( ∑ n = 1 ∞ ∣ Φ x , y { e n } n ∣ p ) 1 p = ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p . \|\Phi_{x,y}\|=\left(\sum_{n=1}^{\infty}|\Phi_{x,y}\{e_n\}_n|^p\right)^{\frac{1}{p}}=\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}. ∥ Φ x , y ∥ = ( n = 1 ∑ ∞ ∣ Φ x , y { e n } n ∣ p ) p 1 = ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 . Now
b ( ∑ n = 1 ∞ ∣ a n ∣ q ) 1 q = b ∥ { a n } n ∥ ≥ ∥ T { a n } n ∥ Lip 0 ≥ sup ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) ≠ ( y , 0 ) ∣ T { a n } n ( x , 0 ) − T { a n } n ( y , 0 ) ∣ d ( x , y ) = sup ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) ≠ ( y , 0 ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( y ) ) ∣ d ( x , y ) = sup ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) ≠ ( y , 0 ) ∣ Φ x , y { a n } n ∣ d ( x , y ) . \begin{aligned}
b\left(\sum_{n=1}^{\infty}|a_n|^q\right)^{\frac{1}{q}}=b\| \{a_n\}_n\| \geq\|T\{a_n\}_n\|_{\operatorname{Lip}_0} \\
&\geq\sup_{\substack{(x,0),(y,0)\in\mathcal{M}\times\mathcal{M}\\(x,0)\ne(y,0)}}\frac{|T\{a_n\}_n(x,0)-T\{a_n\}_n(y,0)|}{d(x,y)} \\
&=\sup_{\substack{(x,0),(y,0)\in\mathcal{M}\times\mathcal{M}\\(x,0)\ne(y,0)}}\frac{\left|\sum_{n=1}^{\infty}a_n(f_n(x)-f_n(y))\right|}{d(x,y)} \\
&=\sup_{\substack{(x,0),(y,0)\in\mathcal{M}\times\mathcal{M}\\(x,0)\ne(y,0)}}\frac{|\Phi_{x,y}\{a_n\}_n|}{d(x,y)}.
\end{aligned} b ( n = 1 ∑ ∞ ∣ a n ∣ q ) q 1 = b ∥ { a n } n ∥ ≥ ∥ T { a n } n ∥ Lip 0 ≥ ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) = ( y , 0 ) sup d ( x , y ) ∣ T { a n } n ( x , 0 ) − T { a n } n ( y , 0 ) ∣ = ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) = ( y , 0 ) sup d ( x , y ) ∣ ∑ n = 1 ∞ a n ( f n ( x ) − f n ( y )) ∣ = ( x , 0 ) , ( y , 0 ) ∈ M × M ( x , 0 ) = ( y , 0 ) sup d ( x , y ) ∣ Φ x , y { a n } n ∣ . which implies
∣ Φ x , y { a n } n ∣ ≤ b ( ∑ n = 1 ∞ ∣ a n ∣ q ) 1 q d ( x , y ) , ∀ x , y ∈ M . |\Phi_{x,y}\{a_n\}_n|\leq b\left(\sum_{n=1}^{\infty}|a_n|^q\right)^{\frac{1}{q}}d(x,y),\qquad\forall x,y\in\mathcal{M}. ∣ Φ x , y { a n } n ∣ ≤ b ( n = 1 ∑ ∞ ∣ a n ∣ q ) q 1 d ( x , y ) , ∀ x , y ∈ M . Using all these, we finally get
( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p = ∥ Φ x , y ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . \left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}=\|\Phi_{x,y}\|\leq bd(x,y),\qquad\forall x,y\in\mathcal{M}. ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 = ∥ Φ x , y ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . Hence { f n } n \{f_n\}_n { f n } n p p p M \mathcal{M} M b b b
In the spirit of definition of X d \mathcal{X}_d X d
Definition 2.1.7. Let M \mathcal{M} M M d \mathcal{M}_d M d { f n } n \{f_n\}_n { f n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K ) { f n } n \{f_n\}_n { f n } n metric M d \mathcal{M}_d M d (or Lipschitz M d \mathcal{M}_d M d M \mathcal{M} M
(i) { f n ( x ) } n ∈ M d \{f_n(x)\}_n\in\mathcal{M}_d { f n ( x ) } n ∈ M d x ∈ M x\in\mathcal{M} x ∈ M
(ii) There exist positive a , b a,b a , b a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ b d ( x , y ) ad(x,y)\leq\|\{f_n(x)-f_n(y)\}_n\|_{\mathcal{M}_d}\leq bd(x,y) a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ b d ( x , y ) ∀ x , y ∈ M \forall x,y\in\mathcal{M} ∀ x , y ∈ M
We say constant a a a lower metric M d \mathcal{M}_d M d and constant b b b upper metric M d \mathcal{M}_d M d . If we do not demand the first inequality, then we say { f n } n \{f_n\}_n { f n } n M d \mathcal{M}_d M d
An easier way of producing metric M d \mathcal{M}_d M d M d \mathcal{M}_d M d { τ n } n \{\tau_n\}_n { τ n } n { f n } n \{f_n\}_n { f n } n { τ n } n \{\tau_n\}_n { τ n } n M \mathcal{M} M A : M → M d A:\mathcal{M}\to\mathcal{M}_d A : M → M d a a a b b b g n : = f n A , ∀ n g_n:=f_nA,\ \forall n g n := f n A , ∀ n g n g_n g n n n n
a d ( x , y ) ≤ ∥ A x − A y ∥ M d = ∥ { f n ( A x − A y ) } n ∥ M d = ∥ { f n ( A x ) − f n ( A y ) } n ∥ M d = ∥ { g n ( x ) − g n ( y ) } n ∥ M d ≤ b d ( x , y ) , ∀ x , y ∈ M . ad(x,y)\leq\|Ax-Ay\|_{\mathcal{M}_d}=\|\{f_n(Ax-Ay)\}_n\|_{\mathcal{M}_d}
=\|\{f_n(Ax)-f_n(Ay)\}_n\|_{\mathcal{M}_d}
=\|\{g_n(x)-g_n(y)\}_n\|_{\mathcal{M}_d}\leq bd(x,y),\quad\forall x,y\in\mathcal{M}. a d ( x , y ) ≤ ∥ A x − A y ∥ M d = ∥ { f n ( A x − A y ) } n ∥ M d = ∥ { f n ( A x ) − f n ( A y ) } n ∥ M d = ∥ { g n ( x ) − g n ( y ) } n ∥ M d ≤ b d ( x , y ) , ∀ x , y ∈ M . Hence { g n } n \{g_n\}_n { g n } n M d \mathcal{M}_d M d M \mathcal{M} M
Following result ensures that metric frames are universal in nature.
Theorem 2.1.8. Every separable metric space M \mathcal{M} M M d \mathcal{M}_d M d
Proof. From Theorem 1.5.2 it follows that there exists a bi-Lipschitz function f : M → c 0 ( N ) f:\mathcal{M}\to c_0(\mathbb{N}) f : M → c 0 ( N ) p n : c 0 ( N ) → K p_n:c_0(\mathbb{N})\to\mathbb{K} p n : c 0 ( N ) → K n n n f n : = p n f f_n:=p_nf f n := p n f n n n { f n } n \{f_n\}_n { f n } n c 0 ( N ) c_0(\mathbb{N}) c 0 ( N ) M \mathcal{M} M
Given metric M d \mathcal{M}_d M d { f n } n \{f_n\}_n { f n } n { g n } n \{g_n\}_n { g n } n λ \lambda λ { f n + λ g n } n \{f_n+\lambda g_n\}_n { f n + λ g n } n { τ n } n \{\tau_n\}_n { τ n } n { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H { τ n + λ ω n } n \{\tau_n+\lambda\omega_n\}_n { τ n + λ ω n } n H \mathcal{H} H H \mathcal{H} H [97 ] . We now obtain similar results for metric spaces.
Theorem 2.1.9. Let { f n } n \{f_n\}_n { f n } n M d \mathcal{M}_d M d M \mathcal{M} M a a a b b b λ \lambda λ
(i) { λ f n } n \{\lambda f_n\}_n { λ f n } n M d \mathcal{M}_d M d M \mathcal{M} M a λ a\lambda aλ b λ b\lambda bλ
(ii) If A : M → M A:\mathcal{M}\to\mathcal{M} A : M → M c c c d d d { f n A } n \{f_nA\}_n { f n A } n M d \mathcal{M}_d M d M \mathcal{M} M a c ac a c b d bd b d
(iii) If { g n } n \{g_n\}_n { g n } n M d \mathcal{M}_d M d M \mathcal{M} M d d d ∣ λ ∣ < a d |\lambda|<\frac{a}{d} ∣ λ ∣ < d a { f n + λ g n } n \{f_n+\lambda g_n\}_n { f n + λ g n } n M d \mathcal{M}_d M d M \mathcal{M} M a − ∣ λ ∣ d a-|\lambda|d a − ∣ λ ∣ d b + ∣ λ ∣ d b+|\lambda|d b + ∣ λ ∣ d
Proof. First two conclusions follow easily. For the upper frame bound of { f n + λ g n } n \{f_n+\lambda g_n\}_n { f n + λ g n } n
∥ { ( f n + λ g n ) ( x ) − ( f n + λ g n ) ( y ) } n ∥ M d = ∥ { f n ( x ) − f n ( y ) + λ ( g n ( x ) − g n ( y ) ) } n ∥ M d ≥ ∥ { f n ( x ) − f n ( y ) } n ∥ M d − ∥ { λ ( g n ( x ) − g n ( y ) ) } n ∥ M d ≥ a d ( x , y ) − ∣ λ ∣ d ( x , y ) = ( a − ∣ λ ∣ ) d ( x , y ) , ∀ x , y ∈ M . \begin{aligned}
\|\{(f_n+\lambda g_n)(x)-(f_n+\lambda g_n)(y)\}_n\|_{\mathcal{M}_d}
=\|\{f_n(x)-f_n(y)+\lambda(g_n(x)-g_n(y))\}_n\|_{\mathcal{M}_d} \\
&\geq\|\{f_n(x)-f_n(y)\}_n\|_{\mathcal{M}_d}
-\|\{\lambda(g_n(x)-g_n(y))\}_n\|_{\mathcal{M}_d} \\
&\geq a d(x,y)-|\lambda|d(x,y)=(a-|\lambda|)d(x,y),\qquad\forall x,y\in\mathcal{M}.
\end{aligned} ∥ {( f n + λ g n ) ( x ) − ( f n + λ g n ) ( y ) } n ∥ M d = ∥ { f n ( x ) − f n ( y ) + λ ( g n ( x ) − g n ( y )) } n ∥ M d ≥ ∥ { f n ( x ) − f n ( y ) } n ∥ M d − ∥ { λ ( g n ( x ) − g n ( y )) } n ∥ M d ≥ a d ( x , y ) − ∣ λ ∣ d ( x , y ) = ( a − ∣ λ ∣ ) d ( x , y ) , ∀ x , y ∈ M . We next define “metric frame” which is stronger than Definition 2.1.7 in light of definition of Banach frame.
Definition 2.1.10. Let M \mathcal{M} M M d \mathcal{M}_d M d { f n } n \{f_n\}_n { f n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K ) S : M d → M S:\mathcal{M}_d\to\mathcal{M} S : M d → M ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) metric frame or Lipschitz metric frame for M \mathcal{M} M
(i) { f n ( x ) } n ∈ M d \{f_n(x)\}_n\in\mathcal{M}_d { f n ( x ) } n ∈ M d x ∈ M x\in\mathcal{M} x ∈ M
(ii) There exist positive a , b a,b a , b a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ b d ( x , y ) ad(x,y)\leq\|\{f_n(x)-f_n(y)\}_n\|_{\mathcal{M}_d}\leq bd(x,y) a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ b d ( x , y ) ∀ x , y ∈ M \forall x,y\in\mathcal{M} ∀ x , y ∈ M
(iii) S S S S ( { f n ( x ) } n ) = x S(\{f_n(x)\}_n)=x S ({ f n ( x ) } n ) = x x ∈ M x\in\mathcal{M} x ∈ M
Mapping S S S a a a lower frame bound and constant b b b upper frame bound . If we do not demand the first inequality, then we say ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) metric Bessel sequence .
We observe that if ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) M \mathcal{M} M
θ f : M ∋ x ↦ θ f x : = { f n ( x ) } n ∈ M d \theta_f:\mathcal{M}\ni x\mapsto\theta_f x:=\{f_n(x)\}_n\in\mathcal{M}_d θ f : M ∋ x ↦ θ f x := { f n ( x ) } n ∈ M d is well-defined and condition (ii) in Definition 2.1.10 tells that θ f \theta_f θ f
a d ( x , y ) ≤ ∥ θ f x − θ f y ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . ad(x,y)\leq\|\theta_f x-\theta_f y\|\leq bd(x,y),\qquad\forall x,y\in\mathcal{M}. a d ( x , y ) ≤ ∥ θ f x − θ f y ∥ ≤ b d ( x , y ) , ∀ x , y ∈ M . Hence θ f \theta_f θ f M d \mathcal{M}_d M d θ f \theta_f θ f S S S S θ f = I M S\theta_f=I_{\mathcal{M}} S θ f = I M
Proposition 2.1.11. If ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) M \mathcal{M} M P f , S : = θ f S : M d → M d P_{f,S}:=\theta_f S:\mathcal{M}_d\to\mathcal{M}_d P f , S := θ f S : M d → M d P f , S ( M d ) = θ f ( M d ) P_{f,S}(\mathcal{M}_d)=\theta_f(\mathcal{M}_d) P f , S ( M d ) = θ f ( M d )
Proposition 2.1.12. Let { f n } n \{f_n\}_n { f n } n M d \mathcal{M}_d M d M \mathcal{M} M S : M d → M S:\mathcal{M}_d\to\mathcal{M} S : M d → M ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) M \mathcal{M} M S S S θ f \theta_f θ f θ f \theta_f θ f S S S
We now give some explicit examples of metric frames.
Example 2.1.13. Let M \mathcal{M} M { f n } n \{f_n\}_n { f n } n a = 1 a=1 a = 1 M d : = ℓ 1 ( { 0 } ∪ N ) \mathcal{M}_d:=\ell^1(\{0\}\cup\mathbb{N}) M d := ℓ 1 ({ 0 } ∪ N )
S : M d ∋ { a n } n ↦ S { a n } n : = 1 + ∣ ∑ n = 1 ∞ a n ∣ ∈ M . S:\mathcal{M}_d\ni\{a_n\}_n\mapsto S\{a_n\}_n:=1+\left|\sum_{n=1}^{\infty}a_n\right|\in\mathcal{M}. S : M d ∋ { a n } n ↦ S { a n } n := 1 + n = 1 ∑ ∞ a n ∈ M . Then
∣ S { a n } n − S { b n } n ∣ = ∣ ∣ ∑ n = 1 ∞ a n ∣ − ∣ ∑ n = 1 ∞ b n ∣ ∣ ≤ ∣ ∑ n = 1 ∞ a n − ∑ n = 1 ∞ b n ∣ \left|S\{a_n\}_n-S\{b_n\}_n\right|=\left|\left|\sum_{n=1}^{\infty}a_n\right|-\left|\sum_{n=1}^{\infty}b_n\right|\right|\leq\left|\sum_{n=1}^{\infty}a_n-\sum_{n=1}^{\infty}b_n\right| ∣ S { a n } n − S { b n } n ∣ = n = 1 ∑ ∞ a n − n = 1 ∑ ∞ b n ≤ n = 1 ∑ ∞ a n − n = 1 ∑ ∞ b n = ∣ ∑ n = 1 ∞ ( a n − b n ) ∣ ≤ ∑ n = 1 ∞ ∣ a n − b n ∣ ≤ ∑ n = 0 ∞ ∣ a n − b n ∣ =\left|\sum_{n=1}^{\infty}(a_n-b_n)\right|\leq\sum_{n=1}^{\infty}|a_n-b_n|\leq\sum_{n=0}^{\infty}|a_n-b_n| = n = 1 ∑ ∞ ( a n − b n ) ≤ n = 1 ∑ ∞ ∣ a n − b n ∣ ≤ n = 0 ∑ ∞ ∣ a n − b n ∣ = ∥ { a n } n − { b n } n ∥ , ∀ { a n } n , { b n } n ∈ ℓ 1 ( { 0 } ∪ N ) . =\left\|\{a_n\}_n-\{b_n\}_n\right\|,\quad\forall\{a_n\}_n,\{b_n\}_n\in\ell^1(\{0\}\cup\mathbb{N}). = ∥ { a n } n − { b n } n ∥ , ∀ { a n } n , { b n } n ∈ ℓ 1 ({ 0 } ∪ N ) . Thus S S S
S ( { f n ( x ) } n ) = 1 + ∣ ∑ n = 1 ∞ f n ( x ) ∣ = 1 + ∑ n = 1 ∞ ( log x ) n n ! = x , ∀ x ∈ M . S(\{f_n(x)\}_n)=1+\left|\sum_{n=1}^{\infty}f_n(x)\right|=1+\sum_{n=1}^{\infty}\frac{(\log x)^n}{n!}=x,\quad\forall x\in\mathcal{M}. S ({ f n ( x ) } n ) = 1 + n = 1 ∑ ∞ f n ( x ) = 1 + n = 1 ∑ ∞ n ! ( log x ) n = x , ∀ x ∈ M . Hence ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) M \mathcal{M} M
T : M d ∋ { a n } n ↦ S { a n } n : = 1 + ∑ n = 1 ∞ ∣ a n ∣ ∈ M , T:\mathcal{M}_d\ni\{a_n\}_n\mapsto S\{a_n\}_n:=1+\sum_{n=1}^{\infty}|a_n|\in\mathcal{M}, T : M d ∋ { a n } n ↦ S { a n } n := 1 + n = 1 ∑ ∞ ∣ a n ∣ ∈ M , then
∣ T { a n } n − T { b n } n ∣ = ∣ ∑ n = 1 ∞ ∣ a n ∣ − ∑ n = 1 ∞ ∣ b n ∣ ∣ = ∣ ∑ n = 1 ∞ ( ∣ a n ∣ − ∣ b n ∣ ) ∣ \left|T\{a_n\}_n-T\{b_n\}_n\right|=\left|\sum_{n=1}^{\infty}|a_n|-\sum_{n=1}^{\infty}|b_n|\right|=\left|\sum_{n=1}^{\infty}(|a_n|-|b_n|)\right| ∣ T { a n } n − T { b n } n ∣ = n = 1 ∑ ∞ ∣ a n ∣ − n = 1 ∑ ∞ ∣ b n ∣ = n = 1 ∑ ∞ ( ∣ a n ∣ − ∣ b n ∣ ) ≤ ∑ n = 1 ∞ ∣ ∣ a n ∣ − ∣ b n ∣ ∣ ≤ ∑ n = 1 ∞ ∣ a n − b n ∣ ≤ ∑ n = 0 ∞ ∣ a n − b n ∣ \leq\sum_{n=1}^{\infty}\left||a_n|-|b_n|\right|\leq\sum_{n=1}^{\infty}|a_n-b_n|\leq\sum_{n=0}^{\infty}|a_n-b_n| ≤ n = 1 ∑ ∞ ∣ ∣ a n ∣ − ∣ b n ∣ ∣ ≤ n = 1 ∑ ∞ ∣ a n − b n ∣ ≤ n = 0 ∑ ∞ ∣ a n − b n ∣ = ∥ { a n } n − { b n } n ∥ , ∀ { a n } n , { b n } n ∈ ℓ 1 ( { 0 } ∪ N ) . =\left\|\{a_n\}_n-\{b_n\}_n\right\|,\quad\forall\{a_n\}_n,\{b_n\}_n\in\ell^1(\{0\}\cup\mathbb{N}). = ∥ { a n } n − { b n } n ∥ , ∀ { a n } n , { b n } n ∈ ℓ 1 ({ 0 } ∪ N ) . Thus T T T ( { f n } n , T ) (\{f_n\}_n,T) ({ f n } n , T ) M \mathcal{M} M
Example 2.1.14. Let f 1 : K → K f_1:\mathbb{K}\to\mathbb{K} f 1 : K → K f 2 , … , f m : K → K f_2,\ldots,f_m:\mathbb{K}\to\mathbb{K} f 2 , … , f m : K → K m m m
f 1 ( x ) + ⋯ + f m ( x ) = x , ∀ x ∈ K . f_1(x)+\cdots+f_m(x)=x,\quad\forall x\in\mathbb{K}. f 1 ( x ) + ⋯ + f m ( x ) = x , ∀ x ∈ K . We now define S : K m ∋ ( x 1 , … , x m ) ↦ ∑ j = 1 m x j ∈ K S : \mathbb{K}^{m} \ni(x_{1},\ldots,x_{m}) \mapsto\sum_{j=1}^{m} x_{j} \in\mathbb{K} S : K m ∋ ( x 1 , … , x m ) ↦ ∑ j = 1 m x j ∈ K ( { f n } n , S ) \left(\{f_{n}\}_{n},S\right) ( { f n } n , S ) K \mathbb{K} K S S S
After the definition of metric frame, the first question which comes is the existence. In Theorem 1.3.13 it was proved that every separable Banach space admits a Banach frame. Even though this result is not known in metric space settings, two results are derived one is close to the definition of metric frame and another gives existence under certain assumptions. To do this we prove a result which we derive from Mc-Shane extension theorem.
Theorem 2.1.15. (Mc-Shane extension theorem) (cf. Weaver (2018)) Let M \mathcal{M} M M 0 \mathcal{M}_{0} M 0 M \mathcal{M} M f 0 : M 0 → R f_{0} : \mathcal{M}_{0} \to\mathbb{R} f 0 : M 0 → R f : M → R f : \mathcal{M} \to\mathbb{R} f : M → R f ∣ M 0 = f 0 f|_{\mathcal{M}_{0}} = f_{0} f ∣ M 0 = f 0 Lip ( f ) = Lip ( f 0 ) \operatorname{Lip}(f) = \operatorname{Lip}(f_{0}) Lip ( f ) = Lip ( f 0 )
Using Theorem 2.1.15 we derive the following.
Corollary 2.1.16. If ( M , 0 ) (\mathcal{M},0) ( M , 0 ) x ∈ M x \in\mathcal{M} x ∈ M f : M → R f : \mathcal{M} \to\mathbb{R} f : M → R f ( x ) = d ( x , 0 ) f(x) = d(x,0) f ( x ) = d ( x , 0 ) f ( 0 ) = 0 f(0) = 0 f ( 0 ) = 0 Lip ( f ) = 1 \operatorname{Lip}(f) = 1 Lip ( f ) = 1
Proof. Case (i) : x ≠ 0 x \ne0 x = 0 M 0 : = { 0 , x } \mathcal{M}_{0} := \{0,x\} M 0 := { 0 , x } f 0 ( 0 ) = 0 f_{0}(0) = 0 f 0 ( 0 ) = 0 f 0 ( x ) = d ( x , 0 ) f_{0}(x) = d(x,0) f 0 ( x ) = d ( x , 0 ) ∣ f 0 ( x ) − f 0 ( 0 ) ∣ = d ( x , 0 ) \lvert f_{0}(x) - f_{0}(0)\rvert= d(x,0) ∣ f 0 ( x ) − f 0 ( 0 )∣ = d ( x , 0 ) f 0 f_{0} f 0
Case (ii) : x = 0 x = 0 x = 0 y ∈ M y \in\mathcal{M} y ∈ M y y y x x x
Theorem 2.1.17. Let M \mathcal{M} M M d \mathcal{M}_{d} M d { f n } n \{f_{n}\}_{n} { f n } n Lip 0 ( M , R ) \operatorname{Lip}_{0}(\mathcal{M},\mathbb{R}) Lip 0 ( M , R ) S : M d → M S : \mathcal{M}_{d} \to\mathcal{M} S : M d → M
(i) { f n ( x ) } n ∈ M d \{f_{n}(x)\}_{n} \in\mathcal{M}_{d} { f n ( x ) } n ∈ M d x ∈ M x \in\mathcal{M} x ∈ M
(ii) ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ d ( x , y ) , ∀ x , y ∈ M \left\|\{f_{n}(x)-f_{n}(y)\}_{n}\right\|_{\mathcal{M}_{d}} \le d(x,y), \forall x,y \in\mathcal{M} ∥ { f n ( x ) − f n ( y ) } n ∥ M d ≤ d ( x , y ) , ∀ x , y ∈ M
(iii) S ( { f n ( x ) } n ) = x S(\{f_{n}(x)\}_{n}) = x S ({ f n ( x ) } n ) = x x ∈ M x \in\mathcal{M} x ∈ M
Proof. Let { x n } n \{x_{n}\}_{n} { x n } n M \mathcal{M} M n ∈ N n \in\mathbb{N} n ∈ N f n : M → R f_{n} : \mathcal{M} \to\mathbb{R} f n : M → R f n ( x n ) = d ( x n , 0 ) f_{n}(x_{n}) = d(x_{n},0) f n ( x n ) = d ( x n , 0 ) f n ( 0 ) = 0 f_{n}(0) = 0 f n ( 0 ) = 0 Lip ( f n ) = 1 \operatorname{Lip}(f_{n}) = 1 Lip ( f n ) = 1 x ∈ M x \in\mathcal{M} x ∈ M n ∈ N n \in\mathbb{N} n ∈ N
∣ f n ( x ) ∣ = ∣ f n ( x ) − f n ( 0 ) ∣ ≤ ∥ f n ∥ Lip 0 d ( x , 0 ) = d ( x , 0 ) \lvert f_{n}(x)\rvert= \lvert f_{n}(x)-f_{n}(0)\rvert\le\|f_{n}\|_{\operatorname{Lip}_{0}}d(x,0) = d(x,0) ∣ f n ( x )∣ = ∣ f n ( x ) − f n ( 0 )∣ ≤ ∥ f n ∥ Lip 0 d ( x , 0 ) = d ( x , 0 ) which gives sup n ∈ N ∣ f n ( x ) ∣ ≤ d ( x , 0 ) \sup_{n\in\mathbb{N}}\lvert f_{n}(x)\rvert\le d(x,0) sup n ∈ N ∣ f n ( x )∣ ≤ d ( x , 0 ) { x n } n \{x_{n}\}_{n} { x n } n { x n k } k \{x_{n_{k}}\}_{k} { x n k } k { x n } n \{x_{n}\}_{n} { x n } n x n k → x x_{n_{k}} \to x x n k → x n → ∞ n \to\infty n → ∞
∣ d ( y , z ) − d ( y , w ) ∣ ≤ d ( z , w ) , ∀ y , z , w ∈ M \lvert d(y,z)-d(y,w)\rvert\le d(z,w), \quad\forall y,z,w \in\mathcal{M} ∣ d ( y , z ) − d ( y , w )∣ ≤ d ( z , w ) , ∀ y , z , w ∈ M we see then that d ( x n k , 0 ) → d ( x , 0 ) d(x_{n_k},0) \to d(x,0) d ( x n k , 0 ) → d ( x , 0 ) n → ∞ n \to\infty n → ∞
d ( x n k , 0 ) = f n k ( x n k ) ≤ ∣ f n k ( x n k ) − f n k ( x ) ∣ + ∣ f n k ( x ) ∣ ≤ 1 ⋅ d ( x n k , x ) + ∣ f n k ( x ) ∣ , ∀ k ∈ N ⟹ lim k → ∞ ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ sup k ∈ N ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ sup k ∈ N ∣ f n k ( x ) ∣ . \begin{aligned}
d(x_{n_k},0) &= f_{n_k}(x_{n_k}) \leq\lvert f_{n_k}(x_{n_k})-f_{n_k}(x)\rvert+\lvert f_{n_k}(x)\rvert\\
&\leq1\cdot d(x_{n_k},x)+\lvert f_{n_k}(x)\rvert,\qquad\forall k\in\mathbb{N}\\
&\Longrightarrow\lim_{k\to\infty}\bigl(d(x_{n_k},0)-d(x_{n_k},x)\bigr)\leq\sup_{k\in\mathbb{N}}\bigl(d(x_{n_k},0)-d(x_{n_k},x)\bigr)\leq\sup_{k\in\mathbb{N}}\lvert f_{n_k}(x)\rvert.
\end{aligned} d ( x n k , 0 ) = f n k ( x n k ) ≤ ∣ f n k ( x n k ) − f n k ( x )∣ + ∣ f n k ( x )∣ ≤ 1 ⋅ d ( x n k , x ) + ∣ f n k ( x )∣ , ∀ k ∈ N ⟹ k → ∞ lim ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ k ∈ N sup ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ k ∈ N sup ∣ f n k ( x )∣ . Therefore
sup n ∈ N ∣ f n ( x ) ∣ ≤ d ( x , 0 ) = lim k → ∞ d ( x n k , 0 ) = lim k → ∞ ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ sup k ∈ N ∣ f n k ( x ) ∣ ≤ sup n ∈ N ∣ f n ( x ) ∣ . \begin{aligned}
\sup_{n\in\mathbb{N}}\lvert f_n(x)\rvert\leq d(x,0)=\lim_{k\to\infty}d(x_{n_k},0)=\lim_{k\to\infty}\bigl(d(x_{n_k},0)-d(x_{n_k},x)\bigr) \\
&\leq\sup_{k\in\mathbb{N}}\lvert f_{n_k}(x)\rvert\leq\sup_{n\in\mathbb{N}}\lvert f_n(x)\rvert.
\end{aligned} n ∈ N sup ∣ f n ( x )∣ ≤ d ( x , 0 ) = k → ∞ lim d ( x n k , 0 ) = k → ∞ lim ( d ( x n k , 0 ) − d ( x n k , x ) ) ≤ k ∈ N sup ∣ f n k ( x )∣ ≤ n ∈ N sup ∣ f n ( x )∣ . So we proved that
d ( x , 0 ) = sup n ∈ N ∣ f n ( x ) ∣ , ∀ x ∈ M . (17) d(x,0)=\sup_{n\in\mathbb{N}}\lvert f_n(x)\rvert,\qquad\forall x\in\mathcal{M}. \tag*{(17)} d ( x , 0 ) = n ∈ N sup ∣ f n ( x )∣ , ∀ x ∈ M . (17) Define M d 0 : = { { f n ( x ) } n : x ∈ M } \mathcal{M}_d^0:=\{\{f_n(x)\}_n:x\in\mathcal{M}\} M d 0 := {{ f n ( x ) } n : x ∈ M } M d 0 \mathcal{M}_d^0 M d 0 ℓ ∞ ( N ) \ell^\infty(\mathbb{N}) ℓ ∞ ( N ) S 0 : M d 0 ∋ { f n ( x ) } n ↦ x ∈ M S_0:\mathcal{M}_d^0\ni\{f_n(x)\}_n\mapsto x\in\mathcal{M} S 0 : M d 0 ∋ { f n ( x ) } n ↦ x ∈ M
d ( S 0 ( { f n ( x ) } n ) , S 0 ( { f n ( y ) } n ) ) = d ( x , y ) ≤ d ( x , 0 ) + d ( 0 , y ) = sup n ∈ N ∣ f n ( x ) ∣ + sup n ∈ N ∣ f n ( y ) ∣ = ∥ { f n ( x ) } n ∥ + ∥ { f n ( y ) } n ∥ , ∀ x , y ∈ M . \begin{aligned}
d\bigl(S_0(\{f_n(x)\}_n),S_0(\{f_n(y)\}_n)\bigr)&=d(x,y)\leq d(x,0)+d(0,y)\\
&=\sup_{n\in\mathbb{N}}\lvert f_n(x)\rvert+\sup_{n\in\mathbb{N}}\lvert f_n(y)\rvert\\
&=\|\{f_n(x)\}_n\|+\|\{f_n(y)\}_n\|,\qquad\forall x,y\in\mathcal{M}.
\end{aligned} d ( S 0 ({ f n ( x ) } n ) , S 0 ({ f n ( y ) } n ) ) = d ( x , y ) ≤ d ( x , 0 ) + d ( 0 , y ) = n ∈ N sup ∣ f n ( x )∣ + n ∈ N sup ∣ f n ( y )∣ = ∥ { f n ( x ) } n ∥ + ∥ { f n ( y ) } n ∥ , ∀ x , y ∈ M . We also have
∥ { f n ( x ) − f n ( y ) } n ∥ M d = sup n ∈ N ∣ f n ( x ) − f n ( y ) ∣ ≤ sup n ∈ N ∥ f n ∥ Lip 0 d ( x , y ) = d ( x , y ) , ∀ x , y ∈ M . \begin{aligned}
\|\{f_n(x)-f_n(y)\}_n\|_{\mathcal{M}_d}&=\sup_{n\in\mathbb{N}}\lvert f_n(x)-f_n(y)\rvert\\
&\leq\sup_{n\in\mathbb{N}}\|f_n\|_{\operatorname{Lip}_0}d(x,y)=d(x,y),\qquad\forall x,y\in\mathcal{M}.
\end{aligned} ∥ { f n ( x ) − f n ( y ) } n ∥ M d = n ∈ N sup ∣ f n ( x ) − f n ( y )∣ ≤ n ∈ N sup ∥ f n ∥ Lip 0 d ( x , y ) = d ( x , y ) , ∀ x , y ∈ M . We can now take S S S S 0 S_0 S 0 ℓ ∞ ( N ) \ell^\infty(\mathbb{N}) ℓ ∞ ( N ) M d = ℓ ∞ ( N ) \mathcal{M}_d=\ell^\infty(\mathbb{N}) M d = ℓ ∞ ( N )
Theorem 2.1.18. If A : M → M d A:\mathcal{M}\to\mathcal{M}_d A : M → M d P : M d → A ( M ) P:\mathcal{M}_d\to A(\mathcal{M}) P : M d → A ( M ) M \mathcal{M} M
Proof. Let { h n } n \{h_n\}_n { h n } n M d \mathcal{M}_d M d f n : = h n A f_n:=h_nA f n := h n A S : = A − 1 P S:=A^{-1}P S := A − 1 P
S ( { f n ( x ) } n ) = A − 1 P ( { h n ( A x ) } n ) = A − 1 P A x = A − 1 A x = x , ∀ x ∈ M . S(\{f_n(x)\}_n)=A^{-1}P(\{h_n(Ax)\}_n)=A^{-1}PAx=A^{-1}Ax=x,\qquad\forall x\in\mathcal{M}. S ({ f n ( x ) } n ) = A − 1 P ({ h n ( A x ) } n ) = A − 1 P A x = A − 1 A x = x , ∀ x ∈ M . Hence ( { f n } n , S ) \left(\{f_n\}_{n},S\right) ( { f n } n , S ) M \mathcal{M} M
It is well-known that Mc-Shane extension theorem fails for complex valued Lipschitz functions. Thus we may ask whether we can take a complex sequence space in Theorem 2.1.17. It is possible for certain metric spaces due to the following theorem.
Theorem 2.1.19. (Kirszbraun extension theorem ) ((cf. Valentine (1945))) Let H \mathcal{H} H M 0 \mathcal{M}_0 M 0 H \mathcal{H} H f 0 : M 0 → K f_0:\mathcal{M}_0\to\mathbb{K} f 0 : M 0 → K f : H → K f:\mathcal{H}\to\mathbb{K} f : H → K f ∣ M 0 = f 0 f|_{\mathcal{M}_0}=f_0 f ∣ M 0 = f 0 Lip ( f ) = Lip ( f 0 ) \operatorname{Lip}(f)=\operatorname{Lip}(f_0) Lip ( f ) = Lip ( f 0 )
Following proposition shows that given a metric frame, we can generate more metric frames.
Proposition 2.1.20. Let ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) M \mathcal{M} M A , B : M → M A,B:\mathcal{M}\to\mathcal{M} A , B : M → M A A A B B B B A = I M BA=I_{\mathcal{M}} B A = I M ( { f n A } n , B S ) \left(\{f_nA\}_n,BS\right) ( { f n A } n , B S ) M \mathcal{M} M A : M → M A:\mathcal{M}\to\mathcal{M} A : M → M ( { f n A } n , A − 1 S ) \left(\{f_nA\}_n,A^{-1}S\right) ( { f n A } n , A − 1 S ) M \mathcal{M} M
Proof. Bi-Lipschitzness of A A A B A = I M BA=I_{\mathcal{M}} B A = I M B S ( { f n A x } n ) = B A x = x , ∀ x ∈ M BS\left(\{f_nAx\}_n\right)=BAx=x,\forall x\in\mathcal{M} B S ( { f n A x } n ) = B A x = x , ∀ x ∈ M
Previous proposition not only helps to generate metric frames from metric frames but also from Banach frames. Since there are large number of examples of Banach frames for a variety of Banach spaces, just by operating with bi-Lipschitz invertible functions on subsets, it produces metric frames for that subset. Next we characterize metric frames using Lipschitz functions. The following theorem precisely says when an M d \mathcal{M}_d M d
Theorem 2.1.21. Let { f n } n \{f_n\}_n { f n } n M d \mathcal{M}_d M d M \mathcal{M} M
(i) There exists a Lipschitz projection P : M d → θ f ( M ) P:\mathcal{M}_d\to\theta_f(\mathcal{M}) P : M d → θ f ( M )
(ii) There exists a Lipschitz map V : M d → M V:\mathcal{M}_d\to\mathcal{M} V : M d → M V ∣ θ f ( M ) = θ f − 1 V|_{\theta_f(\mathcal{M})}=\theta_f^{-1} V ∣ θ f ( M ) = θ f − 1
(iii) There exists a Lipschitz map S : M d → M S:\mathcal{M}_d\to\mathcal{M} S : M d → M ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) M \mathcal{M} M
Proof. (i) ⇒ \Rightarrow ⇒ V : = θ f − 1 P V:=\theta_f^{-1}P V := θ f − 1 P y = θ f ( x ) , x ∈ M y=\theta_f(x),x\in\mathcal{M} y = θ f ( x ) , x ∈ M V y = V θ f ( x ) = θ f − 1 P θ f ( x ) = θ f − 1 θ f ( x ) = θ f − 1 y Vy=V\theta_f(x)=\theta_f^{-1}P\theta_f(x)=\theta_f^{-1}\theta_f(x)=\theta_f^{-1}y V y = V θ f ( x ) = θ f − 1 P θ f ( x ) = θ f − 1 θ f ( x ) = θ f − 1 y
(ii) ⇒ \Rightarrow ⇒ P : = θ f V P:=\theta_fV P := θ f V P 2 = θ f V θ f V = θ f I M V = P P^2=\theta_fV\theta_fV=\theta_fI_{\mathcal{M}}V=P P 2 = θ f V θ f V = θ f I M V = P
(ii) ⇒ \Rightarrow ⇒ S : = V S:=V S := V S ( { f n ( x ) } n ) = S θ f ( x ) = V θ f ( x ) = x S\left(\{f_n(x)\}_n\right)=S\theta_f(x)=V\theta_f(x)=x S ( { f n ( x ) } n ) = S θ f ( x ) = V θ f ( x ) = x x ∈ M x\in\mathcal{M} x ∈ M ( { f n } n , S ) \left(\{f_n\}_n,S\right) ( { f n } n , S ) M \mathcal{M} M
(iii) ⇒ \Rightarrow ⇒ V : = S V:=S V := S V θ f ( x ) = S θ f ( x ) = S ( { f n ( x ) } n ) = x V\theta_f(x)=S\theta_f(x)=S\left(\{f_n(x)\}_n\right)=x V θ f ( x ) = S θ f ( x ) = S ( { f n ( x ) } n ) = x x ∈ M x\in\mathcal{M} x ∈ M
METRIC FRAMES FOR BANACH SPACES
Now we turn onto the representation of elements using metric frames. Naturally, to deal with sums we must look in to Banach space structure. Following theorem can be compared with Theorem 1.3.20.
Theorem 2.2.1. Let { f n } n \{f_n\}_n { f n } n p p p X \mathcal{X} X f n ( 0 ) = 0 f_n(0)=0 f n ( 0 ) = 0 n n n
(i) There exists a bounded linear map V : M d → X V:\mathcal{M}_d\to\mathcal{X} V : M d → X V ∣ θ f ( M ) = θ f − 1 V|_{\theta_f(\mathcal{M})}=\theta_f^{-1} V ∣ θ f ( M ) = θ f − 1
(ii) There exists a bounded linear map S : M d → X S:\mathcal{M}_d\to\mathcal{X} S : M d → X ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) p p p X \mathcal{X} X
(iii) There exists a sequence { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X ∑ n = 1 ∞ c n τ n \sum_{n=1}^{\infty}c_n\tau_n ∑ n = 1 ∞ c n τ n { c n } n ∈ ℓ p ( N ) \{c_n\}_n\in\ell^p(\mathbb{N}) { c n } n ∈ ℓ p ( N ) x = ∑ n = 1 ∞ f n ( x ) τ n x=\sum_{n=1}^{\infty}f_n(x)\tau_n x = ∑ n = 1 ∞ f n ( x ) τ n ∀ x ∈ X \forall x\in\mathcal{X} ∀ x ∈ X
(iv) There exists a q q q { τ n } n \{\tau_n\}_n { τ n } n X ⊆ X ∗ ∗ \mathcal{X}\subseteq\mathcal{X}^{**} X ⊆ X ∗∗ x = ∑ n = 1 ∞ f n ( x ) τ n x=\sum_{n=1}^{\infty}f_n(x)\tau_n x = ∑ n = 1 ∞ f n ( x ) τ n ∀ x ∈ X \forall x\in\mathcal{X} ∀ x ∈ X
(v) There exists a q q q { τ n } n \{\tau_n\}_n { τ n } n X ⊆ X ∗ ∗ \mathcal{X}\subseteq\mathcal{X}^{**} X ⊆ X ∗∗ f = ∑ n = 1 ∞ f ( τ n ) f n f=\sum_{n=1}^{\infty}f(\tau_n)f_n f = ∑ n = 1 ∞ f ( τ n ) f n ∀ f ∈ X ∗ \forall f\in\mathcal{X}^* ∀ f ∈ X ∗
In each of the cases (iv) and (v), { τ n } n \{\tau_n\}_n { τ n } n q q q X ∗ \mathcal{X}^* X ∗
Proof. Proof of (i) ⟺ \Longleftrightarrow ⟺ ⟺ \Longleftrightarrow ⟺ Theorem 2.1.21 .
(iii) ⇒ \Rightarrow ⇒
V : ℓ p ( N ) ∋ { c n } n ↦ ∑ n = 1 ∞ c n τ n ∈ X V:\ell^p(\mathbb{N})\ni\{c_n\}_n\mapsto\sum_{n=1}^{\infty}c_n\tau_n\in\mathcal{X} V : ℓ p ( N ) ∋ { c n } n ↦ n = 1 ∑ ∞ c n τ n ∈ X is well-defined. Banach-Steinhaus theorem now asserts that V V V y = θ f ( x ) y=\theta_f(x) y = θ f ( x ) x ∈ X x\in\mathcal{X} x ∈ X
V y = V θ f ( x ) = V ( { f n ( x ) } n ) = ∑ n = 1 ∞ f n ( x ) τ n = x = θ f − 1 θ f ( x ) = θ f − 1 y . Vy=V\theta_f(x)=V(\{f_n(x)\}_n)=\sum_{n=1}^{\infty}f_n(x)\tau_n=x=\theta_f^{-1}\theta_f(x)=\theta_f^{-1}y. V y = V θ f ( x ) = V ({ f n ( x ) } n ) = n = 1 ∑ ∞ f n ( x ) τ n = x = θ f − 1 θ f ( x ) = θ f − 1 y . (i) ⇒ \Rightarrow ⇒ { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) τ n : = V e n \tau_n:=Ve_n τ n := V e n n n n V V V ∑ n = 1 ∞ c n e n \sum_{n=1}^{\infty}c_ne_n ∑ n = 1 ∞ c n e n { c n } n ∈ ℓ p ( N ) \{c_n\}_n\in\ell^p(\mathbb{N}) { c n } n ∈ ℓ p ( N ) ∑ n = 1 ∞ c n τ n \sum_{n=1}^{\infty}c_n\tau_n ∑ n = 1 ∞ c n τ n { c n } n ∈ ℓ p ( N ) \{c_n\}_n\in\ell^p(\mathbb{N}) { c n } n ∈ ℓ p ( N )
x = V θ f ( x ) = V ( { f n ( x ) } n ) = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X . x=V\theta_f(x)=V(\{f_n(x)\}_n)=\sum_{n=1}^{\infty}f_n(x)\tau_n,\qquad\forall x\in\mathcal{X}. x = V θ f ( x ) = V ({ f n ( x ) } n ) = n = 1 ∑ ∞ f n ( x ) τ n , ∀ x ∈ X . (iii) ⟺ \Longleftrightarrow ⟺ τ n \tau_n τ n X ∗ ∗ \mathcal{X}^{**} X ∗∗ { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X ∑ n = 1 ∞ c n τ n \sum_{n=1}^{\infty} c_n\tau_n ∑ n = 1 ∞ c n τ n { c n } n ∈ ℓ p ( N ) \{c_n\}_n\in\ell^p(\mathbb{N}) { c n } n ∈ ℓ p ( N )
(iv) ⟹ \Longrightarrow ⟹ b b b { τ n } n \{\tau_n\}_n { τ n } n f ∈ X ∗ f\in\mathcal{X}^* f ∈ X ∗ n ∈ N n\in\mathbb{N} n ∈ N
∥ f − ∑ k = 1 n f ( τ k ) f k ∥ Lip 0 = sup x , y ∈ X x ≠ y ∣ ( f − ∑ k = 1 n f ( τ k ) f k ) ( x ) − ( f − ∑ k = 1 n f ( τ k ) f k ) ( y ) ∣ ∥ x − y ∥ = sup x , y ∈ X x ≠ y ∣ f ( ∑ k = 1 ∞ f k ( x ) τ k ) − f ( ∑ k = 1 ∞ f k ( y ) τ k ) − ∑ k = 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y ) ) ∣ ∥ x − y ∥ = sup x , y ∈ X x ≠ y ∣ ∑ k = 1 n f ( τ k ) ( f k ( x ) − f k ( y ) ) − ∑ k = 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y ) ) ∣ ∥ x − y ∥ = sup x , y ∈ X x ≠ y ∣ ∑ k = n + 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y ) ) ∣ ∥ x − y ∥ ≤ sup x , y ∈ X x ≠ y ( ∑ k = n + 1 ∞ ∣ f ( τ k ) ∣ q ) 1 / q ( ∑ k = n + 1 ∞ ∣ f k ( x ) − f k ( y ) ∣ p ) 1 / p ∥ x − y ∥ ≤ b ( ∑ k = n + 1 ∞ ∣ f k ( x ) − f k ( y ) ∣ p ) 1 / p ⟶ 0 as n ⟶ ∞ . \begin{aligned}
\left\|f-\sum_{k=1}^{n}f(\tau_k)f_k\right\|_{\operatorname{Lip}_0}
=\sup_{\substack{x,y\in\mathcal{X}\\x\ne y}}\frac{\left|(f-\sum_{k=1}^{n}f(\tau_k)f_k)(x)-(f-\sum_{k=1}^{n}f(\tau_k)f_k)(y)\right|}{\|x-y\|} \\
&=\sup_{\substack{x,y\in\mathcal{X}\\x\ne y}}\frac{\left|f(\sum_{k=1}^{\infty}f_k(x)\tau_k)-f(\sum_{k=1}^{\infty}f_k(y)\tau_k)-\sum_{k=1}^{\infty}f(\tau_k)(f_k(x)-f_k(y))\right|}{\|x-y\|} \\
&=\sup_{\substack{x,y\in\mathcal{X}\\x\ne y}}\frac{\left|\sum_{k=1}^{n}f(\tau_k)(f_k(x)-f_k(y))-\sum_{k=1}^{\infty}f(\tau_k)(f_k(x)-f_k(y))\right|}{\|x-y\|} \\
&=\sup_{\substack{x,y\in\mathcal{X}\\x\ne y}}\frac{\left|\sum_{k=n+1}^{\infty}f(\tau_k)(f_k(x)-f_k(y))\right|}{\|x-y\|} \\
&\le\sup_{\substack{x,y\in\mathcal{X}\\x\ne y}}\frac{\left(\sum_{k=n+1}^{\infty}|f(\tau_k)|^q\right)^{1/q}\left(\sum_{k=n+1}^{\infty}|f_k(x)-f_k(y)|^p\right)^{1/p}}{\|x-y\|} \\
&\le b\left(\sum_{k=n+1}^{\infty}|f_k(x)-f_k(y)|^p\right)^{1/p}\longrightarrow0\text{ as }n\longrightarrow\infty.
\end{aligned} f − k = 1 ∑ n f ( τ k ) f k Lip 0 = x , y ∈ X x = y sup ∥ x − y ∥ ∣ ( f − ∑ k = 1 n f ( τ k ) f k ) ( x ) − ( f − ∑ k = 1 n f ( τ k ) f k ) ( y ) ∣ = x , y ∈ X x = y sup ∥ x − y ∥ ∣ f ( ∑ k = 1 ∞ f k ( x ) τ k ) − f ( ∑ k = 1 ∞ f k ( y ) τ k ) − ∑ k = 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y )) ∣ = x , y ∈ X x = y sup ∥ x − y ∥ ∣ ∑ k = 1 n f ( τ k ) ( f k ( x ) − f k ( y )) − ∑ k = 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y )) ∣ = x , y ∈ X x = y sup ∥ x − y ∥ ∑ k = n + 1 ∞ f ( τ k ) ( f k ( x ) − f k ( y )) ≤ x , y ∈ X x = y sup ∥ x − y ∥ ( ∑ k = n + 1 ∞ ∣ f ( τ k ) ∣ q ) 1/ q ( ∑ k = n + 1 ∞ ∣ f k ( x ) − f k ( y ) ∣ p ) 1/ p ≤ b ( k = n + 1 ∑ ∞ ∣ f k ( x ) − f k ( y ) ∣ p ) 1/ p ⟶ 0 as n ⟶ ∞. (v) ⟹ \Longrightarrow ⟹ b b b { τ n } n \{\tau_n\}_n { τ n } n x ∈ X x\in\mathcal{X} x ∈ X n ∈ N n\in\mathbb{N} n ∈ N
∥ x − ∑ k = 1 n f k ( x ) τ k ∥ = sup f ∈ X ∗ , ∥ f ∥ = 1 ∣ f ( x ) − ∑ k = 1 n f k ( x ) f ( τ k ) ∣ = sup f ∈ X ∗ , ∥ f ∥ = 1 ∣ ( ∑ k = 1 ∞ f ( τ k ) f k ) ( x ) − ∑ k = 1 n f k ( x ) f ( τ k ) ∣ = sup f ∈ X ∗ , ∥ f ∥ = 1 ∣ ∑ k = n + 1 ∞ f k ( x ) f ( τ k ) ∣ ≤ ( ∑ k = n + 1 ∞ ∣ f ( τ k ) ∣ q ) 1 / q ( ∑ k = n + 1 ∞ ∣ f k ( x ) − f k ( 0 ) ∣ p ) 1 / p ≤ b ( ∑ k = n + 1 ∞ ∣ f k ( x ) ∣ p ) 1 / p ⟶ 0 as n ⟶ ∞ . \begin{aligned}
\left\|x-\sum_{k=1}^{n}f_k(x)\tau_k\right\|
=\sup_{\substack{f\in\mathcal{X}^*,\ \|f\|=1}}\left|f(x)-\sum_{k=1}^{n}f_k(x)f(\tau_k)\right| \\
&=\sup_{\substack{f\in\mathcal{X}^*,\ \|f\|=1}}\left|\left(\sum_{k=1}^{\infty}f(\tau_k)f_k\right)(x)-\sum_{k=1}^{n}f_k(x)f(\tau_k)\right| \\
&=\sup_{\substack{f\in\mathcal{X}^*,\ \|f\|=1}}\left|\sum_{k=n+1}^{\infty}f_k(x)f(\tau_k)\right| \\
&\le\left(\sum_{k=n+1}^{\infty}|f(\tau_k)|^q\right)^{1/q}\left(\sum_{k=n+1}^{\infty}|f_k(x)-f_k(0)|^p\right)^{1/p} \\
&\le b\left(\sum_{k=n+1}^{\infty}|f_k(x)|^p\right)^{1/p}\longrightarrow0\text{ as }n\longrightarrow\infty.
\end{aligned} x − k = 1 ∑ n f k ( x ) τ k = f ∈ X ∗ , ∥ f ∥ = 1 sup f ( x ) − k = 1 ∑ n f k ( x ) f ( τ k ) = f ∈ X ∗ , ∥ f ∥ = 1 sup ( k = 1 ∑ ∞ f ( τ k ) f k ) ( x ) − k = 1 ∑ n f k ( x ) f ( τ k ) = f ∈ X ∗ , ∥ f ∥ = 1 sup k = n + 1 ∑ ∞ f k ( x ) f ( τ k ) ≤ ( k = n + 1 ∑ ∞ ∣ f ( τ k ) ∣ q ) 1/ q ( k = n + 1 ∑ ∞ ∣ f k ( x ) − f k ( 0 ) ∣ p ) 1/ p ≤ b ( k = n + 1 ∑ ∞ ∣ f k ( x ) ∣ p ) 1/ p ⟶ 0 as n ⟶ ∞. Now we are left with proving that { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X f ∈ X ∗ f\in\mathcal{X}^* f ∈ X ∗
∥ f ∥ = sup x ∈ X ∥ x ∥ = 1 ∣ f ( x ) ∣ = sup x ∈ X ∥ x ∥ = 1 ∣ f ( ∑ n = 1 ∞ f n ( x ) τ n ) ∣ = sup x ∈ X , ∥ x ∥ = 1 ∣ ∑ n = 1 ∞ f n ( x ) f ( τ n ) ∣ ≤ b ( ∑ n = 1 ∞ ∣ f n ( x ) ∣ p ) 1 p . \begin{aligned}
\|f\|=\sup_{\substack{x\in\mathcal{X}\\\|x\|=1}}|f(x)|=\sup_{\substack{x\in\mathcal{X}\\\|x\|=1}}\left|f\left(\sum_{n=1}^{\infty}f_n(x)\tau_n\right)\right| \\
&= \sup_{x\in\mathcal{X},\lVert x\rVert=1}\left|\sum_{n=1}^{\infty} f_n(x)f(\tau_n)\right| \le b\left(\sum_{n=1}^{\infty}|f_n(x)|^p\right)^{\frac{1}{p}}.
\end{aligned} ∥ f ∥ = x ∈ X ∥ x ∥ = 1 sup ∣ f ( x ) ∣ = x ∈ X ∥ x ∥ = 1 sup f ( n = 1 ∑ ∞ f n ( x ) τ n ) = x ∈ X , ∥ x ∥ = 1 sup n = 1 ∑ ∞ f n ( x ) f ( τ n ) ≤ b ( n = 1 ∑ ∞ ∣ f n ( x ) ∣ p ) p 1 . Since f f f
Theorem 1.3.20 and Theorem 2.2.1 suggest the following question. For which metric spaces and BK-spaces, does Theorem 2.2.1 hold? We next present a result which demands only reconstruction of elements using Lipschitz functions on a Banach space and not frame conditions. First we see a result for this purpose.
Lemma 2.2.2. [cf. 45 ] Given a Banach space X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X
Y d : = { { a n } n : ∑ n = 1 ∞ a n τ n converges in X } . \mathcal{Y}_d:=\left\{\{a_n\}_n:\sum_{n=1}^{\infty}a_n\tau_n\text{ converges in }\mathcal{X}\right\}. Y d := { { a n } n : n = 1 ∑ ∞ a n τ n converges in X } . Then Y d \mathcal{Y}_d Y d
∥ { a n } n ∥ : = sup m ∥ ∑ n = 1 m a n τ n ∥ . \lVert\{a_n\}_n\rVert:=\sup_m\left\lVert\sum_{n=1}^{m}a_n\tau_n\right\rVert. ∥{ a n } n ∥ := m sup n = 1 ∑ m a n τ n . Further, the canonical unit vectors form a Schauder basis for Y d \mathcal{Y}_d Y d
Theorem 2.2.3. Let X \mathcal{X} X { f n } n \{f_n\}_n { f n } n Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K )
(i) There exists a sequence { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\ \forall x\in\mathcal{X} x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X
(ii) Let { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X S n ( x ) : = ∑ k = 1 n f k ( x ) τ k , ∀ x ∈ X S_n(x):=\sum_{k=1}^{n}f_k(x)\tau_k,\ \forall x\in\mathcal{X} S n ( x ) := ∑ k = 1 n f k ( x ) τ k , ∀ x ∈ X n ∈ N n\in\mathbb{N} n ∈ N sup n ∈ N ∥ S n ∥ Lip 0 < ∞ \sup_{n\in\mathbb{N}}\lVert S_n\rVert_{\operatorname{Lip}_0}<\infty sup n ∈ N ∥ S n ∥ Lip 0 < ∞ M d \mathcal{M}_d M d S : M d → M S:\mathcal{M}_d\to\mathcal{M} S : M d → M ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) X \mathcal{X} X
Further, a choice for τ n \tau_n τ n τ n = S e n \tau_n=Se_n τ n = S e n n ∈ N n\in\mathbb{N} n ∈ N { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
Proof. (ii) ⇒ \Rightarrow ⇒
(i) ⇒ \Rightarrow ⇒ [45 ] . Define A : = { n ∈ N : τ n = 0 } A:=\{n\in\mathbb{N}:\tau_n=0\} A := { n ∈ N : τ n = 0 } B : = N ∖ A B:=\mathbb{N}\setminus A B := N ∖ A c 0 ( A ) c_0(A) c 0 ( A ) A A A { e n } n ∈ A \{e_n\}_{n\in A} { e n } n ∈ A c 0 ( A ) c_0(A) c 0 ( A ) { 1 n ( ∥ f n ∥ Lip 0 + 1 ) e n } n ∈ A \left\{\frac{1}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}e_n\right\}_{n\in A} { n (∥ f n ∥ Lip 0 + 1 ) 1 e n } n ∈ A c 0 ( A ) c_0(A) c 0 ( A )
L d : = { { c n } n ∈ A : ∑ n ∈ A c n n ( ∥ f n ∥ Lip 0 + 1 ) e n converges in A } . \mathcal{L}_d:=\left\{\{c_n\}_{n\in A}:\sum_{n\in A}\frac{c_n}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}e_n\text{ converges in }A\right\}. L d := { { c n } n ∈ A : n ∈ A ∑ n (∥ f n ∥ Lip 0 + 1 ) c n e n converges in A } . We equip L d \mathcal{L}_d L d
∥ { c n } n ∈ A ∥ L d : = ∥ c n n ( ∥ f n ∥ Lip 0 + 1 ) ∥ c 0 ( A ) = sup n ∈ A ∣ c n n ( ∥ f n ∥ Lip 0 + 1 ) ∣ . \left\lVert\{c_n\}_{n\in A}\right\rVert_{\mathcal{L}_d}:=\left\lVert\frac{c_n}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}\right\rVert_{c_0(A)}=\sup_{n\in A}\left|\frac{c_n}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}\right|. ∥ { c n } n ∈ A ∥ L d := n (∥ f n ∥ Lip 0 + 1 ) c n c 0 ( A ) = n ∈ A sup n (∥ f n ∥ Lip 0 + 1 ) c n . Then { e n } n ∈ A \{e_n\}_{n\in A} { e n } n ∈ A L d \mathcal{L}_d L d L d \mathcal{L}_d L d Y d \mathcal{Y}_d Y d B B B M d : = Y d ⊕ L d \mathcal{M}_d:=\mathcal{Y}_d\oplus\mathcal{L}_d M d := Y d ⊕ L d ∥ y ⊕ z ∥ M d : = ∥ y ∥ Y d + ∥ z ∥ L d \|y\oplus z\|_{\mathcal{M}_d}:=\|y\|_{\mathcal{Y}_d}+\|z\|_{\mathcal{L}_d} ∥ y ⊕ z ∥ M d := ∥ y ∥ Y d + ∥ z ∥ L d x ∈ X x\in\mathcal{X} x ∈ X { f n ( x ) } n ∈ B ⊕ { f n ( x ) } n ∈ A ∈ M d \{f_n(x)\}_{n\in B}\oplus\{f_n(x)\}_{n\in A}\in\mathcal{M}_d { f n ( x ) } n ∈ B ⊕ { f n ( x ) } n ∈ A ∈ M d { f n } n \{f_n\}_n { f n } n M d \mathcal{M}_d M d X \mathcal{X} X x , y ∈ X x,y\in\mathcal{X} x , y ∈ X
∥ x − y ∥ = ∥ ∑ n = 1 ∞ ( f n ( x ) − f n ( y ) ) τ n ∥ = lim n → ∞ ∥ ∑ k = 1 n ( f k ( x ) − f k ( y ) ) τ k ∥ ≤ sup n ∈ N ∥ ∑ k = 1 n ( f k ( x ) − f k ( y ) ) τ k ∥ = sup n ∈ B ∥ ∑ k = 1 n ( f k ( x ) − f k ( y ) ) τ k ∥ = ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d ≤ ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d + ∥ { f n ( x ) − f n ( y ) } n ∈ A ∥ L d = ∥ { f n ( x ) − f n ( y ) } n ∈ B ⊕ { f n ( x ) − f n ( y ) } n ∈ A ∥ M d . \begin{aligned}
\lVert x-y\rVert=\left\lVert\sum_{n=1}^{\infty}(f_n(x)-f_n(y))\tau_n\right\rVert=\lim_{n\to\infty}\left\lVert\sum_{k=1}^{n}(f_k(x)-f_k(y))\tau_k\right\rVert\\
&\leq\sup_{n\in\mathbb{N}}\left\lVert\sum_{k=1}^{n}(f_k(x)-f_k(y))\tau_k\right\rVert=\sup_{n\in B}\left\lVert\sum_{k=1}^{n}(f_k(x)-f_k(y))\tau_k\right\rVert\\
&=\left\lVert\{f_n(x)-f_n(y)\}_{n\in B}\right\rVert_{\mathcal{Y}_d} \\
&\leq\left\lVert\{f_n(x)-f_n(y)\}_{n\in B}\right\rVert_{\mathcal{Y}_d}+\left\lVert\{f_n(x)-f_n(y)\}_{n\in A}\right\rVert_{\mathcal{L}_d} \\
&=\left\lVert\{f_n(x)-f_n(y)\}_{n\in B}\oplus\{f_n(x)-f_n(y)\}_{n\in A}\right\rVert_{\mathcal{M}_d}.
\end{aligned} ∥ x − y ∥ = n = 1 ∑ ∞ ( f n ( x ) − f n ( y )) τ n = n → ∞ lim k = 1 ∑ n ( f k ( x ) − f k ( y )) τ k ≤ n ∈ N sup k = 1 ∑ n ( f k ( x ) − f k ( y )) τ k = n ∈ B sup k = 1 ∑ n ( f k ( x ) − f k ( y )) τ k = ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d ≤ ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d + ∥ { f n ( x ) − f n ( y ) } n ∈ A ∥ L d = ∥ { f n ( x ) − f n ( y ) } n ∈ B ⊕ { f n ( x ) − f n ( y ) } n ∈ A ∥ M d . and
∥ { f n ( x ) − f n ( y ) } n ∈ B ⊕ { f n ( x ) − f n ( y ) } n ∈ A ∥ M d = ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d + ∥ { f n ( x ) − f n ( y ) } n ∈ A ∥ L d = sup n ∈ B ∥ ∑ k = 1 n ( f k ( x ) − f k ( y ) ) τ k ∥ + sup n ∈ A ∣ f n ( x ) − f n ( y ) n ( ∥ f n ∥ Lip 0 + 1 ) ∣ = sup n ∈ B ∥ S n ( x ) − S n ( y ) ∥ + sup n ∈ A ∣ f n ( x ) − f n ( y ) n ( ∥ f n ∥ Lip 0 + 1 ) ∣ ≤ sup n ∈ B ∥ S n ∥ Lip 0 ∥ x − y ∥ + sup n ∈ A ∥ f n ∥ Lip 0 n ( ∥ f n ∥ Lip 0 + 1 ) ∥ x − y ∥ ≤ ( sup n ∈ B ∥ S n ∥ Lip 0 + 1 ) ∥ x − y ∥ . \begin{aligned}
&\left\lVert\{f_n(x)-f_n(y)\}_{n\in B}\oplus\{f_n(x)-f_n(y)\}_{n\in A}\right\rVert_{\mathcal{M}_d}\\
&=\left\lVert\{f_n(x)-f_n(y)\}_{n\in B}\right\rVert_{\mathcal{Y}_d}+\left\lVert\{f_n(x)-f_n(y)\}_{n\in A}\right\rVert_{\mathcal{L}_d}\\
&=\sup_{n\in B}\left\lVert\sum_{k=1}^{n}(f_k(x)-f_k(y))\tau_k\right\rVert+\sup_{n\in A}\left|\frac{f_n(x)-f_n(y)}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}\right|\\
&=\sup_{n\in B}\lVert S_n(x)-S_n(y)\rVert+\sup_{n\in A}\left|\frac{f_n(x)-f_n(y)}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}\right|\\
&\leq\sup_{n\in B}\lVert S_n\rVert_{\operatorname{Lip}_0}\lVert x-y\rVert+\sup_{n\in A}\frac{\lVert f_n\rVert_{\operatorname{Lip}_0}}{n(\lVert f_n\rVert_{\operatorname{Lip}_0}+1)}\lVert x-y\rVert\\
&\leq\left(\sup_{n\in B}\lVert S_n\rVert_{\operatorname{Lip}_0}+1\right)\lVert x-y\rVert.
\end{aligned} ∥ { f n ( x ) − f n ( y ) } n ∈ B ⊕ { f n ( x ) − f n ( y ) } n ∈ A ∥ M d = ∥ { f n ( x ) − f n ( y ) } n ∈ B ∥ Y d + ∥ { f n ( x ) − f n ( y ) } n ∈ A ∥ L d = n ∈ B sup k = 1 ∑ n ( f k ( x ) − f k ( y )) τ k + n ∈ A sup n (∥ f n ∥ Lip 0 + 1 ) f n ( x ) − f n ( y ) = n ∈ B sup ∥ S n ( x ) − S n ( y )∥ + n ∈ A sup n (∥ f n ∥ Lip 0 + 1 ) f n ( x ) − f n ( y ) ≤ n ∈ B sup ∥ S n ∥ Lip 0 ∥ x − y ∥ + n ∈ A sup n (∥ f n ∥ Lip 0 + 1 ) ∥ f n ∥ Lip 0 ∥ x − y ∥ ≤ ( n ∈ B sup ∥ S n ∥ Lip 0 + 1 ) ∥ x − y ∥ . We now define
S : M d ∋ { a n } n ∈ B ⊕ { b n } n ∈ A ↦ ∑ n ∈ B a n τ n ∈ X . S:\mathcal{M}_d\ni\{a_n\}_{n\in B}\oplus\{b_n\}_{n\in A}\mapsto\sum_{n\in B}a_n\tau_n\in\mathcal{X}. S : M d ∋ { a n } n ∈ B ⊕ { b n } n ∈ A ↦ n ∈ B ∑ a n τ n ∈ X . Clearly S S S S S S
∥ S ( { a n } n ∈ B ⊕ { b n } n ∈ A ) ∥ = ∥ ∑ n ∈ B a n τ n ∥ ≤ sup n ∈ B ∥ ∑ k = 1 n a k τ k ∥ = ∥ { a n } n ∈ B ∥ Y d ≤ ∥ { a n } n ∈ B ⊕ { b n } n ∈ A ∥ M d . \left\|S\left(\{a_n\}_{n\in B}\oplus\{b_n\}_{n\in A}\right)\right\|
=\left\|\sum_{n\in B}a_n\tau_n\right\|
\leq\sup_{n\in B}\left\|\sum_{k=1}^{n}a_k\tau_k\right\|
=\left\|\{a_n\}_{n\in B}\right\|_{\mathcal{Y}_d}
\leq\left\|\{a_n\}_{n\in B}\oplus\{b_n\}_{n\in A}\right\|_{\mathcal{M}_d}. ∥ S ( { a n } n ∈ B ⊕ { b n } n ∈ A ) ∥ = n ∈ B ∑ a n τ n ≤ n ∈ B sup k = 1 ∑ n a k τ k = ∥ { a n } n ∈ B ∥ Y d ≤ ∥ { a n } n ∈ B ⊕ { b n } n ∈ A ∥ M d . Using Theorem 1.5.6 we derive the following result which tells that given a metric frame for a metric space we can get a metric frame using linear functionals for a subset of the Banach space.
Theorem 2.2.4. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) n ∈ N n\in\mathbb{N} n ∈ N T f n T_{f_n} T f n f n f_n f n e e e F ( M ) \mathcal{F}(\mathcal{M}) F ( M ) Theorem 1.5.6 . Then { f n } n \{f_n\}_n { f n } n M \mathcal{M} M a a a b b b { T f n } n \{T_{f_n}\}_n { T f n } n e ( M ) e(\mathcal{M}) e ( M ) a a a b b b ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) M \mathcal{M} M ( { T f n } n , e S ) (\{T_{f_n}\}_n,eS) ({ T f n } n , e S ) e ( M ) e(\mathcal{M}) e ( M )
Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) u , v ∈ e ( M ) u,v\in e(\mathcal{M}) u , v ∈ e ( M ) u = e ( x ) , v = e ( y ) u=e(x),v=e(y) u = e ( x ) , v = e ( y ) x , y ∈ M x,y\in\mathcal{M} x , y ∈ M e e e
a ∥ u − v ∥ = a ∥ e ( x ) − e ( y ) ∥ = a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ = ∥ { ( T f n e ) ( x ) − ( T f n e ) ( y ) } n ∥ = ∥ { T f n ( e ( x ) ) − T f n ( e ( y ) ) } n ∥ = ∥ { T f n ( u ) − T f n ( v ) } n ∥ ≤ b d ( x , y ) = b ∥ e ( x ) − e ( y ) ∥ = b ∥ u − v ∥ . a\|u-v\|=a\|e(x)-e(y)\|=ad(x,y)\leq\left\|\{f_n(x)-f_n(y)\}_n\right\|
=\left\|\{(T_{f_n}e)(x)-(T_{f_n}e)(y)\}_n\right\|
=\left\|\{T_{f_n}(e(x))-T_{f_n}(e(y))\}_n\right\|
=\left\|\{T_{f_n}(u)-T_{f_n}(v)\}_n\right\|
\leq bd(x,y)=b\|e(x)-e(y)\|=b\|u-v\|. a ∥ u − v ∥ = a ∥ e ( x ) − e ( y ) ∥ = a d ( x , y ) ≤ ∥ { f n ( x ) − f n ( y ) } n ∥ = ∥ {( T f n e ) ( x ) − ( T f n e ) ( y ) } n ∥ = ∥ { T f n ( e ( x )) − T f n ( e ( y )) } n ∥ = ∥ { T f n ( u ) − T f n ( v ) } n ∥ ≤ b d ( x , y ) = b ∥ e ( x ) − e ( y ) ∥ = b ∥ u − v ∥. ( ⇐ ) (\Leftarrow) ( ⇐ ) x , y ∈ M x,y\in\mathcal{M} x , y ∈ M e ( x ) , e ( y ) ∈ e ( M ) e(x),e(y)\in e(\mathcal{M}) e ( x ) , e ( y ) ∈ e ( M )
a d ( x , y ) = a ∥ e ( x ) − e ( y ) ∥ ≤ ∥ { T f n ( e ( x ) ) − T f n ( e ( y ) ) } n ∥ = ∥ { f n ( x ) − f n ( y ) } n ∥ ≤ b ∥ e ( x ) − e ( y ) ∥ = b d ( x , y ) . ad(x,y)=a\|e(x)-e(y)\|\leq\left\|\{T_{f_n}(e(x))-T_{f_n}(e(y))\}_n\right\|
=\left\|\{f_n(x)-f_n(y)\}_n\right\|
\leq b\|e(x)-e(y)\|=bd(x,y). a d ( x , y ) = a ∥ e ( x ) − e ( y ) ∥ ≤ ∥ { T f n ( e ( x )) − T f n ( e ( y )) } n ∥ = ∥ { f n ( x ) − f n ( y ) } n ∥ ≤ b ∥ e ( x ) − e ( y ) ∥ = b d ( x , y ) . Since x , y x,y x , y
PERTURBATIONS
Here we present some stability results. These are important as they say that sequences which are close to metric frames are again metric frames. On the other hand, it asserts that if we perturb a metric frame we again get a metric frame.
Theorem 2.3.1. Let { f n } n \{f_n\}_n { f n } n p p p M \mathcal{M} M a a a b b b { g n } n \{g_n\}_n { g n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K )
(i) There exist α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 β < 1 \beta< 1 β < 1 α < 1 \alpha< 1 α < 1 γ < ( 1 − α ) a \gamma< (1-\alpha)a γ < ( 1 − α ) a
(ii) For all x , y ∈ M x,y \in\mathcal{M} x , y ∈ M m = 1 , 2 , … , m=1,2,\ldots, m = 1 , 2 , … ,
( ∑ n = 1 m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p ≤ α ( ∑ n = 1 m ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 m ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) . (18) \left(\sum_{n=1}^{m}\left|(f_n-g_n)(x)-(f_n-g_n)(y)\right|^p\right)^{\frac{1}{p}}
\le\alpha\left(\sum_{n=1}^{m}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}}
+\beta\left(\sum_{n=1}^{m}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}
+\gamma d(x,y). \tag*{(18)} ( n = 1 ∑ m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 ≤ α ( n = 1 ∑ m ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ m ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) . (18) Then { g n } n \{g_n\}_n { g n } n p p p M \mathcal{M} M ( 1 − α ) a − γ 1 + β \frac{(1-\alpha)a-\gamma}{1+\beta} 1 + β ( 1 − α ) a − γ ( 1 + α ) b + γ 1 − β \frac{(1+\alpha)b+\gamma}{1-\beta} 1 − β ( 1 + α ) b + γ
Proof. Using Minkowski’s inequality and Inequality (2.3.1), we get, for all x , y ∈ M x,y \in\mathcal{M} x , y ∈ M m ∈ N m \in\mathbb{N} m ∈ N
( ∑ n = 1 m ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ ( ∑ n = 1 m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p + ( ∑ n = 1 m ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ ( 1 + α ) ( ∑ n = 1 m ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 m ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) , \begin{aligned}
\left(\sum_{n=1}^{m}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}
\le\left(\sum_{n=1}^{m}|(f_n-g_n)(x)-(f_n-g_n)(y)|^p\right)^{\frac{1}{p}}
+\left(\sum_{n=1}^{m}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}} \\
&\le(1+\alpha)\left(\sum_{n=1}^{m}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}
+\beta\left(\sum_{n=1}^{m}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}
+\gamma d(x,y),
\end{aligned} ( n = 1 ∑ m ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ ( n = 1 ∑ m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 + ( n = 1 ∑ m ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ ( 1 + α ) ( n = 1 ∑ m ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ m ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) , which implies
( 1 − β ) ( ∑ n = 1 m ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ ( 1 + α ) ( ∑ n = 1 m ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + γ d ( x , y ) , (1-\beta)\left(\sum_{n=1}^{m}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}
\le(1+\alpha)\left(\sum_{n=1}^{m}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}
+\gamma d(x,y), ( 1 − β ) ( n = 1 ∑ m ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ ( 1 + α ) ( n = 1 ∑ m ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + γ d ( x , y ) , for all x , y ∈ M x,y \in\mathcal{M} x , y ∈ M ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p \sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p
( ∑ n = 1 ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p ≤ α ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) . (19) \left(\sum_{n=1}^{\infty}\left|(f_n-g_n)(x)-(f_n-g_n)(y)\right|^p\right)^{\frac{1}{p}} \le\alpha\left(\sum_{n=1}^{\infty}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}}+\beta\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}+\gamma d(x,y). \tag*{(19)} ( n = 1 ∑ ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 ≤ α ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) . (19) By doing a similar calculation and using Inequality (2.3.2) we get for all x , y ∈ M x,y \in\mathcal{M} x , y ∈ M
( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ ( ∑ n = 1 ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p + ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ ( 1 + α ) ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) ≤ ( 1 + α ) b d ( x , y ) + β ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) = ( ( 1 + α ) b + γ ) d ( x , y ) + β ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p \begin{aligned}
\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}} \\
&\le\left(\sum_{n=1}^{\infty}\left|(f_n-g_n)(x)-(f_n-g_n)(y)\right|^p\right)^{\frac{1}{p}}+\left(\sum_{n=1}^{\infty}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}} \\
&\le(1+\alpha)\left(\sum_{n=1}^{\infty}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}}+\beta\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}+\gamma d(x,y) \\
&\le(1+\alpha)bd(x,y)+\beta\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}+\gamma d(x,y) \\
&=((1+\alpha)b+\gamma)d(x,y)+\beta\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}
\end{aligned} ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ ( n = 1 ∑ ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 + ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ ( 1 + α ) ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) ≤ ( 1 + α ) b d ( x , y ) + β ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) = (( 1 + α ) b + γ ) d ( x , y ) + β ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 which gives
( 1 − β ) ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ ( ( 1 + α ) b + γ ) d ( x , y ) , ∀ x , y ∈ M (1-\beta)\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}\le((1+\alpha)b+\gamma)d(x,y),\quad\forall x,y\in\mathcal{M} ( 1 − β ) ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ (( 1 + α ) b + γ ) d ( x , y ) , ∀ x , y ∈ M i.e.,
( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ ( 1 + α ) b + γ 1 − β d ( x , y ) , ∀ x , y ∈ M . \left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}\le\frac{(1+\alpha)b+\gamma}{1-\beta}d(x,y),\quad\forall x,y\in\mathcal{M}. ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ 1 − β ( 1 + α ) b + γ d ( x , y ) , ∀ x , y ∈ M . Hence we obtained upper frame bound for { g n } n \{g_n\}_n { g n } n x , y ∈ M x,y\in\mathcal{M} x , y ∈ M
( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ ( ∑ n = 1 ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p + ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p ≤ α ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + ( 1 + β ) ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) \begin{aligned}
\left(\sum_{n=1}^{\infty}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}} \\
&\le\left(\sum_{n=1}^{\infty}\left|(f_n-g_n)(x)-(f_n-g_n)(y)\right|^p\right)^{\frac{1}{p}}+\left(\sum_{n=1}^{\infty}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}} \\
&\le\alpha\left(\sum_{n=1}^{\infty} |f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}} +(1+\beta)\left(\sum_{n=1}^{\infty} |g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}+\gamma d(x,y)
\end{aligned} ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ ( n = 1 ∑ ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 + ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 ≤ α ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + ( 1 + β ) ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) which implies
( 1 − α ) a d ( x , y ) ≤ ( 1 − α ) ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ ( 1 + β ) ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) , ∀ x , y ∈ M \begin{aligned}
(1-\alpha)ad(x,y)\le(1-\alpha)\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}} \\
&\le(1+\beta)\left(\sum_{n=1}^{\infty}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}+\gamma d(x,y),\quad\forall x,y\in\mathcal{M}
\end{aligned} ( 1 − α ) a d ( x , y ) ≤ ( 1 − α ) ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ ( 1 + β ) ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) , ∀ x , y ∈ M i.e., ( ( 1 − α ) a − γ ) 1 + β ≤ ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p , ∀ x , y ∈ M . \text{i.e.,}\quad\frac{((1-\alpha)a-\gamma)}{1+\beta}\le\left(\sum_{n=1}^{\infty}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}},\quad\forall x,y\in\mathcal{M}. i.e., 1 + β (( 1 − α ) a − γ ) ≤ ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 , ∀ x , y ∈ M . Using Theorem 2.3.1 we obtain the following result.
Corollary 2.3.2. Let { f n } n \{f_n\}_n { f n } n p p p M \mathcal{M} M a a a b b b { g n } n \{g_n\}_n { g n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K )
r : = ( ∑ n = 1 ∞ Lip ( f n − g n ) p ) 1 p < a . r:=\left(\sum_{n=1}^{\infty}\operatorname{Lip}(f_n-g_n)^p\right)^{\frac{1}{p}}<a. r := ( n = 1 ∑ ∞ Lip ( f n − g n ) p ) p 1 < a . Then { g n } n \{g_n\}_n { g n } n p p p M \mathcal{M} M a − r a-r a − r b + r b+r b + r
Proof. Define α : = 0 \alpha:=0 α := 0 β : = 0 \beta:=0 β := 0 γ : = r \gamma:=r γ := r x , y ∈ M x,y\in\mathcal{M} x , y ∈ M
( ∑ n = 1 ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p ≤ ( ∑ n = 1 ∞ Lip ( f n − g n ) p d ( x , y ) p ) 1 p = ( ∑ n = 1 ∞ Lip ( f n − g n ) p ) 1 p d ( x , y ) = r d ( x , y ) = α ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) . \begin{aligned}
\left(\sum_{n=1}^{\infty}|(f_n-g_n)(x)-(f_n-g_n)(y)|^p\right)^{\frac{1}{p}}\le\left(\sum_{n=1}^{\infty}\operatorname{Lip}(f_n-g_n)^p d(x,y)^p\right)^{\frac{1}{p}} \\
&=\left(\sum_{n=1}^{\infty}\operatorname{Lip}(f_n-g_n)^p\right)^{\frac{1}{p}}d(x,y)=r d(x,y) \\
&= \alpha\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}+\beta\left(\sum_{n=1}^{\infty}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}+\gamma d(x,y).
\end{aligned} ( n = 1 ∑ ∞ ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 ≤ ( n = 1 ∑ ∞ Lip ( f n − g n ) p d ( x , y ) p ) p 1 = ( n = 1 ∑ ∞ Lip ( f n − g n ) p ) p 1 d ( x , y ) = r d ( x , y ) = α ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) . Thus the hypothesis in Theorem 2.3.1 holds. Hence the corollary.
Corollary 2.3.3. Let { f n } n \{f_n\}_n { f n } n p p p M \mathcal{M} M b b b { g n } n \{g_n\}_n { g n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K )
There exist α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 β < 1 \beta<1 β < 1
(ii) For all x , y ∈ M x,y \in\mathcal{M} x , y ∈ M m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , …
( ∑ n = 1 m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) 1 p ≤ α ( ∑ n = 1 m ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p + β ( ∑ n = 1 m ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p + γ d ( x , y ) . \left(\sum_{n=1}^{m}\left|(f_n-g_n)(x)-(f_n-g_n)(y)\right|^p\right)^{\frac{1}{p}} \leq\alpha\left(\sum_{n=1}^{m}\left|f_n(x)-f_n(y)\right|^p\right)^{\frac{1}{p}}+\beta\left(\sum_{n=1}^{m}\left|g_n(x)-g_n(y)\right|^p\right)^{\frac{1}{p}}+\gamma d(x,y). ( n = 1 ∑ m ∣ ( f n − g n ) ( x ) − ( f n − g n ) ( y ) ∣ p ) p 1 ≤ α ( n = 1 ∑ m ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 + β ( n = 1 ∑ m ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 + γ d ( x , y ) . Then { g n } n \{g_n\}_n { g n } n p p p M \mathcal{M} M ( 1 + α ) b + γ 1 − β \frac{(1+\alpha)b+\gamma}{1-\beta} 1 − β ( 1 + α ) b + γ
We next derive a stability result in which we perturb the Lipschitz functions and then derive the existence of reconstruction operator. This is motivated from Theorem 1.3.21 .
Theorem 2.3.4. Let ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) X \mathcal{X} X f n ( 0 ) = 0 f_n(0)=0 f n ( 0 ) = 0 n ∈ N n \in\mathbb{N} n ∈ N S ( 0 ) = 0 S(0)=0 S ( 0 ) = 0 { g n } n \{g_n\}_n { g n } n Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K )
(i) There exist α , γ ≥ 0 \alpha,\gamma\geq0 α , γ ≥ 0
∥ { ( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ ≤ α ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . (20) \left\|\{(f_n-g_n)(x)-(f_n-g_n)(y)\}_n\right\| \leq\alpha\left\|\{f_n(x)-f_n(y)\}_n\right\|+\gamma\|x-y\|,\quad\forall x,y \in\mathcal{X}. \tag*{(20)} ∥ {( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ ≤ α ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . (20) (ii) α ∥ θ f ∥ Lip 0 + γ ≤ ∥ S ∥ Lip 0 − 1 \alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma\leq\|S\|_{\operatorname{Lip}_0}^{-1} α ∥ θ f ∥ Lip 0 + γ ≤ ∥ S ∥ Lip 0 − 1
Then there exists a reconstruction Lipschitz operator T T T ( { f n } n , T ) (\{f_n\}_n,T) ({ f n } n , T ) X \mathcal{X} X ∥ S ∥ Lip 0 − 1 − ( α ∥ θ f ∥ Lip 0 + γ ) \|S\|_{\operatorname{Lip}_0}^{-1}-(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma) ∥ S ∥ Lip 0 − 1 − ( α ∥ θ f ∥ Lip 0 + γ ) ∥ θ f ∥ Lip 0 + ( α ∥ θ f ∥ Lip 0 + γ ) \|\theta_f\|_{\operatorname{Lip}_0}+(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma) ∥ θ f ∥ Lip 0 + ( α ∥ θ f ∥ Lip 0 + γ )
Proof. Let x ∈ X x \in\mathcal{X} x ∈ X g n ( 0 ) = 0 g_n(0)=0 g n ( 0 ) = 0 f n ( 0 ) = 0 f_n(0)=0 f n ( 0 ) = 0 n ∈ N n \in\mathbb{N} n ∈ N
∥ { g n ( x ) } n ∥ ≤ ∥ { ( f n − g n ) ( x ) } n ∥ + ∥ { f n ( x ) } n ∥ ≤ ( α + 1 ) ∥ { f n ( x ) } n ∥ + γ ∥ x ∥ . \begin{aligned}
\|\{g_n(x)\}_n\| \leq\|\{(f_n-g_n)(x)\}_n\|+\|\{f_n(x)\}_n\| \\
&\leq(\alpha+1)\|\{f_n(x)\}_n\|+\gamma\|x\|.
\end{aligned} ∥ { g n ( x ) } n ∥ ≤ ∥ {( f n − g n ) ( x ) } n ∥ + ∥ { f n ( x ) } n ∥ ≤ ( α + 1 ) ∥ { f n ( x ) } n ∥ + γ ∥ x ∥. Therefore if we define θ g : X ∋ x ↦ { g n ( x ) } n ∈ M d \theta_g:\mathcal{X}\ni x\mapsto\{g_n(x)\}_n\in\mathcal{M}_d θ g : X ∋ x ↦ { g n ( x ) } n ∈ M d θ g \theta_g θ g x , y ∈ X x,y \in\mathcal{X} x , y ∈ X
∥ θ g x − θ g y ∥ = ∥ { g n ( x ) − g n ( y ) } n ∥ = ∥ { − g n ( x ) + g n ( y ) } n ∥ ≤ ∥ { ( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ + ∥ { f n ( x ) − f n ( y ) } n ∥ ≤ ( 1 + α ) ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ = ( 1 + α ) ∥ θ f x − θ f y ∥ + γ ∥ x − y ∥ ≤ ( 1 + α ) ∥ θ f ∥ Lip 0 ∥ x − y ∥ + γ ∥ x − y ∥ = ( ( 1 + α ) ∥ θ f ∥ Lip 0 + γ ) ∥ x − y ∥ . \begin{aligned}
\|\theta_gx-\theta_gy\|=\|\{g_n(x)-g_n(y)\}_n\|=\|\{-g_n(x)+g_n(y)\}_n\| \\
&\leq\|\{(f_n-g_n)(x)-(f_n-g_n)(y)\}_n\|+\|\{f_n(x)-f_n(y)\}_n\| \\
&\leq(1+\alpha)\|\{f_n(x)-f_n(y)\}_n\|+\gamma\|x-y\|=(1+\alpha)\|\theta_fx-\theta_fy\|+\gamma\|x-y\| \\
&\leq(1+\alpha)\|\theta_f\|_{\operatorname{Lip}_0}\|x-y\|+\gamma\|x-y\|=((1+\alpha)\|\theta_f\|_{\operatorname{Lip}_0}+\gamma)\|x-y\|.
\end{aligned} ∥ θ g x − θ g y ∥ = ∥ { g n ( x ) − g n ( y ) } n ∥ = ∥ { − g n ( x ) + g n ( y ) } n ∥ ≤ ∥ {( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ + ∥ { f n ( x ) − f n ( y ) } n ∥ ≤ ( 1 + α ) ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ = ( 1 + α ) ∥ θ f x − θ f y ∥ + γ ∥ x − y ∥ ≤ ( 1 + α ) ∥ θ f ∥ Lip 0 ∥ x − y ∥ + γ ∥ x − y ∥ = (( 1 + α ) ∥ θ f ∥ Lip 0 + γ ) ∥ x − y ∥. Thus ∥ θ g ∥ Lip 0 ≤ ( 1 + α ) ∥ θ f ∥ Lip 0 + γ \|\theta_g\|_{\operatorname{Lip}_0} \le(1+\alpha)\|\theta_f\|_{\operatorname{Lip}_0}+\gamma ∥ θ g ∥ Lip 0 ≤ ( 1 + α ) ∥ θ f ∥ Lip 0 + γ ( ( 1 + α ) ∥ θ f ∥ Lip 0 + γ ) ((1+\alpha)\|\theta_f\|_{\operatorname{Lip}_0}+\gamma) (( 1 + α ) ∥ θ f ∥ Lip 0 + γ )
∥ ( θ f − θ g ) x − ( θ f − θ g ) y ∥ ≤ α ∥ θ f x − θ f y ∥ + γ ∥ x − y ∥ ≤ ( α ∥ θ f ∥ Lip 0 + γ ) ∥ x − y ∥ , ∀ x , y ∈ X . (21) \begin{aligned}
\|(\theta_f-\theta_g)x-(\theta_f-\theta_g)y\| &\le\alpha\|\theta_f x-\theta_f y\|+\gamma\|x-y\| \\
&\le(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma)\|x-y\|,\quad\forall x,y\in\mathcal{X}. \tag*{(21)}
\end{aligned} ∥ ( θ f − θ g ) x − ( θ f − θ g ) y ∥ ≤ α ∥ θ f x − θ f y ∥ + γ ∥ x − y ∥ ≤ ( α ∥ θ f ∥ Lip 0 + γ ) ∥ x − y ∥ , ∀ x , y ∈ X . (21) Now noting S θ f = I X S\theta_f=I_{\mathcal{X}} S θ f = I X
∥ I X − S θ g ∥ Lip 0 = ∥ S θ f − S θ g ∥ Lip 0 ≤ ∥ S ∥ Lip 0 ∥ θ f − θ g ∥ Lip 0 ≤ ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) < 1. \begin{aligned}
\|I_{\mathcal{X}}-S\theta_g\|_{\operatorname{Lip}_0} &= \|S\theta_f-S\theta_g\|_{\operatorname{Lip}_0} \\
&\le\|S\|_{\operatorname{Lip}_0}\|\theta_f-\theta_g\|_{\operatorname{Lip}_0} \\
&\le\|S\|_{\operatorname{Lip}_0}(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma)<1.
\end{aligned} ∥ I X − S θ g ∥ Lip 0 = ∥ S θ f − S θ g ∥ Lip 0 ≤ ∥ S ∥ Lip 0 ∥ θ f − θ g ∥ Lip 0 ≤ ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) < 1. Since Lip 0 ( X ) \operatorname{Lip}_0(\mathcal{X}) Lip 0 ( X ) Theorem 1.5.5 ), last inequality tells that S θ g S\theta_g S θ g
∥ ( S θ g ) − 1 ∥ Lip 0 ≤ 1 1 − ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) . \|(S\theta_g)^{-1}\|_{\operatorname{Lip}_0}\le\frac{1}{1-\|S\|_{\operatorname{Lip}_0}(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma)}. ∥ ( S θ g ) − 1 ∥ Lip 0 ≤ 1 − ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) 1 . Define T : = ( S θ g ) − 1 S T:=(S\theta_g)^{-1}S T := ( S θ g ) − 1 S T θ g = I X T\theta_g=I_{\mathcal{X}} T θ g = I X
∥ x − y ∥ = ∥ T θ g x − T θ g y ∥ ≤ ∥ T ∥ Lip 0 ∥ θ g x − θ g y ∥ ≤ 1 1 − ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) ∥ θ g x − θ g y ∥ , ∀ x , y ∈ X \begin{aligned}
\|x-y\| &= \|T\theta_gx-T\theta_gy\|\le\|T\|_{\operatorname{Lip}_0}\|\theta_gx-\theta_gy\| \\
&\le\frac{1}{1-\|S\|_{\operatorname{Lip}_0}(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma)}\|\theta_gx-\theta_gy\|,\quad\forall x,y\in\mathcal{X}
\end{aligned} ∥ x − y ∥ = ∥ T θ g x − T θ g y ∥ ≤ ∥ T ∥ Lip 0 ∥ θ g x − θ g y ∥ ≤ 1 − ∥ S ∥ Lip 0 ( α ∥ θ f ∥ Lip 0 + γ ) 1 ∥ θ g x − θ g y ∥ , ∀ x , y ∈ X which gives the lower bound stated in the theorem. □
Corollary 2.3.5. Let ( { f n } n , S ) (\{f_n\}_n,S) ({ f n } n , S ) X \mathcal{X} X f n ( 0 ) = 0 f_n(0)=0 f n ( 0 ) = 0 n ∈ N n\in\mathbb{N} n ∈ N S ( 0 ) = 0 S(0)=0 S ( 0 ) = 0 { g n } n \{g_n\}_n { g n } n Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) α , γ ≥ 0 \alpha,\gamma\ge0 α , γ ≥ 0
∥ { ( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ ≤ α ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . \|\{(f_n-g_n)(x)-(f_n-g_n)(y)\}_n\|\le\alpha\|\{f_n(x)-f_n(y)\}_n\|+\gamma\|x-y\|,\quad\forall x,y\in\mathcal{X}. ∥ {( f n − g n ) ( x ) − ( f n − g n ) ( y ) } n ∥ ≤ α ∥ { f n ( x ) − f n ( y ) } n ∥ + γ ∥ x − y ∥ , ∀ x , y ∈ X . Then there exists a reconstruction Lipschitz operator T T T ( { f n } n , T ) (\{f_n\}_n,T) ({ f n } n , T ) X \mathcal{X} X ∥ θ f ∥ Lip 0 + ( α ∥ θ f ∥ Lip 0 + γ ) \|\theta_f\|_{\operatorname{Lip}_0}+(\alpha\|\theta_f\|_{\operatorname{Lip}_0}+\gamma) ∥ θ f ∥ Lip 0 + ( α ∥ θ f ∥ Lip 0 + γ )
CHAPTER 3
MULTIPLIERS FOR METRIC SPACES
DEFINITION AND BASIC PROPERTIES OF MULTIPLIERS
In this chapter, we introduce and study multipliers for metric spaces. We use the following notation in this chapter. Let M \mathcal{M} M X \mathcal{X} X f ∈ Lip ( M , K ) f \in\operatorname{Lip}(\mathcal{M},\mathbb{K}) f ∈ Lip ( M , K ) τ ∈ X \tau\in\mathcal{X} τ ∈ X
τ ⊗ f : M ∋ x ↦ ( τ ⊗ f ) ( x ) : = f ( x ) τ ∈ X . \tau\otimes f:\mathcal{M}\ni x\mapsto(\tau\otimes f)(x):=f(x)\tau\in\mathcal{X}. τ ⊗ f : M ∋ x ↦ ( τ ⊗ f ) ( x ) := f ( x ) τ ∈ X . Then it follows that τ ⊗ f \tau\otimes f τ ⊗ f Lip ( τ ⊗ f ) = ∥ τ ∥ Lip ( f ) \operatorname{Lip}(\tau\otimes f)=\|\tau\|\operatorname{Lip}(f) Lip ( τ ⊗ f ) = ∥ τ ∥ Lip ( f )
We first derive a result which allows us to define multipliers for metric spaces. In the sequel, 1 < p < ∞ 1<p<\infty 1 < p < ∞ q q q p p p
Theorem 3.1.1. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) p p p ( M , 0 ) (\mathcal{M},0) ( M , 0 ) b b b { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) d d d { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N )
T : M ∋ x ↦ ∑ n = 1 ∞ λ n ( τ n ⊗ f n ) x ∈ X T:\mathcal{M}\ni x\mapsto\sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes f_n)x\in\mathcal{X} T : M ∋ x ↦ n = 1 ∑ ∞ λ n ( τ n ⊗ f n ) x ∈ X is a well-defined Lipschitz operator such that T ( 0 ) = 0 T(0)=0 T ( 0 ) = 0 b d ∥ { λ n } n ∥ ∞ bd\|\{\lambda_n\}_n\|_\infty b d ∥ { λ n } n ∥ ∞
Proof. Let n , m ∈ N n,m\in\mathbb{N} n , m ∈ N n ≤ m n\leq m n ≤ m x ∈ M x\in\mathcal{M} x ∈ M
∥ ∑ k = n m λ k ( τ k ⊗ f k ) ( x ) ∥ = ∥ ∑ k = n m λ k f k ( x ) τ k ∥ = sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ϕ ( ∑ k = n m λ k f k ( x ) τ k ) ∣ = sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ∑ k = n m λ k f k ( x ) ϕ ( τ k ) ∣ \begin{aligned}
\left\|\sum_{k=n}^{m}\lambda_k(\tau_k\otimes f_k)(x)\right\|
&=\left\|\sum_{k=n}^{m}\lambda_kf_k(x)\tau_k\right\|\\
&=\sup_{\phi\in\mathcal{X}^*,\|\phi\|\leq1}\left|\phi\left(\sum_{k=n}^{m}\lambda_kf_k(x)\tau_k\right)\right|\\
&=\sup_{\phi\in\mathcal{X}^*,\|\phi\|\leq1}\left|\sum_{k=n}^{m}\lambda_kf_k(x)\phi(\tau_k)\right|
\end{aligned} k = n ∑ m λ k ( τ k ⊗ f k ) ( x ) = k = n ∑ m λ k f k ( x ) τ k = ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ϕ ( k = n ∑ m λ k f k ( x ) τ k ) = ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup k = n ∑ m λ k f k ( x ) ϕ ( τ k ) ≤ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ∑ k = n m λ k ∣ f k ( x ) ∣ ∣ ϕ ( τ k ) ∣ ∣ ≤ sup n ∈ N ∣ λ n ∣ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∑ k = n m ∣ f k ( x ) ∣ ∣ ϕ ( τ k ) ∣ ≤ sup n ∈ N ∣ λ n ∣ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ( ∑ k = n m ∣ f k ( x ) ∣ p ) 1 p ( ∑ k = n m ∣ ϕ ( τ k ) ∣ q ) 1 q ≤ sup n ∈ N ∣ λ n ∣ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ( ∑ k = n m ∣ f k ( x ) ∣ p ) 1 p d ∥ ϕ ∥ = d sup n ∈ N ∣ λ n ∣ ( ∑ k = n m ∣ f k ( x ) ∣ p ) 1 p . \begin{aligned}
&\le\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left|\sum_{k=n}^{m}\lambda_k|f_k(x)||\phi(\tau_k)|\right| \\
&\le\sup_{n\in\mathbb{N}}|\lambda_n|\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\sum_{k=n}^{m}|f_k(x)||\phi(\tau_k)| \\
&\le\sup_{n\in\mathbb{N}}|\lambda_n|\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left(\sum_{k=n}^{m}|f_k(x)|^p\right)^{\frac{1}{p}}\left(\sum_{k=n}^{m}|\phi(\tau_k)|^q\right)^{\frac{1}{q}} \\
&\le\sup_{n\in\mathbb{N}}|\lambda_n|\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left(\sum_{k=n}^{m}|f_k(x)|^p\right)^{\frac{1}{p}}d\|\phi\| \\
&=d\sup_{n\in\mathbb{N}}|\lambda_n|\left(\sum_{k=n}^{m}|f_k(x)|^p\right)^{\frac{1}{p}}.
\end{aligned} ≤ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup k = n ∑ m λ k ∣ f k ( x ) ∣∣ ϕ ( τ k ) ∣ ≤ n ∈ N sup ∣ λ n ∣ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup k = n ∑ m ∣ f k ( x ) ∣∣ ϕ ( τ k ) ∣ ≤ n ∈ N sup ∣ λ n ∣ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ( k = n ∑ m ∣ f k ( x ) ∣ p ) p 1 ( k = n ∑ m ∣ ϕ ( τ k ) ∣ q ) q 1 ≤ n ∈ N sup ∣ λ n ∣ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ( k = n ∑ m ∣ f k ( x ) ∣ p ) p 1 d ∥ ϕ ∥ = d n ∈ N sup ∣ λ n ∣ ( k = n ∑ m ∣ f k ( x ) ∣ p ) p 1 . Since ( ∑ k = 1 ∞ ∣ f k ( x ) ∣ p ) 1 p \left(\sum_{k=1}^{\infty}|f_k(x)|^p\right)^{\frac{1}{p}} ( ∑ k = 1 ∞ ∣ f k ( x ) ∣ p ) p 1 ∑ k = 1 ∞ λ k ( τ k ⊗ f k ) ( x ) \sum_{k=1}^{\infty}\lambda_k(\tau_k\otimes f_k)(x) ∑ k = 1 ∞ λ k ( τ k ⊗ f k ) ( x ) x , y ∈ M x,y\in\mathcal{M} x , y ∈ M
∥ T x − T y ∥ = ∥ ∑ n = 1 ∞ λ n f n ( x ) τ n − ∑ n = 1 ∞ λ n f n ( y ) τ n ∥ = ∥ ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) τ n ∥ = sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ϕ ( ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) τ k ) ∣ = sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) ϕ ( τ k ) ∣ ≤ sup n ∈ N ∣ λ n ∣ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ( ∑ n = 1 ∞ ∣ ϕ ( τ n ) ∣ q ) 1 q ≤ sup n ∈ N ∣ λ n ∣ sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p d ∥ ϕ ∥ = d sup n ∈ N ∣ λ n ∣ ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ b d sup n ∈ N ∣ λ n ∣ d ( x , y ) . \begin{aligned}
\|Tx-Ty\|&=\left\|\sum_{n=1}^{\infty}\lambda_nf_n(x)\tau_n-\sum_{n=1}^{\infty}\lambda_nf_n(y)\tau_n\right\|=\left\|\sum_{n=1}^{\infty}\lambda_n(f_n(x)-f_n(y))\tau_n\right\| \\
&=\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left|\phi\left(\sum_{n=1}^{\infty}\lambda_n(f_n(x)-f_n(y))\tau_k\right)\right| \\
&=\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left|\sum_{n=1}^{\infty}\lambda_n(f_n(x)-f_n(y))\phi(\tau_k)\right| \\
&\le\sup_{n\in\mathbb{N}}|\lambda_n|\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}\left(\sum_{n=1}^{\infty}|\phi(\tau_n)|^q\right)^{\frac{1}{q}} \\
&\le\sup_{n\in\mathbb{N}}|\lambda_n|\sup_{\phi\in\mathcal{X}^{*},\|\phi\|\le1}\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}d\|\phi\| \\
&=d\sup_{n\in\mathbb{N}}|\lambda_n|\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}\le bd\sup_{n\in\mathbb{N}}|\lambda_n|d(x,y).
\end{aligned} ∥ T x − T y ∥ = n = 1 ∑ ∞ λ n f n ( x ) τ n − n = 1 ∑ ∞ λ n f n ( y ) τ n = n = 1 ∑ ∞ λ n ( f n ( x ) − f n ( y )) τ n = ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ϕ ( n = 1 ∑ ∞ λ n ( f n ( x ) − f n ( y )) τ k ) = ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup n = 1 ∑ ∞ λ n ( f n ( x ) − f n ( y )) ϕ ( τ k ) ≤ n ∈ N sup ∣ λ n ∣ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ( n = 1 ∑ ∞ ∣ ϕ ( τ n ) ∣ q ) q 1 ≤ n ∈ N sup ∣ λ n ∣ ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 d ∥ ϕ ∥ = d n ∈ N sup ∣ λ n ∣ ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ b d n ∈ N sup ∣ λ n ∣ d ( x , y ) . Hence
∥ T ∥ Lip 0 = sup x , y ∈ M , x ≠ y ∥ T x − T y ∥ d ( x , y ) ≤ b d sup n ∈ N ∣ λ n ∣ . \|T\|_{\operatorname{Lip}_0}=\sup_{\substack{x,y\in\mathcal{M},\,x\ne y}}\frac{\|Tx-Ty\|}{d(x,y)}\le bd\sup_{n\in\mathbb{N}}|\lambda_n|. ∥ T ∥ Lip 0 = x , y ∈ M , x = y sup d ( x , y ) ∥ T x − T y ∥ ≤ b d n ∈ N sup ∣ λ n ∣. Corollary 3.1.2. Let { f n } n \{f_n\}_n { f n } n Lip ( M , K ) \operatorname{Lip}(\mathcal{M},\mathbb{K}) Lip ( M , K ) p p p M \mathcal{M} M b b b { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q
sequence for Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) d d d { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n \in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N ) z ∈ M z \in\mathcal{M} z ∈ M
T : M ∋ x ↦ ∑ n = 1 ∞ λ n ( τ n ⊗ ( f n − f ( z ) ) ) x ∈ X T:\mathcal{M}\ni x\mapsto\sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes(f_n-f(z)))x\in\mathcal{X} T : M ∋ x ↦ n = 1 ∑ ∞ λ n ( τ n ⊗ ( f n − f ( z ))) x ∈ X is a well-defined Lipschitz operator with Lipschitz number at most b d ∥ { λ n } n ∥ ∞ bd\|\{\lambda_n\}_n\|_\infty b d ∥ { λ n } n ∥ ∞
Proof. Define g n : = f n − f ( z ) , ∀ n ∈ N g_n:=f_n-f(z),\ \forall n\in\mathbb{N} g n := f n − f ( z ) , ∀ n ∈ N x , y ∈ M x,y\in\mathcal{M} x , y ∈ M
( ∑ n = 1 ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) 1 p = ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ≤ b d ( x , y ) . \left(\sum_{n=1}^{\infty}|g_n(x)-g_n(y)|^p\right)^{\frac{1}{p}}=\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}\leq bd(x,y). ( n = 1 ∑ ∞ ∣ g n ( x ) − g n ( y ) ∣ p ) p 1 = ( n = 1 ∑ ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ≤ b d ( x , y ) . Hence { g n } n \{g_n\}_n { g n } n ( M , z ) (\mathcal{M},z) ( M , z ) Theorem 3.1.1 to { g n } n \{g_n\}_n { g n } n
Definition 3.1.3. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) p p p ( M , 0 ) (\mathcal{M},0) ( M , 0 ) { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N )
M λ , f , τ : = ∑ n = 1 ∞ λ n ( τ n ⊗ f n ) M_{\lambda,f,\tau}:=\sum_{n=1}^{\infty}\lambda_n(\tau_n\otimes f_n) M λ , f , τ := n = 1 ∑ ∞ λ n ( τ n ⊗ f n ) is called as the Lipschitz ( p , q ) (p,q) ( p , q ) { λ n } n \{\lambda_n\}_n { λ n } n M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ
We easily see that Definition 3.1.3 generalizes Definition 3.2 in (Rahimi and Balazs (2010)). By varying the symbol and fixing other parameters in the multiplier we get map from ℓ ∞ ( N ) \ell^\infty(\mathbb{N}) ℓ ∞ ( N ) Lip 0 ( M , X ) \operatorname{Lip}_0(\mathcal{M},\mathcal{X}) Lip 0 ( M , X ) [26 ] ) and for Banach spaces it is due to Rahimi and Balazs (Proposition 3.3 in (Rahimi and Balazs (2010))). In the next proposition we study it in the context of metric spaces.
Proposition 3.1.4. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) p p p ( M , 0 ) (\mathcal{M},0) ( M , 0 ) { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N )
T : ℓ ∞ ( N ) ∋ { λ n } n ↦ M λ , f , τ ∈ Lip 0 ( M , X ) T:\ell^\infty(\mathbb{N})\ni\{\lambda_n\}_n\mapsto M_{\lambda,f,\tau}\in\operatorname{Lip}_0(\mathcal{M},\mathcal{X}) T : ℓ ∞ ( N ) ∋ { λ n } n ↦ M λ , f , τ ∈ Lip 0 ( M , X ) is a well-defined injective bounded linear operator.
Proof. From the norm estimate of M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ T T T { λ n } n , { μ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n,\{\mu_n\}_n\in\ell^\infty(\mathbb{N}) { λ n } n , { μ n } n ∈ ℓ ∞ ( N ) M λ , f , τ = T { λ n } n = T { μ n } n = M μ , f , τ M_{\lambda,f,\tau}=T\{\lambda_n\}_n=T\{\mu_n\}_n=M_{\mu,f,\tau} M λ , f , τ = T { λ n } n = T { μ n } n = M μ , f , τ ∑ n = 1 ∞ λ n f n ( x ) τ n = M λ , f , τ x = M μ , f , τ x = ∑ n = 1 ∞ μ n f n ( x ) τ n \sum_{n=1}^{\infty}\lambda_n f_n(x)\tau_n=M_{\lambda,f,\tau}x=M_{\mu,f,\tau}x=\sum_{n=1}^{\infty}\mu_n f_n(x)\tau_n ∑ n = 1 ∞ λ n f n ( x ) τ n = M λ , f , τ x = M μ , f , τ x = ∑ n = 1 ∞ μ n f n ( x ) τ n ∀ x ∈ M ⇒ ∑ n = 1 ∞ ( λ n − μ n ) f n ( x ) τ n = 0 \forall x\in\mathcal{M}\Rightarrow\sum_{n=1}^{\infty}(\lambda_n-\mu_n)f_n(x)\tau_n=0 ∀ x ∈ M ⇒ ∑ n = 1 ∞ ( λ n − μ n ) f n ( x ) τ n = 0 ∀ x ∈ M \forall x\in\mathcal{M} ∀ x ∈ M
a ( ∑ n = 1 ∞ ∣ ( λ n − μ n ) f n ( x ) ∣ q ) 1 q ≤ ∥ ∑ n = 1 ∞ ( λ n − μ n ) f n ( x ) τ n ∥ = 0 , ∀ x ∈ M a\left(\sum_{n=1}^{\infty}|(\lambda_n-\mu_n)f_n(x)|^q\right)^{\frac{1}{q}}\leq\left\|\sum_{n=1}^{\infty}(\lambda_n-\mu_n)f_n(x)\tau_n\right\|=0,\qquad\forall x\in\mathcal{M} a ( n = 1 ∑ ∞ ∣ ( λ n − μ n ) f n ( x ) ∣ q ) q 1 ≤ n = 1 ∑ ∞ ( λ n − μ n ) f n ( x ) τ n = 0 , ∀ x ∈ M ⟹ ( λ n − μ n ) f n ( x ) = 0 , ∀ n ∈ N , ∀ x ∈ M . \Longrightarrow(\lambda_n-\mu_n)f_n(x)=0,\qquad\forall n\in\mathbb{N},\forall x\in\mathcal{M}. ⟹ ( λ n − μ n ) f n ( x ) = 0 , ∀ n ∈ N , ∀ x ∈ M . Let n ∈ N n\in\mathbb{N} n ∈ N f n ≠ 0 f_n\ne0 f n = 0 x ∈ M x\in\mathcal{M} x ∈ M f n ( x ) ≠ 0 f_n(x)\ne0 f n ( x ) = 0 λ n − μ n = 0 \lambda_n-\mu_n=0 λ n − μ n = 0 n ∈ N n\in\mathbb{N} n ∈ N λ n = μ n \lambda_n=\mu_n λ n = μ n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N T T T
CONTINUITY PROPERTIES OF MULTIPLIERS
In Proposition 1.4.3, it was obtained that whenever the symbol is in c 0 ( N ) c_0(\mathbb{N}) c 0 ( N )
Proposition 3.2.1. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) p p p ( M , 0 ) (\mathcal{M},0) ( M , 0 ) b b b { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) d d d { λ n } n ∈ c 0 ( N ) \{\lambda_n\}_n\in c_0(\mathbb{N}) { λ n } n ∈ c 0 ( N ) M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ
Proof. For each m ∈ N m\in\mathbb{N} m ∈ N M λ m , f , τ : = ∑ n = 1 m λ n ( τ n ⊗ f n ) M_{\lambda_m,f,\tau}:=\sum_{n=1}^{m}\lambda_n(\tau_n\otimes f_n) M λ m , f , τ := ∑ n = 1 m λ n ( τ n ⊗ f n ) M λ m , f , τ M_{\lambda_m,f,\tau} M λ m , f , τ Theorem 1.5.14 ). Now
∥ M λ m , f , τ − M λ , f , τ ∥ Lip 0 = sup x , y ∈ M , x ≠ y ∥ ( M λ m , f , τ − M λ , f , τ ) x − ( M λ m , f , τ − M λ , f , τ ) y ∥ d ( x , y ) = sup x , y ∈ M , x ≠ y ∥ ∑ n = m + 1 ∞ λ n f n ( x ) τ n − ∑ n = m + 1 ∞ λ n f n ( y ) τ n ∥ d ( x , y ) = sup x , y ∈ M , x ≠ y ∥ ∑ n = m + 1 ∞ λ n ( f n ( x ) − f n ( y ) ) τ n ∥ d ( x , y ) ≤ b d sup m + 1 ≤ n < ∞ ∣ λ n ∣ ⟶ 0 as m ⟶ ∞ . \begin{aligned}
\|M_{\lambda_m,f,\tau}-M_{\lambda,f,\tau}\|_{\operatorname{Lip}_0}
&=\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}}\frac{\|(M_{\lambda_m,f,\tau}-M_{\lambda,f,\tau})x-(M_{\lambda_m,f,\tau}-M_{\lambda,f,\tau})y\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}}\frac{\left\|\sum_{n=m+1}^{\infty}\lambda_n f_n(x)\tau_n-\sum_{n=m+1}^{\infty}\lambda_n f_n(y)\tau_n\right\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}}\frac{\left\|\sum_{n=m+1}^{\infty}\lambda_n(f_n(x)-f_n(y))\tau_n\right\|}{d(x,y)}\\
&\leq bd\sup_{m+1\leq n<\infty}|\lambda_n|\longrightarrow0\text{ as }m\longrightarrow\infty.
\end{aligned} ∥ M λ m , f , τ − M λ , f , τ ∥ Lip 0 = x , y ∈ M , x = y sup d ( x , y ) ∥ ( M λ m , f , τ − M λ , f , τ ) x − ( M λ m , f , τ − M λ , f , τ ) y ∥ = x , y ∈ M , x = y sup d ( x , y ) ∑ n = m + 1 ∞ λ n f n ( x ) τ n − ∑ n = m + 1 ∞ λ n f n ( y ) τ n = x , y ∈ M , x = y sup d ( x , y ) ∑ n = m + 1 ∞ λ n ( f n ( x ) − f n ( y )) τ n ≤ b d m + 1 ≤ n < ∞ sup ∣ λ n ∣ ⟶ 0 as m ⟶ ∞. Hence M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ { M λ m , f , τ } m = 1 ∞ \{M_{\lambda_m,f,\tau}\}_{m=1}^{\infty} { M λ m , f , τ } m = 1 ∞ M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ Theorem 1.5.16 it follows that M λ , f , τ M_{\lambda,f,\tau} M λ , f , τ
We now study the properties of multiplier by changing its parameters. Following result extends Theorem 1.4.4 .
Theorem 3.2.2. Let { f n } n \{f_n\}_n { f n } n Lip 0 ( M , K ) \operatorname{Lip}_0(\mathcal{M},\mathbb{K}) Lip 0 ( M , K ) p p p M \mathcal{M} M b b b { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K ) d d d { λ n } n ∈ ℓ ∞ ( N ) \{\lambda_n\}_n \in\ell^\infty(\mathbb{N}) { λ n } n ∈ ℓ ∞ ( N ) k ∈ N k \in\mathbb{N} k ∈ N λ ( k ) = { λ 1 ( k ) , λ 2 ( k ) , … } \lambda^{(k)}=\{\lambda_1^{(k)},\lambda_2^{(k)},\ldots\} λ ( k ) = { λ 1 ( k ) , λ 2 ( k ) , … } λ = { λ 1 , λ 2 , … } \lambda=\{\lambda_1,\lambda_2,\ldots\} λ = { λ 1 , λ 2 , … } τ ( k ) = { τ 1 ( k ) , τ 2 ( k ) , … } \tau^{(k)}=\{\tau_1^{(k)},\tau_2^{(k)},\ldots\} τ ( k ) = { τ 1 ( k ) , τ 2 ( k ) , … } τ n ( k ) ∈ X \tau_n^{(k)}\in\mathcal{X} τ n ( k ) ∈ X τ = { τ 1 , τ 2 , … } \tau=\{\tau_1,\tau_2,\ldots\} τ = { τ 1 , τ 2 , … } k k k λ ( k ) ∈ ℓ ∞ ( N ) \lambda^{(k)}\in\ell^\infty(\mathbb{N}) λ ( k ) ∈ ℓ ∞ ( N ) τ ( k ) \tau^{(k)} τ ( k ) q q q Lip 0 ( X , K ) \operatorname{Lip}_0(\mathcal{X},\mathbb{K}) Lip 0 ( X , K )
(i) If λ ( k ) → λ \lambda^{(k)}\to\lambda λ ( k ) → λ k → ∞ k\to\infty k → ∞ p p p
∥ M λ ( k ) , f , τ − M λ , f , τ ∥ Lip 0 → 0 as k → ∞ . \|M_{\lambda^{(k)},f,\tau}-M_{\lambda,f,\tau}\|_{\operatorname{Lip}_0}\to0\text{ as }k\to\infty. ∥ M λ ( k ) , f , τ − M λ , f , τ ∥ Lip 0 → 0 as k → ∞. (ii) If { λ n } n ∈ ℓ p ( N ) \{\lambda_n\}_n\in\ell^p(\mathbb{N}) { λ n } n ∈ ℓ p ( N ) ∑ n = 1 ∞ ∥ τ n ( k ) − τ n ∥ q → 0 \sum_{n=1}^{\infty}\|\tau_n^{(k)}-\tau_n\|^q\to0 ∑ n = 1 ∞ ∥ τ n ( k ) − τ n ∥ q → 0 k → ∞ k\to\infty k → ∞
∥ M λ , f , τ ( k ) − M λ , f , τ ∥ Lip 0 → 0 as k → ∞ . \|M_{\lambda,f,\tau^{(k)}}-M_{\lambda,f,\tau}\|_{\operatorname{Lip}_0}\to0\text{ as }k\to\infty. ∥ M λ , f , τ ( k ) − M λ , f , τ ∥ Lip 0 → 0 as k → ∞. Proof. (i) Using Theorem 3.1.1,
∥ M λ ( k ) , f , τ − M λ , f , τ ∥ Lip 0 = sup x , y ∈ M x ≠ y ∥ ( M λ ( k ) , f , τ − M λ , f , τ ) x − ( M λ ( k ) , f , τ − M λ , f , τ ) y ∥ d ( x , y ) = sup x , y ∈ M x ≠ y ∥ ∑ n = 1 ∞ ( λ n ( k ) − λ n ) f n ( x ) τ n − ∑ n = 1 ∞ ( λ n ( k ) − λ n ) f n ( y ) τ n ∥ d ( x , y ) = sup x , y ∈ M x ≠ y ∥ ∑ n = 1 ∞ ( λ n ( k ) − λ n ) ( f n ( x ) − f n ( y ) ) τ n ∥ d ( x , y ) ≤ b d sup n ∈ N ∣ λ n ( k ) − λ n ∣ = b d ∥ { λ n ( k ) − λ n } n ∥ ∞ ≤ b d ∥ { λ n ( k ) − λ n } n ∥ p → 0 as k → ∞ . \begin{aligned}
\|M_{\lambda^{(k)},f,\tau}-M_{\lambda,f,\tau}\|_{\operatorname{Lip}_0}
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|(M_{\lambda^{(k)},f,\tau}-M_{\lambda,f,\tau})x-(M_{\lambda^{(k)},f,\tau}-M_{\lambda,f,\tau})y\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|\sum_{n=1}^{\infty}(\lambda_n^{(k)}-\lambda_n)f_n(x)\tau_n-\sum_{n=1}^{\infty}(\lambda_n^{(k)}-\lambda_n)f_n(y)\tau_n\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|\sum_{n=1}^{\infty}(\lambda_n^{(k)}-\lambda_n)(f_n(x)-f_n(y))\tau_n\|}{d(x,y)}\\
&\le bd\sup_{n\in\mathbb{N}}|\lambda_n^{(k)}-\lambda_n|=bd\|\{\lambda_n^{(k)}-\lambda_n\}_n\|_\infty\\
&\le bd\|\{\lambda_n^{(k)}-\lambda_n\}_n\|_p\to0\text{ as }k\to\infty.
\end{aligned} ∥ M λ ( k ) , f , τ − M λ , f , τ ∥ Lip 0 = x , y ∈ M x = y sup d ( x , y ) ∥ ( M λ ( k ) , f , τ − M λ , f , τ ) x − ( M λ ( k ) , f , τ − M λ , f , τ ) y ∥ = x , y ∈ M x = y sup d ( x , y ) ∥ ∑ n = 1 ∞ ( λ n ( k ) − λ n ) f n ( x ) τ n − ∑ n = 1 ∞ ( λ n ( k ) − λ n ) f n ( y ) τ n ∥ = x , y ∈ M x = y sup d ( x , y ) ∥ ∑ n = 1 ∞ ( λ n ( k ) − λ n ) ( f n ( x ) − f n ( y )) τ n ∥ ≤ b d n ∈ N sup ∣ λ n ( k ) − λ n ∣ = b d ∥ { λ n ( k ) − λ n } n ∥ ∞ ≤ b d ∥ { λ n ( k ) − λ n } n ∥ p → 0 as k → ∞. (ii) Using Holder’s inequality,
∥ M λ , f , τ ( k ) − M λ , f , τ ∥ Lip 0 = sup x , y ∈ M x ≠ y ∥ ( M λ , f , τ ( k ) − M λ , f , τ ) x − ( M λ , f , τ ( k ) − M λ , f , τ ) y ∥ d ( x , y ) = sup x , y ∈ M x ≠ y ∥ ∑ n = 1 ∞ λ n f n ( x ) ( τ n ( k ) − τ n ) − ∑ n = 1 ∞ λ n f n ( y ) ( τ n ( k ) − τ n ) ∥ d ( x , y ) = sup x , y ∈ M x ≠ y ∥ ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) ( τ n ( k ) − τ n ) ∥ d ( x , y ) \begin{aligned}
\|M_{\lambda,f,\tau^{(k)}}-M_{\lambda,f,\tau}\|_{\operatorname{Lip}_0}
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|(M_{\lambda,f,\tau^{(k)}}-M_{\lambda,f,\tau})x-(M_{\lambda,f,\tau^{(k)}}-M_{\lambda,f,\tau})y\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|\sum_{n=1}^{\infty}\lambda_n f_n(x)(\tau_n^{(k)}-\tau_n)-\sum_{n=1}^{\infty}\lambda_n f_n(y)(\tau_n^{(k)}-\tau_n)\|}{d(x,y)}\\
&=\sup_{\substack{x,y\in\mathcal{M}\\x\ne y}}\frac{\|\sum_{n=1}^{\infty}\lambda_n(f_n(x)-f_n(y))(\tau_n^{(k)}-\tau_n)\|}{d(x,y)}
\end{aligned} ∥ M λ , f , τ ( k ) − M λ , f , τ ∥ Lip 0 = x , y ∈ M x = y sup d ( x , y ) ∥ ( M λ , f , τ ( k ) − M λ , f , τ ) x − ( M λ , f , τ ( k ) − M λ , f , τ ) y ∥ = x , y ∈ M x = y sup d ( x , y ) ∥ ∑ n = 1 ∞ λ n f n ( x ) ( τ n ( k ) − τ n ) − ∑ n = 1 ∞ λ n f n ( y ) ( τ n ( k ) − τ n ) ∥ = x , y ∈ M x = y sup d ( x , y ) ∥ ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y )) ( τ n ( k ) − τ n ) ∥ :::
= sup x , y ∈ M , x ≠ y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ∣ ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) ϕ ( τ n ( k ) − τ n ) ∣ d ( x , y ) ≤ sup x , y ∈ M , x ≠ y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 ( ∑ n = 1 ∞ ∣ λ n ( f n ( x ) − f n ( y ) ) ∣ p ) 1 p ( ∑ n = 1 ∞ ∣ ϕ ( τ n ( k ) − τ n ) ∣ q ) 1 q d ( x , y ) ≤ sup x , y ∈ M , x ≠ y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 { ( ∑ n = 1 ∞ ∣ λ n ∣ p ) 1 p ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) 1 p ( ∑ n = 1 ∞ ∣ ϕ ( τ n ( k ) − τ n ) ∣ q ) 1 q d ( x , y ) } ≤ b ∥ { λ n } ∥ p ( ∑ n = 1 ∞ ∥ τ n ( k ) − τ n ∥ q ) 1 q ⟶ 0 as k ⟶ ∞ . \begin{aligned}
&= \sup_{\substack{x,y\in\mathcal{M},\ x\ne y}} \sup_{\substack{\phi\in\mathcal{X}^{*},\ \lVert\phi\rVert\le1}}
\frac{\left|\sum_{n=1}^{\infty}\lambda_n\bigl(f_n(x)-f_n(y)\bigr)\phi\bigl(\tau_n^{(k)}-\tau_n\bigr)\right|}{d(x,y)} \\
&\le\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}} \sup_{\substack{\phi\in\mathcal{X}^{*},\ \lVert\phi\rVert\le1}}
\frac{\left(\sum_{n=1}^{\infty}\left|\lambda_n\bigl(f_n(x)-f_n(y)\bigr)\right|^p\right)^{\frac{1}{p}}
\left(\sum_{n=1}^{\infty}\left|\phi\bigl(\tau_n^{(k)}-\tau_n\bigr)\right|^q\right)^{\frac{1}{q}}}{d(x,y)} \\
&\le\sup_{\substack{x,y\in\mathcal{M},\ x\ne y}} \sup_{\substack{\phi\in\mathcal{X}^{*},\ \lVert\phi\rVert\le1}}
\left\{\frac{\left(\sum_{n=1}^{\infty}|\lambda_n|^p\right)^{\frac{1}{p}}
\left(\sum_{n=1}^{\infty}|f_n(x)-f_n(y)|^p\right)^{\frac{1}{p}}
\left(\sum_{n=1}^{\infty}|\phi(\tau_n^{(k)}-\tau_n)|^q\right)^{\frac{1}{q}}}{d(x,y)}\right\} \\
&\le b\lVert\{\lambda_n\}\rVert_p
\left(\sum_{n=1}^{\infty}\lVert\tau_n^{(k)}-\tau_n\rVert^q\right)^{\frac{1}{q}}
\longrightarrow0\quad\text{as }k\longrightarrow\infty.
\end{aligned} = x , y ∈ M , x = y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup d ( x , y ) ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) ϕ ( τ n ( k ) − τ n ) ≤ x , y ∈ M , x = y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup d ( x , y ) ( ∑ n = 1 ∞ λ n ( f n ( x ) − f n ( y ) ) p ) p 1 ( ∑ n = 1 ∞ ϕ ( τ n ( k ) − τ n ) q ) q 1 ≤ x , y ∈ M , x = y sup ϕ ∈ X ∗ , ∥ ϕ ∥ ≤ 1 sup ⎩ ⎨ ⎧ d ( x , y ) ( ∑ n = 1 ∞ ∣ λ n ∣ p ) p 1 ( ∑ n = 1 ∞ ∣ f n ( x ) − f n ( y ) ∣ p ) p 1 ( ∑ n = 1 ∞ ∣ ϕ ( τ n ( k ) − τ n ) ∣ q ) q 1 ⎭ ⎬ ⎫ ≤ b ∥{ λ n } ∥ p ( n = 1 ∑ ∞ ∥ τ n ( k ) − τ n ∥ q ) q 1 ⟶ 0 as k ⟶ ∞. CHAPTER 4
p-APPROXIMATE SCHAUDER FRAMES FOR BANACH SPACES
p-APPROXIMATE SCHAUDER FRAMES
Let X \mathcal{X} X X ∗ \mathcal{X}^{*} X ∗ X \mathcal{X} X
Definition 4.1.1. ([51 ] ) Let { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^{*} X ∗ ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) Schauder frame for X \mathcal{X} X if
x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X . (22) x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\qquad\forall x\in\mathcal{X}. \tag*{(22)} x = n = 1 ∑ ∞ f n ( x ) τ n , ∀ x ∈ X . (22) Definition 4.1.1 was generalized for R n \mathbb{R}^{n} R n
Definition 4.1.2. ([111 ] (); Thomas 2012)) Let { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^{*} X ∗ ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) approximate Schauder frame (ASF) for X \mathcal{X} X
(Frame operator) S f , τ : X ∋ x ↦ S f , τ x : = ∑ n = 1 ∞ f n ( x ) τ n ∈ X (23) \textbf{(Frame operator)}\quad S_{f,\tau}:\mathcal{X}\ni x\mapsto S_{f,\tau}x:=\sum_{n=1}^{\infty}f_n(x)\tau_n\in\mathcal{X} \tag*{(23)} (Frame operator) S f , τ : X ∋ x ↦ S f , τ x := n = 1 ∑ ∞ f n ( x ) τ n ∈ X (23) is a well-defined bounded linear, invertible operator.
Note that whenever S f , τ = I X S_{f,\tau}=I_{\mathcal{X}} S f , τ = I X X \mathcal{X} X S f , τ S_{f,\tau} S f , τ a , b > 0 a,b>0 a , b > 0
a ∥ x ∥ ≤ ∥ ∑ n = 1 ∞ f n ( x ) τ n ∥ ≤ b ∥ x ∥ , ∀ x ∈ X . a\lVert x\rVert\leq\left\lVert\sum_{n=1}^{\infty}f_n(x)\tau_n\right\rVert\leq b\lVert x\rVert,\qquad\forall x\in\mathcal{X}. a ∥ x ∥ ≤ n = 1 ∑ ∞ f n ( x ) τ n ≤ b ∥ x ∥ , ∀ x ∈ X . We call a a a lower ASF bound and b b b upper ASF bound . Supremum (resp. infimum) of the set of all lower (resp. upper) ASF bounds is called optimal lower (resp. optimal upper ) ASF bound. From the theory of bounded linear operators between Banach spaces, one sees that optimal lower frame bound is ∥ S f , τ − 1 ∥ − 1 \lVert S_{f,\tau}^{-1}\rVert^{-1} ∥ S f , τ − 1 ∥ − 1 ∥ S f , τ ∥ \lVert S_{f,\tau}\rVert ∥ S f , τ ∥ (22) .
Example 4.1.3. ([111 ] ) Let 2 < p < ∞ 2 < p < \infty 2 < p < ∞ { λ n } n \{\lambda_n\}_n { λ n } n a ∈ R a \in\mathbb{R} a ∈ R
T a : L p ( R ) ∋ f ↦ T a f ∈ L p ( R ) ; T a f : R ∋ x ↦ ( T a f ) ( x ) : = f ( x − a ) ∈ C . T_a : \mathcal{L}^p(\mathbb{R}) \ni f \mapsto T_a f \in\mathcal{L}^p(\mathbb{R});\quad T_a f : \mathbb{R} \ni x \mapsto(T_a f)(x) := f(x-a) \in\mathbb{C}. T a : L p ( R ) ∋ f ↦ T a f ∈ L p ( R ) ; T a f : R ∋ x ↦ ( T a f ) ( x ) := f ( x − a ) ∈ C . Then there exist ϕ ∈ L p ( R ) \phi\in\mathcal{L}^p(\mathbb{R}) ϕ ∈ L p ( R ) { f n } n \{f_n\}_n { f n } n f n ∈ ( L p ( R ) ) ∗ f_n \in(\mathcal{L}^p(\mathbb{R}))^* f n ∈ ( L p ( R ) ) ∗ ∀ n ∈ N \forall n \in\mathbb{N} ∀ n ∈ N ( { f n } n , { T λ n ϕ } n ) \left(\{f_n\}_n,\{T_{\lambda_n}\phi\}_n\right) ( { f n } n , { T λ n ϕ } n ) L p ( R ) \mathcal{L}^p(\mathbb{R}) L p ( R )
Definition 4.1.4. An ASF ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X p p p p p p p ∈ [ 1 , ∞ ) p \in[1,\infty) p ∈ [ 1 , ∞ )
(Analysis operator) θ f : X ∋ x ↦ θ f x : = { f n ( x ) } n ∈ ℓ p ( N ) and (24) \text{(Analysis operator)}\quad\theta_f:\mathcal{X} \ni x \mapsto\theta_f x:=\{f_n(x)\}_n \in\ell^p(\mathbb{N}) \text{ and} \tag*{(24)} (Analysis operator) θ f : X ∋ x ↦ θ f x := { f n ( x ) } n ∈ ℓ p ( N ) and (24) (Synthesis operator) θ τ : ℓ p ( N ) ∋ { a n } n ↦ θ τ { a n } n : = ∑ n = 1 ∞ a n τ n ∈ X (25) \text{(Synthesis operator)}\quad\theta_\tau:\ell^p(\mathbb{N}) \ni\{a_n\}_n \mapsto\theta_\tau\{a_n\}_n:=\sum_{n=1}^{\infty}a_n\tau_n \in\mathcal{X} \tag*{(25)} (Synthesis operator) θ τ : ℓ p ( N ) ∋ { a n } n ↦ θ τ { a n } n := n = 1 ∑ ∞ a n τ n ∈ X (25) are well-defined bounded linear operators. A Schauder frame which is a p p p simple p p p or Parseval p p p .
It can be easily observed that a p p p p p p X = K \mathcal{X}=\mathbb{K} X = K τ n : = 1 n 2 \tau_n:=\frac{1}{n^2} τ n := n 2 1 f n ( x ) = x , ∀ x ∈ K , ∀ n ∈ N f_n(x)=x,\forall x\in\mathbb{K},\forall n\in\mathbb{N} f n ( x ) = x , ∀ x ∈ K , ∀ n ∈ N ∑ n = 1 ∞ f n ( x ) τ n = π 2 6 x , ∀ x ∈ K \sum_{n=1}^{\infty}f_n(x)\tau_n=\frac{\pi^2}{6}x,\ \forall x\in\mathbb{K} ∑ n = 1 ∞ f n ( x ) τ n = 6 π 2 x , ∀ x ∈ K ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X x ∈ K x\in\mathbb{K} x ∈ K p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ )
∑ n = 1 m ∣ f n ( x ) ∣ p = m ∣ x ∣ p → ∞ as m → ∞ . \sum_{n=1}^{m}|f_n(x)|^p=m|x|^p\to\infty\quad\text{as }m\to\infty. n = 1 ∑ m ∣ f n ( x ) ∣ p = m ∣ x ∣ p → ∞ as m → ∞. Thus { f n ( x ) } n ∉ ℓ p ( N ) \{f_n(x)\}_n\notin\ell^p(\mathbb{N}) { f n ( x ) } n ∈ / ℓ p ( N ) p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) [111 ] ). We observe that, in terms of inequalities, (4.1.3) and (4.1.4) say that there exist c , d > 0 c,d>0 c , d > 0
( ∑ n = 1 ∞ ∣ f n ( x ) ∣ p ) 1 p ≤ c ∥ x ∥ , ∀ x ∈ X and (26) \left(\sum_{n=1}^{\infty}|f_n(x)|^p\right)^{\frac{1}{p}}\le c\|x\|,\quad\forall x\in\mathcal{X}\text{ and} \tag*{(26)} ( n = 1 ∑ ∞ ∣ f n ( x ) ∣ p ) p 1 ≤ c ∥ x ∥ , ∀ x ∈ X and (26) ∥ ∑ n = 1 ∞ a n τ n ∥ ≤ d ( ∑ n = 1 ∞ ∣ a n ∣ p ) 1 p , ∀ { a n } n ∈ ℓ p ( N ) . (27) \left\|\sum_{n=1}^{\infty}a_n\tau_n\right\|\le d\left(\sum_{n=1}^{\infty}|a_n|^p\right)^{\frac{1}{p}},\quad\forall\{a_n\}_n\in\ell^p(\mathbb{N}). \tag*{(27)} n = 1 ∑ ∞ a n τ n ≤ d ( n = 1 ∑ ∞ ∣ a n ∣ p ) p 1 , ∀ { a n } n ∈ ℓ p ( N ) . (27) We now give various examples of p-ASFs.
Example 4.1.5. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X V U VU V U { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n
f n : = ζ n U , τ n : = V e n , ∀ n ∈ N . f_n:=\zeta_nU,\qquad\tau_n:=Ve_n,\qquad\forall n\in\mathbb{N}. f n := ζ n U , τ n := V e n , ∀ n ∈ N . Then ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X U : ℓ p ( N ) → ℓ p ( N ) U:\ell^p(\mathbb{N})\to\ell^p(\mathbb{N}) U : ℓ p ( N ) → ℓ p ( N ) ( { ζ n U } n , { U − 1 e n } n ) (\{\zeta_nU\}_n,\{U^{-1}e_n\}_n) ({ ζ n U } n , { U − 1 e n } n ) p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
Example 4.1.6. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) { τ n } n = 1 m \{\tau_n\}_{n=1}^m { τ n } n = 1 m X \mathcal{X} X { f n } n = 1 m \{f_n\}_{n=1}^m { f n } n = 1 m X ∗ \mathcal{X}^* X ∗ ( { f n } n = 1 m , { τ n } n = 1 m ) (\{f_n\}_{n=1}^m,\{\tau_n\}_{n=1}^m) ({ f n } n = 1 m , { τ n } n = 1 m ) p p p X \mathcal{X} X X \mathcal{X} X X ∋ x ↦ ∑ n = 1 m f n ( x ) τ n ∈ X \mathcal{X}\ni x\mapsto\sum_{n=1}^m f_n(x)\tau_n\in\mathcal{X} X ∋ x ↦ ∑ n = 1 m f n ( x ) τ n ∈ X x ∈ X x\in\mathcal{X} x ∈ X ∑ n = 1 m f n ( x ) τ n = 0 \sum_{n=1}^m f_n(x)\tau_n=0 ∑ n = 1 m f n ( x ) τ n = 0 { τ n } n = 1 m \{\tau_n\}_{n=1}^m { τ n } n = 1 m X \mathcal{X} X f 1 ( x ) = ⋯ = f n ( x ) = 0 f_1(x)=\cdots=f_n(x)=0 f 1 ( x ) = ⋯ = f n ( x ) = 0 f ( x ) = 0 , ∀ f ∈ X ∗ f(x)=0,\forall f\in\mathcal{X}^* f ( x ) = 0 , ∀ f ∈ X ∗ x = 0 x=0 x = 0 ( { f n } n = 1 m , { τ n } n = 1 m ) (\{f_n\}_{n=1}^m,\{\tau_n\}_{n=1}^m) ({ f n } n = 1 m , { τ n } n = 1 m ) p p p X \mathcal{X} X
Example 4.1.7. Recall that a spanning set is a frame for a finite dimensional Hilbert space (Han et al. [128 ] ). We now generalize this for p-ASFs. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) X \mathcal{X} X { τ n } n = 1 m \{\tau_n\}_{n=1}^m { τ n } n = 1 m X \mathcal{X} X { f n } n = 1 m \{f_n\}_{n=1}^m { f n } n = 1 m X ∗ \mathcal{X}^* X ∗ ( { f n } n = 1 m , { τ n } n = 1 m ) (\{f_n\}_{n=1}^m,\{\tau_n\}_{n=1}^m) ({ f n } n = 1 m , { τ n } n = 1 m ) p p p X \mathcal{X} X { τ n } n = 1 m \{\tau_n\}_{n=1}^m { τ n } n = 1 m X \mathcal{X} X { τ n } n = 1 m \{\tau_n\}_{n=1}^m { τ n } n = 1 m { τ n } n = 1 r \{\tau_n\}_{n=1}^r { τ n } n = 1 r X \mathcal{X} X { f n } n = 1 r \{f_n\}_{n=1}^r { f n } n = 1 r { τ n } n = 1 r \{\tau_n\}_{n=1}^r { τ n } n = 1 r U , V : X → X U,V:\mathcal{X}\to\mathcal{X} U , V : X → X V U VU V U f r + 1 = ⋯ = f m = 0 f_{r+1}=\cdots=f_m=0 f r + 1 = ⋯ = f m = 0 ( { f n U } n = 1 m , { V τ n } n = 1 m ) (\{f_nU\}_{n=1}^m,\{V\tau_n\}_{n=1}^m) ({ f n U } n = 1 m , { V τ n } n = 1 m ) p p p X \mathcal{X} X
Example 4.1.8. Let X \mathcal{X} X { ω n } n \{\omega_n\}_n { ω n } n { g n } n \{g_n\}_n { g n } n { e n } n \{e_n\}_n { e n } n U , V : X → X U,V:\mathcal{X}\to\mathcal{X} U , V : X → X
bounded linear operators such that V U VU V U
f n : = g n U , τ n : = V ω n , ∀ n ∈ N . f_n := g_nU,\qquad\tau_n := V\omega_n,\qquad\forall n\in\mathbb{N}. f n := g n U , τ n := V ω n , ∀ n ∈ N . Then ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X V U = I X VU=I_{\mathcal{X}} V U = I X ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X
Example 4.1.9. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X V U VU V U { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n
f n : = ζ n U , τ n : = V e n , ∀ n ∈ N . f_n := \zeta_nU,\qquad\tau_n := Ve_n,\qquad\forall n\in\mathbb{N}. f n := ζ n U , τ n := V e n , ∀ n ∈ N . Then ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X
Now we have Banach space analogous of Theorem 1.2.36.
Theorem 4.1.10. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X
(i) We have
x = ∑ n = 1 ∞ ( f n S f , τ − 1 ) ( x ) τ n = ∑ n = 1 ∞ f n ( x ) S f , τ − 1 τ n , ∀ x ∈ X . (28) x=\sum_{n=1}^{\infty}(f_nS_{f,\tau}^{-1})(x)\tau_n=\sum_{n=1}^{\infty}f_n(x)S_{f,\tau}^{-1}\tau_n,\qquad\forall x\in\mathcal{X}. \tag*{(28)} x = n = 1 ∑ ∞ ( f n S f , τ − 1 ) ( x ) τ n = n = 1 ∑ ∞ f n ( x ) S f , τ − 1 τ n , ∀ x ∈ X . (28) (ii) ( { f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) \left(\{f_nS_{f,\tau}^{-1}\}_n,\{S_{f,\tau}^{-1}\tau_n\}_n\right) ( { f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) p p p X \mathcal{X} X
(iii) The analysis operator θ f : X ∋ x ↦ { f n ( x ) } n ∈ ℓ p ( N ) \theta_f:\mathcal{X}\ni x\mapsto\{f_n(x)\}_n\in\ell^p(\mathbb{N}) θ f : X ∋ x ↦ { f n ( x ) } n ∈ ℓ p ( N )
(iv) The synthesis operator θ τ : ℓ p ( N ) ∋ { a n } n ↦ ∑ n = 1 ∞ a n τ n ∈ X \theta_\tau:\ell^p(\mathbb{N})\ni\{a_n\}_n\mapsto\sum_{n=1}^{\infty}a_n\tau_n\in\mathcal{X} θ τ : ℓ p ( N ) ∋ { a n } n ↦ ∑ n = 1 ∞ a n τ n ∈ X
(v) Frame operator splits as S f , τ = θ τ θ f S_{f,\tau}=\theta_\tau\theta_f S f , τ = θ τ θ f
(vi) P f , τ : = θ f S f , τ − 1 θ τ : ℓ p ( N ) → ℓ p ( N ) P_{f,\tau}:=\theta_fS_{f,\tau}^{-1}\theta_\tau:\ell^p(\mathbb{N})\to\ell^p(\mathbb{N}) P f , τ := θ f S f , τ − 1 θ τ : ℓ p ( N ) → ℓ p ( N ) θ f ( X ) \theta_f(\mathcal{X}) θ f ( X )
Proof. First follows from the continuity and linearity of S f , τ − 1 S_{f,\tau}^{-1} S f , τ − 1 S f , τ S_{f,\tau} S f , τ S f , τ S_{f,\tau} S f , τ θ f \theta_f θ f θ τ \theta_\tau θ τ
Now we can derive a generalization of Theorem 1.2.39 for Banach spaces.
Theorem 4.1.11. Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X
f n = ζ n U , τ n = V e n , ∀ n ∈ N , f_n=\zeta_nU,\qquad\tau_n=Ve_n,\qquad\forall n\in\mathbb{N}, f n = ζ n U , τ n = V e n , ∀ n ∈ N , where U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X V U VU V U
Proof. (⇐ \Leftarrow ⇐ θ f \theta_f θ f θ τ \theta_\tau θ τ x ∈ X x\in\mathcal{X} x ∈ X
S f , τ x = ∑ n = 1 ∞ f n ( x ) τ n = ∑ n = 1 ∞ ζ n ( U x ) V e n = V ( ∑ n = 1 ∞ ζ n ( U x ) e n ) = V U x . (29) S_{f,\tau}x=\sum_{n=1}^{\infty}f_n(x)\tau_n=\sum_{n=1}^{\infty}\zeta_n(Ux)Ve_n=V\left(\sum_{n=1}^{\infty}\zeta_n(Ux)e_n\right)=VUx. \tag*{(29)} S f , τ x = n = 1 ∑ ∞ f n ( x ) τ n = n = 1 ∑ ∞ ζ n ( U x ) V e n = V ( n = 1 ∑ ∞ ζ n ( U x ) e n ) = V U x . (29) Hence S f , τ S_{f,\tau} S f , τ
(⇒ \Rightarrow ⇒ U : = θ f U:=\theta_f U := θ f V : = θ τ V:=\theta_\tau V := θ τ ζ n U x = ζ n θ f x = ζ n ( { f k ( x ) } k ) = f n ( x ) \zeta_nUx=\zeta_n\theta_f x=\zeta_n(\{f_k(x)\}_k)=f_n(x) ζ n U x = ζ n θ f x = ζ n ({ f k ( x ) } k ) = f n ( x ) ∀ x ∈ X \forall x\in\mathcal{X} ∀ x ∈ X V e n = θ τ e n = τ n Ve_n=\theta_\tau e_n=\tau_n V e n = θ τ e n = τ n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N V U = θ τ θ f = S f , τ VU=\theta_\tau\theta_f=S_{f,\tau} V U = θ τ θ f = S f , τ
Note that Theorem 4.1.11 generalizes Theorem 1.2.39. In fact, in the case of Hilbert spaces, Theorem 4.1.11 reads as “A sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H \mathcal{H} H T : ℓ 2 ( N ) → H T:\ell^2(\mathbb{N})\to\mathcal{H} T : ℓ 2 ( N ) → H T e n = τ n Te_n=\tau_n T e n = τ n n ∈ N n\in\mathbb{N} n ∈ N T T ∗ TT^* T T ∗ T T ∗ TT^* T T ∗ T T T
Since every separable Hilbert space admits an orthonormal basis, the existence of orthonormal basis in Theorem 1.2.40 is automatic. On the other hand, Enflo showed that there are separable Banach spaces which do not have Schauder basis [94 , 134 ] . Thus to obtain analogous of Theorem 1.2.40 for Banach spaces, we need to impose condition on X \mathcal{X} X
Theorem 4.1.12. Assume that X \mathcal{X} X { ω n } n \{\omega_n\}_n { ω n } n { g n } n \{g_n\}_n { g n } n { ω n } n \{\omega_n\}_n { ω n } n
{ g n ( x ) } n ∈ ℓ p ( N ) , ∀ x ∈ X . (30) \{g_n(x)\}_n\in\ell^p(\mathbb{N}),\quad\forall x\in\mathcal{X}. \tag*{(30)} { g n ( x ) } n ∈ ℓ p ( N ) , ∀ x ∈ X . (30) Then a pair ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
f n = g n U , τ n = V ω n , ∀ n ∈ N , f_n=g_nU,\quad\tau_n=V\omega_n,\quad\forall n\in\mathbb{N}, f n = g n U , τ n = V ω n , ∀ n ∈ N , where U , V : X → X U,V:\mathcal{X}\to\mathcal{X} U , V : X → X V U VU V U
Proof. (⇐ \Leftarrow ⇐
(⇒ \Rightarrow ⇒ T T T
T : X ∋ ∑ n = 1 ∞ a n ω n ↦ ∑ n = 1 ∞ a n e n ∈ ℓ p ( N ) . T:\mathcal{X}\ni\sum_{n=1}^{\infty}a_n\omega_n\mapsto\sum_{n=1}^{\infty}a_ne_n\in\ell^p(\mathbb{N}). T : X ∋ n = 1 ∑ ∞ a n ω n ↦ n = 1 ∑ ∞ a n e n ∈ ℓ p ( N ) . Assumption (4.1.9) then says that T T T T − 1 T^{-1} T − 1
ℓ p ( N ) ∋ ∑ n = 1 ∞ b n e n ↦ ∑ n = 1 ∞ b n ω n ∈ X . \ell^p(\mathbb{N})\ni\sum_{n=1}^{\infty}b_ne_n\mapsto\sum_{n=1}^{\infty}b_n\omega_n\in\mathcal{X}. ℓ p ( N ) ∋ n = 1 ∑ ∞ b n e n ↦ n = 1 ∑ ∞ b n ω n ∈ X . Define U : = T − 1 θ f U:=T^{-1}\theta_f U := T − 1 θ f V : = θ τ T V:=\theta_\tau T V := θ τ T U , V U,V U , V V U = ( θ τ T ) ( T − 1 θ f ) = θ τ θ f = S f , τ VU=(\theta_\tau T)(T^{-1}\theta_f)=\theta_\tau\theta_f=S_{f,\tau} V U = ( θ τ T ) ( T − 1 θ f ) = θ τ θ f = S f , τ x ∈ X x\in\mathcal{X} x ∈ X
( g n U ) ( x ) = g n ( T − 1 θ f x ) = g n ( T − 1 ( { f k ( x ) } k ) ) = g n ( ∑ k = 1 ∞ f k ( x ) T − 1 e k ) = g n ( ∑ k = 1 ∞ f k ( x ) ω k ) = ∑ k = 1 ∞ f k ( x ) g n ( ω k ) = f n ( x ) , ∀ x ∈ X \begin{aligned}
(g_nU)(x)&=g_n(T^{-1}\theta_fx)=g_n\left(T^{-1}(\{f_k(x)\}_k)\right)=g_n\left(\sum_{k=1}^{\infty}f_k(x)T^{-1}e_k\right)\\
&=g_n\left(\sum_{k=1}^{\infty}f_k(x)\omega_k\right)=\sum_{k=1}^{\infty}f_k(x)g_n(\omega_k)=f_n(x),\qquad\forall x\in\mathcal{X}
\end{aligned} ( g n U ) ( x ) = g n ( T − 1 θ f x ) = g n ( T − 1 ({ f k ( x ) } k ) ) = g n ( k = 1 ∑ ∞ f k ( x ) T − 1 e k ) = g n ( k = 1 ∑ ∞ f k ( x ) ω k ) = k = 1 ∑ ∞ f k ( x ) g n ( ω k ) = f n ( x ) , ∀ x ∈ X and V ω n = θ τ T ω n = θ τ e n = τ n , ∀ n ∈ N V\omega_n=\theta_\tau T\omega_n=\theta_\tau e_n=\tau_n,\forall n\in\mathbb{N} V ω n = θ τ T ω n = θ τ e n = τ n , ∀ n ∈ N
DUAL FRAMES FOR p-APPROXIMATE SCHAUDER FRAMES
Equation (4.1.7) motivates us to define the notion of dual frame as follows.
Definition 4.2.1. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X p p p ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) X \mathcal{X} X dual p-ASF for ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n )
x = ∑ n = 1 ∞ g n ( x ) τ n = ∑ n = 1 ∞ f n ( x ) ω n , ∀ x ∈ X . x=\sum_{n=1}^{\infty}g_n(x)\tau_n=\sum_{n=1}^{\infty}f_n(x)\omega_n,\qquad\forall x\in\mathcal{X}. x = n = 1 ∑ ∞ g n ( x ) τ n = n = 1 ∑ ∞ f n ( x ) ω n , ∀ x ∈ X . Note that dual frames always exist. In fact, the Equation (4.1.7) shows that the frame ( { f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) (\{f_nS_{f,\tau}^{-1}\}_n,\{S_{f,\tau}^{-1}\tau_n\}_n) ({ f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) ( { f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) (\{f_nS_{f,\tau}^{-1}\}_n,\{S_{f,\tau}^{-1}\tau_n\}_n) ({ f n S f , τ − 1 } n , { S f , τ − 1 τ n } n ) canonical dual for ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n )
Theorem 4.2.2. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X a a a b b b
(i) The canonical dual p p p p p p ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n )
(ii) 1 b , 1 a \frac{1}{b},\frac{1}{a} b 1 , a 1 ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n )
(iii) If a , b a,b a , b ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) 1 b , 1 a \frac{1}{b},\frac{1}{a} b 1 , a 1
One can naturally ask when a p p p
Proposition 4.2.3. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X f k ( τ n ) = δ k , n , ∀ k , n ∈ N f_k(\tau_n)=\delta_{k,n},\forall k,n\in\mathbb{N} f k ( τ n ) = δ k , n , ∀ k , n ∈ N ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n )
Proof. Let ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { u n } n , { ρ n } n ) \left(\{u_n\}_n,\{\rho_n\}_n\right) ( { u n } n , { ρ n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
∑ n = 1 ∞ ( g n ( x ) − u n ( x ) ) τ n = 0 = ∑ n = 1 ∞ f n ( x ) ( ω n − ρ n ) , ∀ x ∈ X . \sum_{n=1}^{\infty}(g_n(x)-u_n(x))\tau_n=0=\sum_{n=1}^{\infty}f_n(x)(\omega_n-\rho_n),\qquad\forall x\in\mathcal{X}. n = 1 ∑ ∞ ( g n ( x ) − u n ( x )) τ n = 0 = n = 1 ∑ ∞ f n ( x ) ( ω n − ρ n ) , ∀ x ∈ X . First equality gives g n = u n , ∀ n ∈ N g_n=u_n,\forall n\in\mathbb{N} g n = u n , ∀ n ∈ N τ k \tau_k τ k ω k = ρ k \omega_k=\rho_k ω k = ρ k k k k
We now characterize dual frames by using analysis and synthesis operators.
Proposition 4.2.4. For two p-ASFs ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X
(i) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
(ii) θ τ θ g = θ ω θ f = I X \theta_\tau\theta_g=\theta_\omega\theta_f=I_{\mathcal{X}} θ τ θ g = θ ω θ f = I X
Like Lemmas 1.2.42, 1.2.43 and Theorem 1.2.44, we now characterize dual frames using standard Schauder basis for ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
Lemma 4.2.5. Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
g n = ζ n U , ω n = V e n , ∀ n ∈ N , g_n=\zeta_nU,\qquad\omega_n=Ve_n,\qquad\forall n\in\mathbb{N}, g n = ζ n U , ω n = V e n , ∀ n ∈ N , where U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) θ τ \theta_\tau θ τ V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X θ f \theta_f θ f V U VU V U
Proof. (⇐ \Leftarrow ⇐ ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) θ τ θ g = θ τ U = I X \theta_\tau\theta_g=\theta_\tau U=I_{\mathcal{X}} θ τ θ g = θ τ U = I X θ ω θ f = V θ f = I X \theta_\omega\theta_f=V\theta_f=I_{\mathcal{X}} θ ω θ f = V θ f = I X
(⇒ \Rightarrow ⇒ ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) θ τ θ g = I X \theta_\tau\theta_g=I_{\mathcal{X}} θ τ θ g = I X θ ω θ f = I X \theta_\omega\theta_f=I_{\mathcal{X}} θ ω θ f = I X U : = θ g , V : = θ ω U:=\theta_g,V:=\theta_\omega U := θ g , V := θ ω U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) θ τ \theta_\tau θ τ V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X θ f \theta_f θ f V U = θ ω θ g = S g , ω VU=\theta_\omega\theta_g=S_{g,\omega} V U = θ ω θ g = S g , ω
( ζ n U ) x = ζ n ( ∑ k = 1 ∞ g k ( x ) e k ) = ∑ k = 1 ∞ g k ( x ) ζ n ( e k ) = g n ( x ) , ∀ x ∈ X (\zeta_nU)x=\zeta_n\left(\sum_{k=1}^{\infty}g_k(x)e_k\right)=\sum_{k=1}^{\infty}g_k(x)\zeta_n(e_k)=g_n(x),\qquad\forall x\in\mathcal{X} ( ζ n U ) x = ζ n ( k = 1 ∑ ∞ g k ( x ) e k ) = k = 1 ∑ ∞ g k ( x ) ζ n ( e k ) = g n ( x ) , ∀ x ∈ X and V e n = θ ω e n = ω n , ∀ n ∈ N Ve_n=\theta_\omega e_n=\omega_n,\forall n\in\mathbb{N} V e n = θ ω e n = ω n , ∀ n ∈ N
Lemma 4.2.6. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X
(i) R : X → ℓ p ( N ) R:\mathcal{X}\to\ell^p(\mathbb{N}) R : X → ℓ p ( N ) θ τ \theta_\tau θ τ
R = θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U R=\theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U R = θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U where U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N )
(ii) L : ℓ p ( N ) → X L:\ell^p(\mathbb{N})\to\mathcal{X} L : ℓ p ( N ) → X θ f \theta_f θ f
L = S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) , L=S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right), L = S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) , where V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X
Proof. (i) ( ⇐ ) (\Leftarrow) ( ⇐ ) θ τ ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) = I X + θ τ U − I X θ τ U = I X \theta_\tau\left(\theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U\right)=I_{\mathcal{X}}+\theta_\tau U-I_{\mathcal{X}}\theta_\tau U=I_{\mathcal{X}} θ τ ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) = I X + θ τ U − I X θ τ U = I X θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U \theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U θ τ \theta_\tau θ τ
( ⇒ ) (\Rightarrow) ( ⇒ ) U : = R U:=R U := R θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U = θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) R = θ f S f , τ − 1 + R − θ f S f , τ − 1 = R \theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U=\theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)R=\theta_fS_{f,\tau}^{-1}+R-\theta_fS_{f,\tau}^{-1}=R θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U = θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) R = θ f S f , τ − 1 + R − θ f S f , τ − 1 = R
(ii) ( ⇐ ) (\Leftarrow) ( ⇐ ) ( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) θ f = I X + V θ f − V θ f I X = I X \left(S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)\right)\theta_f=I_{\mathcal{X}}+V\theta_f-V\theta_fI_{\mathcal{X}}=I_{\mathcal{X}} ( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) θ f = I X + V θ f − V θ f I X = I X S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right) S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) θ f \theta_f θ f
( ⇒ ) (\Rightarrow) ( ⇒ ) V : = L V:=L V := L S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) = S f , τ − 1 θ τ + L ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) = S f , τ − 1 θ τ + L − S f , τ − 1 θ τ = L S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)=S_{f,\tau}^{-1}\theta_\tau+L\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)=S_{f,\tau}^{-1}\theta_\tau+L-S_{f,\tau}^{-1}\theta_\tau=L S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) = S f , τ − 1 θ τ + L ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) = S f , τ − 1 θ τ + L − S f , τ − 1 θ τ = L
:::
Theorem 4.2.7. Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X p p p ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
g n = f n S f , τ − 1 + ζ n U − f n S f , τ − 1 θ τ U , g_n=f_nS_{f,\tau}^{-1}+\zeta_nU-f_nS_{f,\tau}^{-1}\theta_\tau U, g n = f n S f , τ − 1 + ζ n U − f n S f , τ − 1 θ τ U , ω n = S f , τ − 1 τ n + V e n − V θ f S f , τ − 1 τ n , ∀ n ∈ N \omega_n=S_{f,\tau}^{-1}\tau_n+Ve_n-V\theta_fS_{f,\tau}^{-1}\tau_n,\qquad\forall n\in\mathbb{N} ω n = S f , τ − 1 τ n + V e n − V θ f S f , τ − 1 τ n , ∀ n ∈ N such that the operator
S f , τ − 1 + V U − V θ f S f , τ − 1 θ τ U S_{f,\tau}^{-1}+VU-V\theta_fS_{f,\tau}^{-1}\theta_\tau U S f , τ − 1 + V U − V θ f S f , τ − 1 θ τ U is bounded invertible, where U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X
Proof. Lemmas 4.2.5 and 4.2.6 give the characterization of dual frame as
g n = ζ n θ f S f , τ − 1 + ζ n U − ζ n θ f S f , τ − 1 θ τ U = f n S f , τ − 1 + ζ n U − f n S f , τ − 1 θ τ U , g_n=\zeta_n\theta_fS_{f,\tau}^{-1}+\zeta_nU-\zeta_n\theta_fS_{f,\tau}^{-1}\theta_\tau U=f_nS_{f,\tau}^{-1}+\zeta_nU-f_nS_{f,\tau}^{-1}\theta_\tau U, g n = ζ n θ f S f , τ − 1 + ζ n U − ζ n θ f S f , τ − 1 θ τ U = f n S f , τ − 1 + ζ n U − f n S f , τ − 1 θ τ U , ω n = S f , τ − 1 θ f e n + V e n − V θ f S f , τ − 1 θ f e n = S f , τ − 1 τ n + V e n − V θ f S f , τ − 1 τ n , ∀ n ∈ N \omega_n=S_{f,\tau}^{-1}\theta_f e_n+Ve_n-V\theta_fS_{f,\tau}^{-1}\theta_f e_n=S_{f,\tau}^{-1}\tau_n+Ve_n-V\theta_fS_{f,\tau}^{-1}\tau_n,\qquad\forall n\in\mathbb{N} ω n = S f , τ − 1 θ f e n + V e n − V θ f S f , τ − 1 θ f e n = S f , τ − 1 τ n + V e n − V θ f S f , τ − 1 τ n , ∀ n ∈ N such that the operator
( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) \left(S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)\right)\left(\theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U\right) ( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) is bounded invertible, where U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X
( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) = S f , τ − 1 + V U − V θ f S f , τ − 1 θ τ U . \begin{aligned}
&\left(S_{f,\tau}^{-1}\theta_\tau+V\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)\right)\left(\theta_fS_{f,\tau}^{-1}+\left(I_{\ell^p(\mathbb{N})}-\theta_fS_{f,\tau}^{-1}\theta_\tau\right)U\right)\\
&=S_{f,\tau}^{-1}+VU-V\theta_fS_{f,\tau}^{-1}\theta_\tau U.
\end{aligned} ( S f , τ − 1 θ τ + V ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) ) ( θ f S f , τ − 1 + ( I ℓ p ( N ) − θ f S f , τ − 1 θ τ ) U ) = S f , τ − 1 + V U − V θ f S f , τ − 1 θ τ U . □ \square □
We know that a bounded linear operator from ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) H \mathcal{H} H Theorem 1.2.58 ). Thus, for Hilbert spaces, Theorem 4.2.7 becomes Theorem 1.2.44 .
SIMILARITY FOR p-APPROXIMATE SCHAUDER FRAMES
We define Definition 1.2.45 to Banach spaces as follows.
Definition 4.3.1. Two p p p ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X
g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N . g_n=f_nT_{f,g},\qquad\omega_n=T_{\tau,\omega}\tau_n,\qquad\forall n\in\mathbb{N}. g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N . Since the operators giving similarity are bounded invertible, the notion of similarity is symmetric. Further, a routine calculation shows that it is an equivalence relation (hence the name equivalent) on the set
{ ( { f n } n , { τ n } n ) : ( { f n } n , { τ n } n ) is a p -ASF for X } . \left\{\left(\{f_n\}_n,\{\tau_n\}_n\right):\left(\{f_n\}_n,\{\tau_n\}_n\right)\text{ is a }p\text{-ASF for }\mathcal{X}\right\}. { ( { f n } n , { τ n } n ) : ( { f n } n , { τ n } n ) is a p -ASF for X } . We now characterize similarity using just operators. In the sequel, given a p p p ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) P f , τ : = θ f S f , τ − 1 θ τ P_{f,\tau}:=\theta_fS_{f,\tau}^{-1}\theta_\tau P f , τ := θ f S f , τ − 1 θ τ
Theorem 4.3.2. For two p p p ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X
(i) g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N g_n=f_nT_{f,g},\ \omega_n=T_{\tau,\omega}\tau_n,\ \forall n\in\mathbb{N} g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X
(ii) θ g = θ f T f , g , θ ω = T τ , ω θ τ \theta_g=\theta_fT_{f,g},\ \theta_\omega=T_{\tau,\omega}\theta_\tau θ g = θ f T f , g , θ ω = T τ , ω θ τ T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X
(iii) P g , ω = P f , τ P_{g,\omega}=P_{f,\tau} P g , ω = P f , τ
If one of the above conditions is satisfied, then invertible operators in (i) and (ii) are unique and are given by T f , g = S f , τ − 1 θ τ θ g , T τ , ω = θ ω θ f S f , τ − 1 T_{f,g}=S_{f,\tau}^{-1}\theta_\tau\theta_g,T_{\tau,\omega}=\theta_\omega\theta_fS_{f,\tau}^{-1} T f , g = S f , τ − 1 θ τ θ g , T τ , ω = θ ω θ f S f , τ − 1 ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) p p p T τ , ω T f , g = I X T_{\tau,\omega}T_{f,g}=I_{\mathcal{X}} T τ , ω T f , g = I X T f , g T τ , ω = I X T_{f,g}T_{\tau,\omega}=I_{\mathcal{X}} T f , g T τ , ω = I X
Proof. (i) ⇒ \Rightarrow ⇒ θ g x = { g n ( x ) } n = { f n ( T f , g x ) } n = θ f ( T f , g x ) , ∀ x ∈ X , θ ω ( { a n } n ) = ∑ n = 1 ∞ a n ω n = ∑ n = 1 ∞ a n T τ , ω τ n = T τ , ω ( θ τ ( { a n } n ) ) , ∀ { a n } n ∈ ℓ p ( N ) \theta_gx=\{g_n(x)\}_n=\{f_n(T_{f,g}x)\}_n=\theta_f(T_{f,g}x),\forall x\in\mathcal{X},\ \theta_\omega(\{a_n\}_n)=\sum_{n=1}^{\infty}a_n\omega_n=\sum_{n=1}^{\infty}a_nT_{\tau,\omega}\tau_n=T_{\tau,\omega}(\theta_\tau(\{a_n\}_n)),\forall\{a_n\}_n\in\ell^p(\mathbb{N}) θ g x = { g n ( x ) } n = { f n ( T f , g x ) } n = θ f ( T f , g x ) , ∀ x ∈ X , θ ω ({ a n } n ) = ∑ n = 1 ∞ a n ω n = ∑ n = 1 ∞ a n T τ , ω τ n = T τ , ω ( θ τ ({ a n } n )) , ∀ { a n } n ∈ ℓ p ( N )
(ii) ⇒ \Rightarrow ⇒ S g , ω = θ ω θ g = T τ , ω θ τ θ f T f , g = T τ , ω S f , τ T f , g S_{g,\omega}=\theta_\omega\theta_g=T_{\tau,\omega}\theta_\tau\theta_fT_{f,g}=T_{\tau,\omega}S_{f,\tau}T_{f,g} S g , ω = θ ω θ g = T τ , ω θ τ θ f T f , g = T τ , ω S f , τ T f , g
P g , ω = θ g S g , ω − 1 θ ω = ( θ f T f , g ) ( T τ , ω S f , τ T f , g ) − 1 ( T τ , ω θ τ ) = P f , τ . P_{g,\omega}=\theta_gS_{g,\omega}^{-1}\theta_\omega=(\theta_fT_{f,g})(T_{\tau,\omega}S_{f,\tau}T_{f,g})^{-1}(T_{\tau,\omega}\theta_\tau)=P_{f,\tau}. P g , ω = θ g S g , ω − 1 θ ω = ( θ f T f , g ) ( T τ , ω S f , τ T f , g ) − 1 ( T τ , ω θ τ ) = P f , τ . (ii) ⇒ \Rightarrow ⇒ ∑ n = 1 ∞ g n ( x ) e n = θ g ( x ) = θ f ( T f , g x ) = ∑ n = 1 ∞ f n ( T f , g x ) e n , ∀ x ∈ X \sum_{n=1}^{\infty}g_n(x)e_n=\theta_g(x)=\theta_f(T_{f,g}x)=\sum_{n=1}^{\infty}f_n(T_{f,g}x)e_n,\forall x\in\mathcal{X} ∑ n = 1 ∞ g n ( x ) e n = θ g ( x ) = θ f ( T f , g x ) = ∑ n = 1 ∞ f n ( T f , g x ) e n , ∀ x ∈ X
(iii) ⇒ \Rightarrow ⇒ θ g = P g , ω θ g = P f , τ θ g = θ f ( S f , τ − 1 θ τ θ g ) \theta_g=P_{g,\omega}\theta_g=P_{f,\tau}\theta_g=\theta_f(S_{f,\tau}^{-1}\theta_\tau\theta_g) θ g = P g , ω θ g = P f , τ θ g = θ f ( S f , τ − 1 θ τ θ g )
θ ω = θ ω P g , ω = θ ω P f , τ = ( θ ω θ f S f , τ − 1 ) θ τ . \theta_\omega=\theta_\omega P_{g,\omega}=\theta_\omega P_{f,\tau}=(\theta_\omega\theta_fS_{f,\tau}^{-1})\theta_\tau. θ ω = θ ω P g , ω = θ ω P f , τ = ( θ ω θ f S f , τ − 1 ) θ τ . We show that S f , τ − 1 θ τ θ g S_{f,\tau}^{-1}\theta_\tau\theta_g S f , τ − 1 θ τ θ g θ ω θ f S f , τ − 1 \theta_\omega\theta_fS_{f,\tau}^{-1} θ ω θ f S f , τ − 1
( S f , τ − 1 θ τ θ g ) ( S g , ω − 1 θ ω θ f ) = S f , τ − 1 θ τ P g , ω θ f = S f , τ − 1 θ τ P f , τ θ f = I X , (S_{f,\tau}^{-1}\theta_\tau\theta_g)(S_{g,\omega}^{-1}\theta_\omega\theta_f)=S_{f,\tau}^{-1}\theta_\tau P_{g,\omega}\theta_f=S_{f,\tau}^{-1}\theta_\tau P_{f,\tau}\theta_f=I_{\mathcal{X}}, ( S f , τ − 1 θ τ θ g ) ( S g , ω − 1 θ ω θ f ) = S f , τ − 1 θ τ P g , ω θ f = S f , τ − 1 θ τ P f , τ θ f = I X , ( S g , ω − 1 θ ω θ f ) ( S f , τ − 1 θ τ θ g ) = S g , ω − 1 θ ω P f , τ θ g = S g , ω − 1 θ ω P g , ω θ g = I X (S_{g,\omega}^{-1}\theta_\omega\theta_f)(S_{f,\tau}^{-1}\theta_\tau\theta_g)=S_{g,\omega}^{-1}\theta_\omega P_{f,\tau}\theta_g=S_{g,\omega}^{-1}\theta_\omega P_{g,\omega}\theta_g=I_{\mathcal{X}} ( S g , ω − 1 θ ω θ f ) ( S f , τ − 1 θ τ θ g ) = S g , ω − 1 θ ω P f , τ θ g = S g , ω − 1 θ ω P g , ω θ g = I X and
( θ ω θ f S f , τ − 1 ) ( θ τ θ g S g , ω − 1 ) = θ ω P f , τ θ g S g , ω − 1 = θ ω P g , ω θ g S g , ω − 1 = I X , (\theta_\omega\theta_fS_{f,\tau}^{-1})(\theta_\tau\theta_gS_{g,\omega}^{-1})=\theta_\omega P_{f,\tau}\theta_gS_{g,\omega}^{-1}=\theta_\omega P_{g,\omega}\theta_gS_{g,\omega}^{-1}=I_{\mathcal{X}}, ( θ ω θ f S f , τ − 1 ) ( θ τ θ g S g , ω − 1 ) = θ ω P f , τ θ g S g , ω − 1 = θ ω P g , ω θ g S g , ω − 1 = I X , ( θ τ θ g S g , ω − 1 ) ( θ ω θ f S f , τ − 1 ) = θ τ P g , ω θ f S f , τ − 1 = θ τ P f , τ θ f S f , τ − 1 = I X . (\theta_\tau\theta_gS_{g,\omega}^{-1})(\theta_\omega\theta_fS_{f,\tau}^{-1})=\theta_\tau P_{g,\omega}\theta_fS_{f,\tau}^{-1}=\theta_\tau P_{f,\tau}\theta_fS_{f,\tau}^{-1}=I_{\mathcal{X}}. ( θ τ θ g S g , ω − 1 ) ( θ ω θ f S f , τ − 1 ) = θ τ P g , ω θ f S f , τ − 1 = θ τ P f , τ θ f S f , τ − 1 = I X . Let T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N g_n=f_nT_{f,g},\omega_n=T_{\tau,\omega}\tau_n,\forall n\in\mathbb{N} g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N θ g = θ f T f , g \theta_g=\theta_fT_{f,g} θ g = θ f T f , g θ τ θ g = θ τ θ f T f , g = S f , τ T f , g \theta_\tau\theta_g=\theta_\tau\theta_fT_{f,g}=S_{f,\tau}T_{f,g} θ τ θ g = θ τ θ f T f , g = S f , τ T f , g T f , g = S f , τ − 1 θ τ θ g T_{f,g}=S_{f,\tau}^{-1}\theta_\tau\theta_g T f , g = S f , τ − 1 θ τ θ g θ ω = T τ , ω θ τ \theta_\omega=T_{\tau,\omega}\theta_\tau θ ω = T τ , ω θ τ θ ω θ f = T τ , ω θ τ θ f = T τ , ω S f , τ \theta_\omega\theta_f=T_{\tau,\omega}\theta_\tau\theta_f=T_{\tau,\omega}S_{f,\tau} θ ω θ f = T τ , ω θ τ θ f = T τ , ω S f , τ T τ , ω = θ ω θ f S f , τ − 1 T_{\tau,\omega}=\theta_\omega\theta_fS_{f,\tau}^{-1} T τ , ω = θ ω θ f S f , τ − 1
It is easy to see that for Hilbert spaces, Theorem 4.3.2 reduces to Theorem 1.2.46.
Definition 4.2.1 introduced the notion of dual frames. A twin notion associated is the notion of orthogonality.
Definition 4.3.3. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X p p p ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
0 = ∑ n = 1 ∞ g n ( x ) τ n = ∑ n = 1 ∞ f n ( x ) ω n , ∀ x ∈ X . 0=\sum_{n=1}^{\infty}g_n(x)\tau_n=\sum_{n=1}^{\infty}f_n(x)\omega_n,\quad\forall x\in\mathcal{X}. 0 = n = 1 ∑ ∞ g n ( x ) τ n = n = 1 ∑ ∞ f n ( x ) ω n , ∀ x ∈ X . Unlike duality, the notion orthogonality is symmetric but not reflexive. Further, dual p p p p p p ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { f n } n , { ω n } n ) \left(\{f_n\}_n,\{\omega_n\}_n\right) ( { f n } n , { ω n } n ) ( { g n } n , { τ n } n ) \left(\{g_n\}_n,\{\tau_n\}_n\right) ( { g n } n , { τ n } n ) p p p
Proposition 4.3.4. For two p p p ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) X \mathcal{X} X
(i) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
(ii) θ τ θ g = θ ω θ f = 0 \theta_\tau\theta_g=\theta_\omega\theta_f=0 θ τ θ g = θ ω θ f = 0
Usefulness of orthogonal frames is that we have interpolation result, i.e., these frames can be stitched along certain curves (in particular, on the unit circle centered at the origin) to get new frames.
Theorem 4.3.5. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) p p p X \mathcal{X} X A , B , C , D : X → X A,B,C,D:\mathcal{X}\to\mathcal{X} A , B , C , D : X → X C A + D B = I X CA+DB=I_{\mathcal{X}} C A + D B = I X
( { f n A + g n B } n , { C τ n + D ω n } n ) \left(\{f_nA+g_nB\}_n,\{C\tau_n+D\omega_n\}_n\right) ( { f n A + g n B } n , { C τ n + D ω n } n ) is a simple p p p X \mathcal{X} X a , b , c , d a,b,c,d a , b , c , d c a + d b = 1 ca+db=1 c a + d b = 1 ( { a f n + b g n } n , { c τ n + d ω n } n ) \left(\{af_n+bg_n\}_n,\{c\tau_n+d\omega_n\}_n\right) ( { a f n + b g n } n , { c τ n + d ω n } n ) p p p X \mathcal{X} X
Proof. By a calculation we find
θ f A + g B x = { ( f n A + g n B ) ( x ) } n = { f n ( A x ) } n + { g n ( B x ) } n = θ f ( A x ) + θ g ( B x ) , ∀ x ∈ X \theta_{fA+gB}x=\left\{\left(f_nA+g_nB\right)(x)\right\}_n=\{f_n(Ax)\}_n+\{g_n(Bx)\}_n=\theta_f(Ax)+\theta_g(Bx),\quad\forall x\in\mathcal{X} θ f A + g B x = { ( f n A + g n B ) ( x ) } n = { f n ( A x ) } n + { g n ( B x ) } n = θ f ( A x ) + θ g ( B x ) , ∀ x ∈ X and
θ C τ + D ω ( { a n } n ) = ∑ n = 1 ∞ a n ( C τ n + D ω n ) = C θ τ ( { a n } n ) + D θ ω ( { a n } n ) , ∀ { a n } n ∈ ℓ p ( N ) . \theta_{C\tau+D\omega}\left(\{a_n\}_n\right)=\sum_{n=1}^{\infty}a_n\left(C\tau_n+D\omega_n\right)=C\theta_\tau\left(\{a_n\}_n\right)+D\theta_\omega\left(\{a_n\}_n\right),\quad\forall\{a_n\}_n\in\ell^p(\mathbb{N}). θ C τ + D ω ( { a n } n ) = n = 1 ∑ ∞ a n ( C τ n + D ω n ) = C θ τ ( { a n } n ) + D θ ω ( { a n } n ) , ∀ { a n } n ∈ ℓ p ( N ) . So
S f A + g B , C τ + D ω = θ C τ + D ω θ f A + g B = ( C θ τ + D θ ω ) ( θ f A + θ g B ) = C θ τ θ f A + C θ τ θ g B + D θ ω θ f A + D θ ω θ g B = C S f , τ A + 0 + 0 + D S g , ω B = C I X A + D I X B = I X . \begin{aligned}
S_{fA+gB,C\tau+D\omega}=\theta_{C\tau+D\omega}\theta_{fA+gB}=(C\theta_\tau+D\theta_\omega)(\theta_fA+\theta_gB) \\
&= C\theta_\tau\theta_fA+C\theta_\tau\theta_gB+D\theta_\omega\theta_fA+D\theta_\omega\theta_gB \\
&= CS_{f,\tau}A+0+0+DS_{g,\omega}B=CI_{\mathcal{X}}A+DI_{\mathcal{X}}B=I_{\mathcal{X}}.
\end{aligned} S f A + g B , C τ + D ω = θ C τ + D ω θ f A + g B = ( C θ τ + D θ ω ) ( θ f A + θ g B ) = C θ τ θ f A + C θ τ θ g B + D θ ω θ f A + D θ ω θ g B = C S f , τ A + 0 + 0 + D S g , ω B = C I X A + D I X B = I X . Using Theorem 4.3.2 we finally relate three notions duality, similarity and orthogonality.
Proposition 4.3.6. For every p-ASF ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) the canonical dual for ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) is the only dual p-ASF that is similar to ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
Proof. Let us suppose that two p-ASFs ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N g_n=f_nT_{f,g},\omega_n=T_{\tau,\omega}\tau_n,\forall n\in\mathbb{N} g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N
T f , g = S f , τ − 1 θ τ θ g = S f , τ − 1 I X = S f , τ − 1 T_{f,g}=S_{f,\tau}^{-1}\theta_\tau\theta_g=S_{f,\tau}^{-1}I_{\mathcal{X}}=S_{f,\tau}^{-1} T f , g = S f , τ − 1 θ τ θ g = S f , τ − 1 I X = S f , τ − 1 and
T τ , ω = θ ω θ f S f , τ − 1 = I X S f , τ − 1 = S f , τ − 1 . T_{\tau,\omega}=\theta_\omega\theta_fS_{f,\tau}^{-1}=I_{\mathcal{X}}S_{f,\tau}^{-1}=S_{f,\tau}^{-1}. T τ , ω = θ ω θ f S f , τ − 1 = I X S f , τ − 1 = S f , τ − 1 . Hence ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n )
Proposition 4.3.7. Two similar p-ASFs cannot be orthogonal.
Proof. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) T f , g , T τ , ω : X → X T_{f,g},T_{\tau,\omega}:\mathcal{X}\to\mathcal{X} T f , g , T τ , ω : X → X g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N g_n=f_nT_{f,g},\omega_n=T_{\tau,\omega}\tau_n,\forall n\in\mathbb{N} g n = f n T f , g , ω n = T τ , ω τ n , ∀ n ∈ N θ g = θ f T f , g , θ ω = T τ , ω θ τ \theta_g=\theta_fT_{f,g},\theta_\omega=T_{\tau,\omega}\theta_\tau θ g = θ f T f , g , θ ω = T τ , ω θ τ
θ τ θ g = θ τ θ f T f , g = S f , τ T f , g ≠ 0. \theta_\tau\theta_g=\theta_\tau\theta_fT_{f,g}=S_{f,\tau}T_{f,g}\ne0. θ τ θ g = θ τ θ f T f , g = S f , τ T f , g = 0. DILATION THEOREM FOR p-APPROXIMATE SCHAUDER FRAMES
Here we derive a generalization of Theorem 1.2.37 (Naimark-Han-Larson dilation theorem) for frames in Hilbert spaces to p-ASFs for Banach spaces. In order to derive the dilation result we must have a notion of Riesz basis for Banach space. Theorem 1.2.25 gives various characterizations for Riesz bases for Hilbert spaces but all uses (implicitly or explicitly) inner product structures and orthonormal bases. These characterizations lead to the notion of p-Riesz basis for Banach spaces using a single sequence in the Banach space (Definition 1.3.1) but we consider a different notion in this chapter.
To define the notion of Riesz basis, which is compatible with Hilbert space situation, we first derive an operator-theoretic characterization for Riesz basis in Hilbert spaces, which does not use the inner product of Hilbert space. To do so, we need a result from Hilbert space frame theory.
Theorem 4.4.1. For sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(i) { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
(ii) { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
θ τ S τ − 1 θ τ ∗ = I ℓ 2 ( N ) . (31) \theta_\tau S_\tau^{-1}\theta_\tau^*=I_{\ell_2(\mathbb{N})}. \tag*{(31)} θ τ S τ − 1 θ τ ∗ = I ℓ 2 ( N ) . (31) Proof. (i) ⟹ \Longrightarrow ⟹ Theorem 1.2.22 that a Riesz basis is a frame. Now there exist an orthonormal basis { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H T ω n = τ n T\omega_n=\tau_n T ω n = τ n n ∈ N n\in\mathbb{N} n ∈ N
S τ h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n = ∑ n = 1 ∞ ⟨ h , T ω n ⟩ T ω n = T ( ∑ n = 1 ∞ ⟨ T ∗ h , ω n ⟩ ω n ) = T T ∗ h , ∀ h ∈ H . \begin{aligned}
S_\tau h=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n=\sum_{n=1}^{\infty}\langle h,T\omega_n\rangle T\omega_n \\
&=T\left(\sum_{n=1}^{\infty}\langle T^*h,\omega_n\rangle\omega_n\right)=TT^*h,\quad\forall h\in\mathcal{H}.
\end{aligned} S τ h = n = 1 ∑ ∞ ⟨ h , τ n ⟩ τ n = n = 1 ∑ ∞ ⟨ h , T ω n ⟩ T ω n = T ( n = 1 ∑ ∞ ⟨ T ∗ h , ω n ⟩ ω n ) = T T ∗ h , ∀ h ∈ H . Therefore
θ τ S τ − 1 θ τ ∗ { a n } n = θ τ ( T T ∗ ) − 1 θ τ ∗ { a n } n = θ τ ( T ∗ ) − 1 T − 1 θ τ ∗ { a n } n = θ τ ( T ∗ ) − 1 T − 1 ( ∑ n = 1 ∞ a n τ n ) = θ τ ( T ∗ ) − 1 T − 1 ( ∑ n = 1 ∞ a n T ω n ) = θ τ ( ∑ n = 1 ∞ a n ( T ∗ ) − 1 ω n ) = ∑ k = 1 ∞ ⟨ ∑ n = 1 ∞ a n ( T ∗ ) − 1 ω n , τ k ⟩ e k = ∑ k = 1 ∞ ⟨ ∑ n = 1 ∞ a n ( T ∗ ) − 1 ω n , T ω k ⟩ e k = ∑ k = 1 ∞ ⟨ ∑ n = 1 ∞ a n ω n , ω k ⟩ e k = { a k } k , ∀ { a n } n ∈ ℓ 2 ( N ) . \begin{aligned}
\theta_\tau S_\tau^{-1}\theta_\tau^*\{a_n\}_n=\theta_\tau(TT^*)^{-1}\theta_\tau^*\{a_n\}_n=\theta_\tau(T^*)^{-1}T^{-1}\theta_\tau^*\{a_n\}_n \\
&=\theta_\tau(T^*)^{-1}T^{-1}\left(\sum_{n=1}^{\infty}a_n\tau_n\right)=\theta_\tau(T^*)^{-1}T^{-1}\left(\sum_{n=1}^{\infty}a_nT\omega_n\right) \\
&=\theta_\tau\left(\sum_{n=1}^{\infty}a_n(T^*)^{-1}\omega_n\right)=\sum_{k=1}^{\infty}\left\langle\sum_{n=1}^{\infty}a_n(T^*)^{-1}\omega_n,\tau_k\right\rangle e_k \\
&= \sum_{k=1}^{\infty} \left\langle\sum_{n=1}^{\infty} a_n (T^*)^{-1}\omega_n,T\omega_k \right\rangle e_k \\
&= \sum_{k=1}^{\infty} \left\langle\sum_{n=1}^{\infty} a_n\omega_n,\omega_k \right\rangle e_k=\{a_k\}_k,\quad\forall\{a_n\}_n\in\ell^2(\mathbb{N}).
\end{aligned} θ τ S τ − 1 θ τ ∗ { a n } n = θ τ ( T T ∗ ) − 1 θ τ ∗ { a n } n = θ τ ( T ∗ ) − 1 T − 1 θ τ ∗ { a n } n = θ τ ( T ∗ ) − 1 T − 1 ( n = 1 ∑ ∞ a n τ n ) = θ τ ( T ∗ ) − 1 T − 1 ( n = 1 ∑ ∞ a n T ω n ) = θ τ ( n = 1 ∑ ∞ a n ( T ∗ ) − 1 ω n ) = k = 1 ∑ ∞ ⟨ n = 1 ∑ ∞ a n ( T ∗ ) − 1 ω n , τ k ⟩ e k = k = 1 ∑ ∞ ⟨ n = 1 ∑ ∞ a n ( T ∗ ) − 1 ω n , T ω k ⟩ e k = k = 1 ∑ ∞ ⟨ n = 1 ∑ ∞ a n ω n , ω k ⟩ e k = { a k } k , ∀ { a n } n ∈ ℓ 2 ( N ) . (ii) ⟹ \Longrightarrow ⟹ T : ℓ 2 ( N ) → H T:\ell^2(\mathbb{N})\to\mathcal{H} T : ℓ 2 ( N ) → H T e n = τ n Te_n=\tau_n T e n = τ n n ∈ N n\in\mathbb{N} n ∈ N { e n } n \{e_n\}_n { e n } n H \mathcal{H} H T T T H \mathcal{H} H T T T T T T { a n } n ∈ ℓ 2 ( N ) \{a_n\}_n\in\ell^2(\mathbb{N}) { a n } n ∈ ℓ 2 ( N ) { a n } n = θ τ ( S τ − 1 θ τ ∗ { a n } n ) \{a_n\}_n=\theta_\tau(S_\tau^{-1}\theta_\tau^*\{a_n\}_n) { a n } n = θ τ ( S τ − 1 θ τ ∗ { a n } n ) θ τ \theta_\tau θ τ
θ τ h = ∑ n = 1 ∞ ⟨ h , τ n ⟩ e n = ∑ n = 1 ∞ ⟨ h , T e n ⟩ e n = T ∗ h , ∀ h ∈ H . \theta_\tau h=\sum_{n=1}^{\infty}\langle h,\tau_n\rangle e_n=\sum_{n=1}^{\infty}\langle h,Te_n\rangle e_n=T^*h,\quad\forall h\in\mathcal{H}. θ τ h = n = 1 ∑ ∞ ⟨ h , τ n ⟩ e n = n = 1 ∑ ∞ ⟨ h , T e n ⟩ e n = T ∗ h , ∀ h ∈ H . Therefore
Kernel ( T ) = T ∗ ( H ) ⊥ = θ τ ( H ) ⊥ = H ⊥ = { 0 } . \operatorname{Kernel}(T)=T^*(\mathcal{H})^\perp=\theta_\tau(\mathcal{H})^\perp=\mathcal{H}^\perp=\{0\}. Kernel ( T ) = T ∗ ( H ) ⊥ = θ τ ( H ) ⊥ = H ⊥ = { 0 } . Hence T T T
Definition 4.4.2. A pair ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p-approximate Riesz basis for X \mathcal{X} X X \mathcal{X} X θ f S f − 1 θ τ = I ℓ p ( N ) \theta_fS_f^{-1}\theta_\tau=I_{\ell^p(\mathbb{N})} θ f S f − 1 θ τ = I ℓ p ( N )
Example 4.4.3. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) U : X → ℓ p ( N ) U:\mathcal{X}\to\ell^p(\mathbb{N}) U : X → ℓ p ( N ) V : ℓ p ( N ) → X V:\ell^p(\mathbb{N})\to\mathcal{X} V : ℓ p ( N ) → X { e n } n \{e_n\}_n { e n } n { ζ n } n \{\zeta_n\}_n { ζ n } n { f n } n \{f_n\}_n { f n } n { τ n } n \{\tau_n\}_n { τ n } n ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
We now derive the dilation theorem.
Theorem 4.4.4. (Dilation theorem for p-approximate Schauder frames) Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X X 1 \mathcal{X}_1 X 1 X \mathcal{X} X ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) X 1 \mathcal{X}_1 X 1
f n = g n P ∣ X , τ n = P ω n , ∀ n ∈ N , f_n=g_nP|_{\mathcal{X}},\qquad\tau_n=P\omega_n,\qquad\forall n\in\mathbb{N}, f n = g n P ∣ X , τ n = P ω n , ∀ n ∈ N , where P : X 1 → X P:\mathcal{X}_1\to\mathcal{X} P : X 1 → X
Proof. Let { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n
X 1 : = X ⊕ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N ) ) , P : X 1 ∋ x ⊕ y ↦ x ⊕ 0 ∈ X 1 X_1 := X \oplus(I_{\ell^p(\mathbb{N})} - P_{f,\tau})(\ell^p(\mathbb{N})), \qquad P : X_1 \ni x \oplus y \mapsto x \oplus0 \in X_1 X 1 := X ⊕ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N )) , P : X 1 ∋ x ⊕ y ↦ x ⊕ 0 ∈ X 1 and
ω n : = τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ∈ X 1 , g n : = f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) ∈ X 1 ∗ , ∀ n ∈ N . \omega_n := \tau_n \oplus(I_{\ell^p(\mathbb{N})} - P_{f,\tau})e_n \in X_1, \qquad g_n := f_n \oplus\zeta_n(I_{\ell^p(\mathbb{N})} - P_{f,\tau}) \in X_1^*, \quad\forall n \in\mathbb{N}. ω n := τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ∈ X 1 , g n := f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) ∈ X 1 ∗ , ∀ n ∈ N . Then clearly X 1 X_1 X 1 X X X P : X 1 → X P : X_1 \to X P : X 1 → X
( g n P ∣ X ) ( x ) = g n ( P ∣ X x ) = g n ( x ) = ( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) ) ( x ⊕ 0 ) = f n ( x ) , ∀ x ∈ X , (g_nP|_X)(x) = g_n(P|_X x) = g_n(x) = (f_n \oplus\zeta_n(I_{\ell^p(\mathbb{N})} - P_{f,\tau}))(x \oplus0) = f_n(x), \quad\forall x \in X, ( g n P ∣ X ) ( x ) = g n ( P ∣ X x ) = g n ( x ) = ( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ )) ( x ⊕ 0 ) = f n ( x ) , ∀ x ∈ X , P ω n = P ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = τ n , ∀ n ∈ N . P\omega_n = P(\tau_n \oplus(I_{\ell^p(\mathbb{N})} - P_{f,\tau})e_n) = \tau_n, \quad\forall n \in\mathbb{N}. P ω n = P ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = τ n , ∀ n ∈ N . Since the operator I ℓ p ( N ) − P f , τ I_{\ell^p(\mathbb{N})} - P_{f,\tau} I ℓ p ( N ) − P f , τ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N ) ) (I_{\ell^p(\mathbb{N})} - P_{f,\tau})(\ell^p(\mathbb{N})) ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N )) ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) X 1 X_1 X 1 x ⊕ y ∈ X 1 x \oplus y \in X_1 x ⊕ y ∈ X 1 y = { a n } n ∈ ℓ p ( N ) y = \{a_n\}_n \in\ell^p(\mathbb{N}) y = { a n } n ∈ ℓ p ( N )
∑ n = 1 ∞ ( ζ n ( I ℓ p ( N ) − P f , τ ) ( y ) ) τ n = ∑ n = 1 ∞ ζ n ( y ) τ n − ∑ n = 1 ∞ ζ n ( P f , τ ( y ) ) τ n = ∑ n = 1 ∞ ζ n ( { a k } k ) τ n − ∑ n = 1 ∞ ζ n ( θ f S f , τ − 1 θ τ ( { a k } k ) ) τ n = ∑ n = 1 ∞ a n τ n − ∑ n = 1 ∞ ζ n ( θ f S f , τ − 1 ( ∑ k = 1 ∞ a k τ k ) ) τ n = ∑ n = 1 ∞ a n τ n − ∑ n = 1 ∞ ζ n ( ∑ k = 1 ∞ a k θ f S f , τ − 1 τ k ) τ n = ∑ n = 1 ∞ a n τ n − ∑ n = 1 ∞ ζ n ( ∑ k = 1 ∞ a k ∑ r = 1 ∞ f r ( S f , τ − 1 τ k ) e r ) τ n = ∑ n = 1 ∞ a n τ n − ∑ n = 1 ∞ ∑ k = 1 ∞ a k ∑ r = 1 ∞ f r ( S f , τ − 1 τ k ) ζ n ( e r ) τ n = ∑ n = 1 ∞ a n τ n − ∑ n = 1 ∞ ∑ k = 1 ∞ a k f n ( S f , τ − 1 τ k ) τ n = ∑ n = 1 ∞ a n τ n − ∑ k = 1 ∞ a k ∑ n = 1 ∞ f n ( S f , τ − 1 τ k ) τ n = ∑ n = 1 ∞ a n τ n − ∑ k = 1 ∞ a k τ k = 0. \begin{aligned}
\sum_{n=1}^{\infty} \bigl(\zeta_n(I_{\ell^p(\mathbb{N})} - P_{f,\tau})(y)\bigr)\tau_n
&= \sum_{n=1}^{\infty} \zeta_n(y)\tau_n - \sum_{n=1}^{\infty} \zeta_n(P_{f,\tau}(y))\tau_n \\
&= \sum_{n=1}^{\infty} \zeta_n(\{a_k\}_k)\tau_n - \sum_{n=1}^{\infty} \zeta_n\bigl(\theta_{f}S_{f,\tau}^{-1}\theta_{\tau}(\{a_k\}_k)\bigr)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{n=1}^{\infty} \zeta_n\left(\theta_f S_{f,\tau}^{-1}\left(\sum_{k=1}^{\infty}a_k\tau_k\right)\right)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{n=1}^{\infty} \zeta_n\left(\sum_{k=1}^{\infty}a_k\theta_f S_{f,\tau}^{-1}\tau_k\right)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{n=1}^{\infty} \zeta_n\left(\sum_{k=1}^{\infty}a_k\sum_{r=1}^{\infty}f_r(S_{f,\tau}^{-1}\tau_k)e_r\right)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{n=1}^{\infty}\sum_{k=1}^{\infty}a_k\sum_{r=1}^{\infty}f_r(S_{f,\tau}^{-1}\tau_k)\zeta_n(e_r)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{n=1}^{\infty}\sum_{k=1}^{\infty}a_k f_n(S_{f,\tau}^{-1}\tau_k)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{k=1}^{\infty}a_k\sum_{n=1}^{\infty}f_n(S_{f,\tau}^{-1}\tau_k)\tau_n \\
&= \sum_{n=1}^{\infty} a_n\tau_n - \sum_{k=1}^{\infty}a_k\tau_k = 0.
\end{aligned} n = 1 ∑ ∞ ( ζ n ( I ℓ p ( N ) − P f , τ ) ( y ) ) τ n = n = 1 ∑ ∞ ζ n ( y ) τ n − n = 1 ∑ ∞ ζ n ( P f , τ ( y )) τ n = n = 1 ∑ ∞ ζ n ({ a k } k ) τ n − n = 1 ∑ ∞ ζ n ( θ f S f , τ − 1 θ τ ({ a k } k ) ) τ n = n = 1 ∑ ∞ a n τ n − n = 1 ∑ ∞ ζ n ( θ f S f , τ − 1 ( k = 1 ∑ ∞ a k τ k ) ) τ n = n = 1 ∑ ∞ a n τ n − n = 1 ∑ ∞ ζ n ( k = 1 ∑ ∞ a k θ f S f , τ − 1 τ k ) τ n = n = 1 ∑ ∞ a n τ n − n = 1 ∑ ∞ ζ n ( k = 1 ∑ ∞ a k r = 1 ∑ ∞ f r ( S f , τ − 1 τ k ) e r ) τ n = n = 1 ∑ ∞ a n τ n − n = 1 ∑ ∞ k = 1 ∑ ∞ a k r = 1 ∑ ∞ f r ( S f , τ − 1 τ k ) ζ n ( e r ) τ n = n = 1 ∑ ∞ a n τ n − n = 1 ∑ ∞ k = 1 ∑ ∞ a k f n ( S f , τ − 1 τ k ) τ n = n = 1 ∑ ∞ a n τ n − k = 1 ∑ ∞ a k n = 1 ∑ ∞ f n ( S f , τ − 1 τ k ) τ n = n = 1 ∑ ∞ a n τ n − k = 1 ∑ ∞ a k τ k = 0. and
∑ n = 1 ∞ f n ( x ) ( I ℓ p ( N ) − P f , τ ) e n = ∑ n = 1 ∞ f n ( x ) e n − ∑ n = 1 ∞ f n ( x ) P f , τ e n = ∑ n = 1 ∞ f n ( x ) e n − ∑ n = 1 ∞ f n ( x ) θ f S f , τ − 1 θ τ e n = ∑ n = 1 ∞ f n ( x ) e n − ∑ n = 1 ∞ f n ( x ) θ f S f , τ − 1 τ n = ∑ n = 1 ∞ f n ( x ) e n − ∑ n = 1 ∞ f n ( x ) ∑ k = 1 ∞ f k ( S f , τ − 1 τ n ) e k = ∑ n = 1 ∞ f n ( x ) e n − ∑ n = 1 ∞ ∑ k = 1 ∞ f n ( x ) f k ( S f , τ − 1 τ n ) e k = ∑ n = 1 ∞ f n ( x ) e n − ∑ k = 1 ∞ ∑ n = 1 ∞ f n ( x ) f k ( S f , τ − 1 τ n ) e k = ∑ n = 1 ∞ f n ( x ) e n − ∑ k = 1 ∞ f k ( ∑ n = 1 n f n ( x ) S f , τ − 1 τ n ) e k = ∑ n = 1 ∞ f n ( x ) e n − ∑ k = 1 ∞ f k ( x ) e k = 0. \begin{aligned}
\sum_{n=1}^{\infty} f_n(x)(I_{\ell_p(\mathbb{N})}-P_{f,\tau})e_n
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{n=1}^{\infty}f_n(x)P_{f,\tau}e_n\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{n=1}^{\infty}f_n(x)\theta_fS_{f,\tau}^{-1}\theta_\tau e_n\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{n=1}^{\infty}f_n(x)\theta_fS_{f,\tau}^{-1}\tau_n\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{n=1}^{\infty}f_n(x)\sum_{k=1}^{\infty}f_k(S_{f,\tau}^{-1}\tau_n)e_k\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}f_n(x)f_k(S_{f,\tau}^{-1}\tau_n)e_k\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}f_n(x)f_k(S_{f,\tau}^{-1}\tau_n)e_k\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{k=1}^{\infty}f_k\left(\sum_{n=1}^{n}f_n(x)S_{f,\tau}^{-1}\tau_n\right)e_k\\
&=\sum_{n=1}^{\infty}f_n(x)e_n-\sum_{k=1}^{\infty}f_k(x)e_k=0.
\end{aligned} n = 1 ∑ ∞ f n ( x ) ( I ℓ p ( N ) − P f , τ ) e n = n = 1 ∑ ∞ f n ( x ) e n − n = 1 ∑ ∞ f n ( x ) P f , τ e n = n = 1 ∑ ∞ f n ( x ) e n − n = 1 ∑ ∞ f n ( x ) θ f S f , τ − 1 θ τ e n = n = 1 ∑ ∞ f n ( x ) e n − n = 1 ∑ ∞ f n ( x ) θ f S f , τ − 1 τ n = n = 1 ∑ ∞ f n ( x ) e n − n = 1 ∑ ∞ f n ( x ) k = 1 ∑ ∞ f k ( S f , τ − 1 τ n ) e k = n = 1 ∑ ∞ f n ( x ) e n − n = 1 ∑ ∞ k = 1 ∑ ∞ f n ( x ) f k ( S f , τ − 1 τ n ) e k = n = 1 ∑ ∞ f n ( x ) e n − k = 1 ∑ ∞ n = 1 ∑ ∞ f n ( x ) f k ( S f , τ − 1 τ n ) e k = n = 1 ∑ ∞ f n ( x ) e n − k = 1 ∑ ∞ f k ( n = 1 ∑ n f n ( x ) S f , τ − 1 τ n ) e k = n = 1 ∑ ∞ f n ( x ) e n − k = 1 ∑ ∞ f k ( x ) e k = 0. By using previous two calculations, we get
S g , ω ( x ⊕ y ) = ∑ n = 1 ∞ g n ( x ⊕ y ) ω n = ∑ n = 1 ∞ ( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) ) ( x ⊕ y ) ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = ∑ n = 1 ∞ ( f n ( x ) + ( ζ n ( I ℓ p ( N ) − P f , τ ) ) ( y ) ) ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = ( ∑ n = 1 ∞ f n ( x ) τ n + ∑ n = 1 ∞ ( ζ n ( I ℓ p ( N ) − P f , τ ) ) ( y ) τ n ) ⊕ ( ∑ n = 1 ∞ f n ( x ) ( I ℓ p ( N ) − P f , τ ) e n + ∑ n = 1 ∞ ( ζ n ( I ℓ p ( N ) − P f , τ ) ) ( y ) ( I ℓ p ( N ) − P f , τ ) e n ) = ( S f , τ x + 0 ) ⊕ ( 0 + ( I ℓ p ( N ) − P f , τ ) ∑ n = 1 ∞ ζ n ( ( I ℓ p ( N ) − P f , τ ) y ) e n ) = S f , τ x ⊕ ( I ℓ p ( N ) − P f , τ ) ( I ℓ p ( N ) − P f , τ ) y = S f , τ x ⊕ ( I ℓ p ( N ) − P f , τ ) y = ( S f , τ ⊕ ( I ℓ p ( N ) − P f , τ ) ) ( x ⊕ y ) . \begin{aligned}
S_{g,\omega}(x\oplus y)
&=\sum_{n=1}^{\infty}g_n(x\oplus y)\omega_n\\
&=\sum_{n=1}^{\infty}(f_n\oplus\zeta_n(I_{\ell_p(\mathbb{N})}-P_{f,\tau}))(x\oplus y)(\tau_n\oplus(I_{\ell_p(\mathbb{N})}-P_{f,\tau})e_n)\\
&=\sum_{n=1}^{\infty}(f_n(x)+(\zeta_n(I_{\ell_p(\mathbb{N})}-P_{f,\tau}))(y))(\tau_n\oplus(I_{\ell_p(\mathbb{N})}-P_{f,\tau})e_n)\\
&=\left(\sum_{n=1}^{\infty}f_n(x)\tau_n+\sum_{n=1}^{\infty}(\zeta_n(I_{\ell_p(\mathbb{N})}-P_{f,\tau}))(y)\tau_n\right)\oplus\\
&\qquad\left(\sum_{n=1}^{\infty}f_n(x)(I_{\ell_p(\mathbb{N})}-P_{f,\tau})e_n+\sum_{n=1}^{\infty}(\zeta_n(I_{\ell_p(\mathbb{N})}-P_{f,\tau}))(y)(I_{\ell_p(\mathbb{N})}-P_{f,\tau})e_n\right)\\
&=(S_{f,\tau}x+0)\oplus\left(0+(I_{\ell_p(\mathbb{N})}-P_{f,\tau})\sum_{n=1}^{\infty}\zeta_n((I_{\ell_p(\mathbb{N})}-P_{f,\tau})y)e_n\right)\\
&=S_{f,\tau}x\oplus(I_{\ell_p(\mathbb{N})}-P_{f,\tau})(I_{\ell_p(\mathbb{N})}-P_{f,\tau})y\\
&=S_{f,\tau}x\oplus(I_{\ell_p(\mathbb{N})}-P_{f,\tau})y\\
&=(S_{f,\tau}\oplus(I_{\ell_p(\mathbb{N})}-P_{f,\tau}))(x\oplus y).
\end{aligned} S g , ω ( x ⊕ y ) = n = 1 ∑ ∞ g n ( x ⊕ y ) ω n = n = 1 ∑ ∞ ( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ )) ( x ⊕ y ) ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = n = 1 ∑ ∞ ( f n ( x ) + ( ζ n ( I ℓ p ( N ) − P f , τ )) ( y )) ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = ( n = 1 ∑ ∞ f n ( x ) τ n + n = 1 ∑ ∞ ( ζ n ( I ℓ p ( N ) − P f , τ )) ( y ) τ n ) ⊕ ( n = 1 ∑ ∞ f n ( x ) ( I ℓ p ( N ) − P f , τ ) e n + n = 1 ∑ ∞ ( ζ n ( I ℓ p ( N ) − P f , τ )) ( y ) ( I ℓ p ( N ) − P f , τ ) e n ) = ( S f , τ x + 0 ) ⊕ ( 0 + ( I ℓ p ( N ) − P f , τ ) n = 1 ∑ ∞ ζ n (( I ℓ p ( N ) − P f , τ ) y ) e n ) = S f , τ x ⊕ ( I ℓ p ( N ) − P f , τ ) ( I ℓ p ( N ) − P f , τ ) y = S f , τ x ⊕ ( I ℓ p ( N ) − P f , τ ) y = ( S f , τ ⊕ ( I ℓ p ( N ) − P f , τ )) ( x ⊕ y ) . Since the operator I ℓ p ( N ) − P f , τ I_{\ell_p(\mathbb{N})}-P_{f,\tau} I ℓ p ( N ) − P f , τ I ℓ p ( N ) − P f , τ I_{\ell_p(\mathbb{N})}-P_{f,\tau} I ℓ p ( N ) − P f , τ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N ) ) (I_{\ell^p(\mathbb{N})}-P_{f,\tau})(\ell^p(\mathbb{N})) ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N )) S g , ω = S f , τ ⊕ ( I ℓ p ( N ) − P f , τ ) S_{g,\omega}=S_{f,\tau}\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau}) S g , ω = S f , τ ⊕ ( I ℓ p ( N ) − P f , τ ) X 1 \mathcal{X}_1 X 1 ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) p p p X 1 \mathcal{X}_1 X 1 θ g \theta_g θ g θ ω \theta_\omega θ ω
θ g ( x ⊕ y ) = { g n ( x ⊕ y ) } n = { ( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) ) ( x ⊕ y ) } n = { f n ( x ) + ζ n ( ( I ℓ p ( N ) − P f , τ ) y ) } n = { f n ( x ) } n + { ζ n ( ( I ℓ p ( N ) − P f , τ ) y ) } n = θ f x + ∑ n = 1 ∞ ζ n ( ( I ℓ p ( N ) − P f , τ ) y ) e n = θ f x + ( I ℓ p ( N ) − P f , τ ) y , ∀ x ⊕ y ∈ X 1 \begin{aligned}
\theta_g(x\oplus y)&=\{g_n(x\oplus y)\}_n=\{(f_n\oplus\zeta_n(I_{\ell^p(\mathbb{N})}-P_{f,\tau}))(x\oplus y)\}_n\\
&=\{f_n(x)+\zeta_n((I_{\ell^p(\mathbb{N})}-P_{f,\tau})y)\}_n=\{f_n(x)\}_n+\{\zeta_n((I_{\ell^p(\mathbb{N})}-P_{f,\tau})y)\}_n\\
&=\theta_f x+\sum_{n=1}^{\infty}\zeta_n((I_{\ell^p(\mathbb{N})}-P_{f,\tau})y)e_n=\theta_f x+(I_{\ell^p(\mathbb{N})}-P_{f,\tau})y,\quad\forall x\oplus y\in\mathcal{X}_1
\end{aligned} θ g ( x ⊕ y ) = { g n ( x ⊕ y ) } n = {( f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ )) ( x ⊕ y ) } n = { f n ( x ) + ζ n (( I ℓ p ( N ) − P f , τ ) y ) } n = { f n ( x ) } n + { ζ n (( I ℓ p ( N ) − P f , τ ) y ) } n = θ f x + n = 1 ∑ ∞ ζ n (( I ℓ p ( N ) − P f , τ ) y ) e n = θ f x + ( I ℓ p ( N ) − P f , τ ) y , ∀ x ⊕ y ∈ X 1 and
θ ω { a n } n = ∑ n = 1 ∞ a n ω n = ∑ n = 1 ∞ a n ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = ( ∑ n = 1 ∞ a n τ n ) ⊕ ( ∑ n = 1 ∞ a n ( I ℓ p ( N ) − P f , τ ) e n ) = θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) ( ∑ n = 1 ∞ a n e n ) = θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n , ∀ { a n } n ∈ ℓ p ( N ) . \begin{aligned}
\theta_\omega\{a_n\}_n&=\sum_{n=1}^{\infty}a_n\omega_n=\sum_{n=1}^{\infty}a_n\left(\tau_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})e_n\right)\\
&=\left(\sum_{n=1}^{\infty}a_n\tau_n\right)\oplus\left(\sum_{n=1}^{\infty}a_n(I_{\ell^p(\mathbb{N})}-P_{f,\tau})e_n\right)\\
&=\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\left(\sum_{n=1}^{\infty}a_ne_n\right)\\
&=\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n,\quad\forall\{a_n\}_n\in\ell^p(\mathbb{N}).
\end{aligned} θ ω { a n } n = n = 1 ∑ ∞ a n ω n = n = 1 ∑ ∞ a n ( τ n ⊕ ( I ℓ p ( N ) − P f , τ ) e n ) = ( n = 1 ∑ ∞ a n τ n ) ⊕ ( n = 1 ∑ ∞ a n ( I ℓ p ( N ) − P f , τ ) e n ) = θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) ( n = 1 ∑ ∞ a n e n ) = θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n , ∀ { a n } n ∈ ℓ p ( N ) . Therefore
P g , ω { a n } n = θ g S g , ω − 1 θ ω { a n } n = θ g S g , ω − 1 ( θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ g ( S f , τ − 1 ⊕ ( I ℓ p ( N ) − P f , τ ) ) ( θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ g ( S f , τ − 1 θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) 2 { a n } n ) = θ g ( S f , τ − 1 θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ f ( S f , τ − 1 θ τ { a n } n ) + ( I ℓ p ( N ) − P f , τ ) ( I ℓ p ( N ) − P f , τ ) { a n } n = P f , τ { a n } n + ( I ℓ p ( N ) − P f , τ ) { a n } n = { a n } n , ∀ { a n } n ∈ ℓ p ( N ) . \begin{aligned}
P_{g,\omega}\{a_n\}_n&=\theta_gS_{g,\omega}^{-1}\theta_\omega\{a_n\}_n=\theta_gS_{g,\omega}^{-1}\left(\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n\right)\\
&=\theta_g\left(S_{f,\tau}^{-1}\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\right)\left(\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n\right)\\
&=\theta_g\left(S_{f,\tau}^{-1}\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})^2\{a_n\}_n\right)\\
&=\theta_g\left(S_{f,\tau}^{-1}\theta_\tau\{a_n\}_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n\right)\\
&=\theta_f\left(S_{f,\tau}^{-1}\theta_\tau\{a_n\}_n\right)+(I_{\ell^p(\mathbb{N})}-P_{f,\tau})(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n\\
&=P_{f,\tau}\{a_n\}_n+(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\{a_n\}_n=\{a_n\}_n,\quad\forall\{a_n\}_n\in\ell^p(\mathbb{N}).
\end{aligned} P g , ω { a n } n = θ g S g , ω − 1 θ ω { a n } n = θ g S g , ω − 1 ( θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ g ( S f , τ − 1 ⊕ ( I ℓ p ( N ) − P f , τ ) ) ( θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ g ( S f , τ − 1 θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) 2 { a n } n ) = θ g ( S f , τ − 1 θ τ { a n } n ⊕ ( I ℓ p ( N ) − P f , τ ) { a n } n ) = θ f ( S f , τ − 1 θ τ { a n } n ) + ( I ℓ p ( N ) − P f , τ ) ( I ℓ p ( N ) − P f , τ ) { a n } n = P f , τ { a n } n + ( I ℓ p ( N ) − P f , τ ) { a n } n = { a n } n , ∀ { a n } n ∈ ℓ p ( N ) . Corollary 4.4.5. [129 , 148 ] Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H H 1 \mathcal{H}_1 H 1 H \mathcal{H} H { ω n } n \{\omega_n\}_n { ω n } n H 1 \mathcal{H}_1 H 1
τ n = P ω n , ∀ n ∈ N , \tau_n=P\omega_n,\quad\forall n\in\mathbb{N}, τ n = P ω n , ∀ n ∈ N , where P P P H 1 \mathcal{H}_1 H 1 H \mathcal{H} H
Proof. Let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
f n : H ∋ h ↦ f n ( h ) : = ⟨ h , τ n ⟩ ∈ K , ∀ n ∈ N . f_n:\mathcal{H}\ni h\mapsto f_n(h):=\langle h,\tau_n\rangle\in\mathbb{K},\quad\forall n\in\mathbb{N}. f n : H ∋ h ↦ f n ( h ) := ⟨ h , τ n ⟩ ∈ K , ∀ n ∈ N . Then θ f = θ τ \theta_f=\theta_\tau θ f = θ τ ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) H \mathcal{H} H X 1 \mathcal{X}_1 X 1 H \mathcal{H} H ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) X 1 = H ⊕ ( I ℓ 2 ( N ) − P τ ) ( ℓ 2 ( N ) ) \mathcal{X}_1=\mathcal{H}\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)(\ell^2(\mathbb{N})) X 1 = H ⊕ ( I ℓ 2 ( N ) − P τ ) ( ℓ 2 ( N ))
f n = g n P ∣ H , τ n = P ω n , ∀ n ∈ N . f_n=g_nP|_{\mathcal{H}},\quad\tau_n=P\omega_n,\quad\forall n\in\mathbb{N}. f n = g n P ∣ H , τ n = P ω n , ∀ n ∈ N . where P : X 1 → H P:\mathcal{X}_1\to\mathcal{H} P : X 1 → H ( I ℓ 2 ( N ) − P τ ) ( ℓ 2 ( N ) ) (I_{\ell^2(\mathbb{N})}-P_\tau)(\ell^2(\mathbb{N})) ( I ℓ 2 ( N ) − P τ ) ( ℓ 2 ( N )) ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) X 1 \mathcal{X}_1 X 1 P P P { ω n } n \{\omega_n\}_n { ω n } n X 1 \mathcal{X}_1 X 1 { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H a , b > 0 a,b>0 a , b > 0
a ∥ h ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . a\|h\|^2\leq\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2\leq b\|h\|^2,\quad\forall h\in\mathcal{H}. a ∥ h ∥ 2 ≤ n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 ≤ b ∥ h ∥ 2 , ∀ h ∈ H . Let h ⊕ ( I ℓ 2 ( N ) − P f , τ ) { a k } k ∈ X 1 h\oplus(I_{\ell^2(\mathbb{N})}-P_{f,\tau})\{a_k\}_k\in\mathcal{X}_1 h ⊕ ( I ℓ 2 ( N ) − P f , τ ) { a k } k ∈ X 1 b ≥ 1 b\geq1 b ≥ 1
∑ n = 1 ∞ ∣ ⟨ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k , ω n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k , τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + ∑ n = 1 ∞ ∣ ⟨ ( I ℓ 2 ( N ) − P τ ) { a k } k , ( I ℓ 2 ( N ) − P τ ) e n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + ∑ n = 1 ∞ ∣ ⟨ ( I ℓ 2 ( N ) − P τ ) ( I ℓ 2 ( N ) − P τ ) { a k } k , e n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + ∑ n = 1 ∞ ∣ ⟨ ( I ℓ 2 ( N ) − P τ ) { a k } k , e n ⟩ ∣ 2 = ∑ n = 1 ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ≤ b ∥ h ∥ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ≤ b ( ∥ h ∥ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ) = b ∥ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 . \begin{aligned}
\sum_{n=1}^{\infty}\left|\left\langle h\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k,\omega_n\right\rangle\right|^2
&=\sum_{n=1}^{\infty}\left|\left\langle h\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k,\tau_n\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)e_n\right\rangle\right|^2\\
&=\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2+\sum_{n=1}^{\infty}\left|\left\langle(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k,(I_{\ell^2(\mathbb{N})}-P_\tau)e_n\right\rangle\right|^2\\
&=\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2+\sum_{n=1}^{\infty}\left|\left\langle(I_{\ell^2(\mathbb{N})}-P_\tau)(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k,e_n\right\rangle\right|^2\\
&=\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2+\sum_{n=1}^{\infty}\left|\left\langle(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k,e_n\right\rangle\right|^2\\
&=\sum_{n=1}^{\infty}|\langle h,\tau_n\rangle|^2+\|(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k\|^2\\
&\leq b\|h\|^2+\|(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k\|^2\\
&\leq b\left(\|h\|^2+\|(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k\|^2\right)\\
&=b\|h\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)\{a_k\}_k\|^2.
\end{aligned} n = 1 ∑ ∞ ⟨ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k , ω n ⟩ 2 = n = 1 ∑ ∞ ⟨ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k , τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ⟩ 2 = n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + n = 1 ∑ ∞ ⟨ ( I ℓ 2 ( N ) − P τ ) { a k } k , ( I ℓ 2 ( N ) − P τ ) e n ⟩ 2 = n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + n = 1 ∑ ∞ ⟨ ( I ℓ 2 ( N ) − P τ ) ( I ℓ 2 ( N ) − P τ ) { a k } k , e n ⟩ 2 = n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + n = 1 ∑ ∞ ⟨ ( I ℓ 2 ( N ) − P τ ) { a k } k , e n ⟩ 2 = n = 1 ∑ ∞ ∣ ⟨ h , τ n ⟩ ∣ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ≤ b ∥ h ∥ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ≤ b ( ∥ h ∥ 2 + ∥ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 ) = b ∥ h ⊕ ( I ℓ 2 ( N ) − P τ ) { a k } k ∥ 2 . Previous calculation tells that { ω n } n \{\omega_n\}_n { ω n } n X 1 \mathcal{X}_1 X 1 x ⊕ { a k } k ↦ ∑ n = 1 ∞ ⟨ x ⊕ { a k } k , ω n ⟩ ω n ∈ X 1 x \oplus\{a_k\}_k \mapsto\sum_{n=1}^{\infty} \langle x \oplus\{a_k\}_k,\omega_n\rangle\omega_n \in\mathcal{X}_1 x ⊕ { a k } k ↦ ∑ n = 1 ∞ ⟨ x ⊕ { a k } k , ω n ⟩ ω n ∈ X 1
Next we claim that
g n ( x ⊕ { a k } k ) = ⟨ x ⊕ { a k } k , ω n ⟩ , ∀ x ⊕ { a k } k ∈ X 1 , ∀ n ∈ N . (32) g_n(x \oplus\{a_k\}_k)=\langle x \oplus\{a_k\}_k,\omega_n\rangle,\qquad\forall x \oplus\{a_k\}_k \in\mathcal{X}_1,\ \forall n \in\mathbb{N}. \tag*{(32)} g n ( x ⊕ { a k } k ) = ⟨ x ⊕ { a k } k , ω n ⟩ , ∀ x ⊕ { a k } k ∈ X 1 , ∀ n ∈ N . (32) Consider
g n ( x ⊕ { a k } k ) = ( f n ⊕ ζ n ( I ℓ 2 ( N ) − P τ ) ) ( x ⊕ { a k } k ) = f n ( x ) + ζ n ( ( I ℓ 2 ( N ) − P τ ) { a k } k ) = f n ( x ) + ζ n ( { a k } k ) − ζ n ( P τ { a k } k ) = f n ( x ) + ζ n ( { a k } k ) − ζ n ( θ τ S τ − 1 θ τ ∗ { a k } k ) = f n ( x ) + a n − ζ n ( θ τ S τ − 1 ( ∑ k = 1 ∞ a k τ k ) ) = f n ( x ) + a n − ζ n ( ∑ k = 1 ∞ a k θ τ S τ − 1 τ k ) = f n ( x ) + a n − ζ n ( ∑ k = 1 ∞ a k ∑ r = 1 ∞ ⟨ S τ − 1 τ k , τ r ⟩ e r ) = f n ( x ) + a n − ∑ k = 1 ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ = ⟨ x , τ n ⟩ + a n − ∑ k = 1 ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ \begin{aligned}
g_n(x \oplus\{a_k\}_k)
&=(f_n\oplus\zeta_n(I_{\ell_2(\mathbb{N})}-P_\tau))(x\oplus\{a_k\}_k)\\
&=f_n(x)+\zeta_n((I_{\ell_2(\mathbb{N})}-P_\tau)\{a_k\}_k)
=f_n(x)+\zeta_n(\{a_k\}_k)-\zeta_n(P_\tau\{a_k\}_k)\\
&=f_n(x)+\zeta_n(\{a_k\}_k)-\zeta_n(\theta_\tau S_\tau^{-1}\theta_\tau^*\{a_k\}_k)
=f_n(x)+a_n-\zeta_n\left(\theta_\tau S_\tau^{-1}\left(\sum_{k=1}^{\infty}a_k\tau_k\right)\right)\\
&=f_n(x)+a_n-\zeta_n\left(\sum_{k=1}^{\infty}a_k\theta_\tau S_\tau^{-1}\tau_k\right)
=f_n(x)+a_n-\zeta_n\left(\sum_{k=1}^{\infty}a_k\sum_{r=1}^{\infty}\langle S_\tau^{-1}\tau_k,\tau_r\rangle e_r\right)\\
&=f_n(x)+a_n-\sum_{k=1}^{\infty}a_k\langle S_\tau^{-1}\tau_k,\tau_n\rangle
=\langle x,\tau_n\rangle+a_n-\sum_{k=1}^{\infty}a_k\langle S_\tau^{-1}\tau_k,\tau_n\rangle
\end{aligned} g n ( x ⊕ { a k } k ) = ( f n ⊕ ζ n ( I ℓ 2 ( N ) − P τ )) ( x ⊕ { a k } k ) = f n ( x ) + ζ n (( I ℓ 2 ( N ) − P τ ) { a k } k ) = f n ( x ) + ζ n ({ a k } k ) − ζ n ( P τ { a k } k ) = f n ( x ) + ζ n ({ a k } k ) − ζ n ( θ τ S τ − 1 θ τ ∗ { a k } k ) = f n ( x ) + a n − ζ n ( θ τ S τ − 1 ( k = 1 ∑ ∞ a k τ k ) ) = f n ( x ) + a n − ζ n ( k = 1 ∑ ∞ a k θ τ S τ − 1 τ k ) = f n ( x ) + a n − ζ n ( k = 1 ∑ ∞ a k r = 1 ∑ ∞ ⟨ S τ − 1 τ k , τ r ⟩ e r ) = f n ( x ) + a n − k = 1 ∑ ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ = ⟨ x , τ n ⟩ + a n − k = 1 ∑ ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ and
⟨ x ⊕ { a k } k , ω n ⟩ = ⟨ x ⊕ { a k } k , τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ⟩ = ⟨ x , τ n ⟩ + ⟨ { a k } k , ( I ℓ 2 ( N ) − P τ ) e n ⟩ = ⟨ x , τ n ⟩ + ⟨ { a k } k , e n ⟩ + ⟨ { a k } k , P τ e n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨ { a k } k , θ τ S τ − 1 θ τ ∗ e n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨ { a k } k , θ τ S τ − 1 τ n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨ { a k } k , { ⟨ S τ − 1 τ n , τ k ⟩ } k ⟩ = ⟨ x , τ n ⟩ + a n − ∑ k = 1 ∞ a k ⟨ S τ − 1 τ n , τ k ⟩ = ⟨ x , τ n ⟩ + a n − ∑ k = 1 ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ . \begin{aligned}
\langle x\oplus\{a_k\}_k,\omega_n\rangle
&=\langle x\oplus\{a_k\}_k,\tau_n\oplus(I_{\ell_2(\mathbb{N})}-P_\tau)e_n\rangle\\
&=\langle x,\tau_n\rangle+\langle\{a_k\}_k,(I_{\ell_2(\mathbb{N})}-P_\tau)e_n\rangle\\
&=\langle x,\tau_n\rangle+\langle\{a_k\}_k,e_n\rangle+\langle\{a_k\}_k,P_\tau e_n\rangle\\
&=\langle x,\tau_n\rangle+a_n-\langle\{a_k\}_k,\theta_\tau S_\tau^{-1}\theta_\tau^*e_n\rangle\\
&=\langle x,\tau_n\rangle+a_n-\langle\{a_k\}_k,\theta_\tau S_\tau^{-1}\tau_n\rangle\\
&=\langle x,\tau_n\rangle+a_n-\langle\{a_k\}_k,\{\langle S_\tau^{-1}\tau_n,\tau_k\rangle\}_k\rangle\\
&=\langle x,\tau_n\rangle+a_n-\sum_{k=1}^{\infty}a_k\langle S_\tau^{-1}\tau_n,\tau_k\rangle\\
&=\langle x,\tau_n\rangle+a_n-\sum_{k=1}^{\infty}a_k\langle S_\tau^{-1}\tau_k,\tau_n\rangle.
\end{aligned} ⟨ x ⊕ { a k } k , ω n ⟩ = ⟨ x ⊕ { a k } k , τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ⟩ = ⟨ x , τ n ⟩ + ⟨{ a k } k , ( I ℓ 2 ( N ) − P τ ) e n ⟩ = ⟨ x , τ n ⟩ + ⟨{ a k } k , e n ⟩ + ⟨{ a k } k , P τ e n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨{ a k } k , θ τ S τ − 1 θ τ ∗ e n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨{ a k } k , θ τ S τ − 1 τ n ⟩ = ⟨ x , τ n ⟩ + a n − ⟨{ a k } k , {⟨ S τ − 1 τ n , τ k ⟩ } k ⟩ = ⟨ x , τ n ⟩ + a n − k = 1 ∑ ∞ a k ⟨ S τ − 1 τ n , τ k ⟩ = ⟨ x , τ n ⟩ + a n − k = 1 ∑ ∞ a k ⟨ S τ − 1 τ k , τ n ⟩ . Thus Equation (4.4.2) holds. Therefore for all x ⊕ { a k } k ∈ X 1 x\oplus\{a_k\}_k\in\mathcal{X}_1 x ⊕ { a k } k ∈ X 1
S g , ω ( x ⊕ { a k } k ) = ∑ n = 1 ∞ g n ( x ⊕ { a k } k ) ω n = ∑ n = 1 ∞ ⟨ x ⊕ { a k } k , ω n ⟩ ω n = S ω ( x ⊕ { a k } k ) . S_{g,\omega}(x\oplus\{a_k\}_k)
=\sum_{n=1}^{\infty}g_n(x\oplus\{a_k\}_k)\omega_n
=\sum_{n=1}^{\infty}\langle x\oplus\{a_k\}_k,\omega_n\rangle\omega_n
=S_\omega(x\oplus\{a_k\}_k). S g , ω ( x ⊕ { a k } k ) = n = 1 ∑ ∞ g n ( x ⊕ { a k } k ) ω n = n = 1 ∑ ∞ ⟨ x ⊕ { a k } k , ω n ⟩ ω n = S ω ( x ⊕ { a k } k ) . Since S g , ω S_{g,\omega} S g , ω S ω S_\omega S ω S ω S_\omega S ω
1 ∥ S ω ∥ − 1 ∥ g ∥ 2 ≤ ⟨ S ω g , g ⟩ ≤ ∥ S ω ∥ ∥ g ∥ 2 , ∀ g ∈ X 1 . \frac{1}{\lVert S_\omega\rVert^{-1}}\lVert g\rVert^2
\leq\langle S_\omega g,g\rangle
\leq\lVert S_\omega\rVert\lVert g\rVert^2,\qquad\forall g\in\mathcal{X}_1. ∥ S ω ∥ − 1 1 ∥ g ∥ 2 ≤ ⟨ S ω g , g ⟩ ≤ ∥ S ω ∥ ∥ g ∥ 2 , ∀ g ∈ X 1 . Hence
1 ∥ S ω ∥ − 1 ∥ g ∥ 2 ≤ ∑ n = 1 ∞ ∣ ⟨ g , ω n ⟩ ∣ 2 ≤ ∥ S ω ∥ ∥ g ∥ 2 , ∀ g ∈ X 1 . \frac{1}{\lVert S_\omega\rVert^{-1}}\lVert g\rVert^2
\leq\sum_{n=1}^{\infty}\lvert\langle g,\omega_n\rangle\rvert^2
\leq\lVert S_\omega\rVert\lVert g\rVert^2,\qquad\forall g\in\mathcal{X}_1. ∥ S ω ∥ − 1 1 ∥ g ∥ 2 ≤ n = 1 ∑ ∞ ∣⟨ g , ω n ⟩ ∣ 2 ≤ ∥ S ω ∥ ∥ g ∥ 2 , ∀ g ∈ X 1 . Hence { ω n } n \{\omega_n\}_n { ω n } n X 1 \mathcal{X}_1 X 1
Finally we show Equation (4.4.1) in Theorem 4.4.1 for the frame { ω n } n \{\omega_n\}_n { ω n } n
θ ω S ω − 1 θ ω ∗ { a n } n = θ ω S ω − 1 ( ∑ n = 1 ∞ a n ω n ) = θ ω ( ∑ n = 1 ∞ a n S ω − 1 ω n ) = ∑ k = 1 ∞ ⟨ ∑ n = 1 ∞ a n S ω − 1 ω n , ω k ⟩ = ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ S ω − 1 ω n , ω k ⟩ = ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ ( S τ − 1 ⊕ ( I ℓ 2 ( N ) − P τ ) ) ( τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ) , τ k ⊕ ( I ℓ 2 ( N ) − P τ ) e k ⟩ = ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ S τ − 1 τ n ⊕ ( I ℓ 2 ( N ) − P τ ) 2 e n , τ k ⊕ ( I ℓ 2 ( N ) − P τ ) e k ⟩ = ∑ k = 1 ∞ ∑ n = 1 ∞ a n ( ⟨ S τ − 1 τ n , τ k ⟩ + ⟨ ( I ℓ 2 ( N ) − P τ ) e n , ( I ℓ 2 ( N ) − P τ ) e k ⟩ ) = ∑ k = 1 ∞ ⟨ ∑ n = 1 ∞ a n S τ − 1 τ n , τ k ⟩ + ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ ( I ℓ 2 ( N ) − P f , τ ) e n , ( I ℓ 2 ( N ) − P τ ) e k ⟩ = P τ { a n } n + ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ ( I ℓ 2 ( N ) − P τ ) e n , e k ⟩ = P τ { a n } n + ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ e n , e k ⟩ − ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ P τ e n , e k ⟩ = P τ { a n } n + ∑ k = 1 ∞ a k e k − ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ θ τ S τ − 1 θ τ ∗ e n , e k ⟩ = P τ { a n } n + ∑ k = 1 ∞ a k e k − ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ S τ − 1 τ n , θ τ ∗ e k ⟩ = P τ { a n } n + ∑ k = 1 ∞ a k e k − ∑ k = 1 ∞ ∑ n = 1 ∞ a n ⟨ S τ − 1 τ n , τ k ⟩ = P τ { a n } n + ∑ k = 1 ∞ a k e k − P τ { a n } n = { a n } n , ∀ { a n } n ∈ ℓ 2 ( N ) . \begin{aligned}
\theta_\omega S_\omega^{-1}\theta_\omega^*\{a_n\}_n
&=\theta_\omega S_\omega^{-1}\left(\sum_{n=1}^{\infty}a_n\omega_n\right)
=\theta_\omega\left(\sum_{n=1}^{\infty}a_nS_\omega^{-1}\omega_n\right)\\
&=\sum_{k=1}^{\infty}\left\langle\sum_{n=1}^{\infty}a_nS_\omega^{-1}\omega_n,\omega_k\right\rangle
=\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle S_\omega^{-1}\omega_n,\omega_k\rangle\\
&=\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\left\langle\left(S_\tau^{-1}\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)\right)\left(\tau_n\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)e_n\right),\tau_k\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)e_k\right\rangle\\
&=\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\left\langle S_\tau^{-1}\tau_n\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)^2e_n,\tau_k\oplus(I_{\ell^2(\mathbb{N})}-P_\tau)e_k\right\rangle\\
&=\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\left(\langle S_\tau^{-1}\tau_n,\tau_k\rangle+\langle(I_{\ell^2(\mathbb{N})}-P_\tau)e_n,(I_{\ell^2(\mathbb{N})}-P_\tau)e_k\rangle\right)\\
&=\sum_{k=1}^{\infty}\left\langle\sum_{n=1}^{\infty}a_nS_\tau^{-1}\tau_n,\tau_k\right\rangle+\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle(I_{\ell^2(\mathbb{N})}-P_{f,\tau})e_n,(I_{\ell^2(\mathbb{N})}-P_\tau)e_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle(I_{\ell^2(\mathbb{N})}-P_\tau)e_n,e_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle e_n,e_k\rangle-\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle P_\tau e_n,e_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}a_ke_k-\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle\theta_\tau S_\tau^{-1}\theta_\tau^*e_n,e_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}a_ke_k-\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle S_\tau^{-1}\tau_n,\theta_\tau^*e_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}a_ke_k-\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_n\langle S_\tau^{-1}\tau_n,\tau_k\rangle\\
&=P_\tau\{a_n\}_n+\sum_{k=1}^{\infty}a_ke_k-P_\tau\{a_n\}_n=\{a_n\}_n,\quad\forall\{a_n\}_n\in\ell^2(\mathbb{N}).
\end{aligned} θ ω S ω − 1 θ ω ∗ { a n } n = θ ω S ω − 1 ( n = 1 ∑ ∞ a n ω n ) = θ ω ( n = 1 ∑ ∞ a n S ω − 1 ω n ) = k = 1 ∑ ∞ ⟨ n = 1 ∑ ∞ a n S ω − 1 ω n , ω k ⟩ = k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ S ω − 1 ω n , ω k ⟩ = k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ ( S τ − 1 ⊕ ( I ℓ 2 ( N ) − P τ ) ) ( τ n ⊕ ( I ℓ 2 ( N ) − P τ ) e n ) , τ k ⊕ ( I ℓ 2 ( N ) − P τ ) e k ⟩ = k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ S τ − 1 τ n ⊕ ( I ℓ 2 ( N ) − P τ ) 2 e n , τ k ⊕ ( I ℓ 2 ( N ) − P τ ) e k ⟩ = k = 1 ∑ ∞ n = 1 ∑ ∞ a n ( ⟨ S τ − 1 τ n , τ k ⟩ + ⟨( I ℓ 2 ( N ) − P τ ) e n , ( I ℓ 2 ( N ) − P τ ) e k ⟩ ) = k = 1 ∑ ∞ ⟨ n = 1 ∑ ∞ a n S τ − 1 τ n , τ k ⟩ + k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨( I ℓ 2 ( N ) − P f , τ ) e n , ( I ℓ 2 ( N ) − P τ ) e k ⟩ = P τ { a n } n + k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨( I ℓ 2 ( N ) − P τ ) e n , e k ⟩ = P τ { a n } n + k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ e n , e k ⟩ − k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ P τ e n , e k ⟩ = P τ { a n } n + k = 1 ∑ ∞ a k e k − k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ θ τ S τ − 1 θ τ ∗ e n , e k ⟩ = P τ { a n } n + k = 1 ∑ ∞ a k e k − k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ S τ − 1 τ n , θ τ ∗ e k ⟩ = P τ { a n } n + k = 1 ∑ ∞ a k e k − k = 1 ∑ ∞ n = 1 ∑ ∞ a n ⟨ S τ − 1 τ n , τ k ⟩ = P τ { a n } n + k = 1 ∑ ∞ a k e k − P τ { a n } n = { a n } n , ∀ { a n } n ∈ ℓ 2 ( N ) . Thus { ω n } n \{\omega_n\}_n { ω n } n X 1 \mathcal{X}_1 X 1
We now illustrate Theorem 4.4.4 with an example.
Example 4.4.6. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n
R : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( 0 , x 1 , x 2 , … ) ∈ ℓ p ( N ) , R:\ell^p(\mathbb{N})\ni(x_n)_{n=1}^{\infty}\mapsto(0,x_1,x_2,\ldots)\in\ell^p(\mathbb{N}), R : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( 0 , x 1 , x 2 , … ) ∈ ℓ p ( N ) , L : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( x 2 , x 3 , x 4 , … ) ∈ ℓ p ( N ) . L:\ell^p(\mathbb{N})\ni(x_n)_{n=1}^{\infty}\mapsto(x_2,x_3,x_4,\ldots)\in\ell^p(\mathbb{N}). L : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( x 2 , x 3 , x 4 , … ) ∈ ℓ p ( N ) . Then L R = I ℓ p ( N ) LR=I_{\ell^p(\mathbb{N})} L R = I ℓ p ( N ) Example 4.4.3 says that ( { f n : = ζ n R } n , { τ n : = L e n } n ) (\{f_n:=\zeta_nR\}_n,\{\tau_n:=Le_n\}_n) ({ f n := ζ n R } n , { τ n := L e n } n ) p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) θ f = R \theta_f=R θ f = R θ τ = L \theta_\tau=L θ τ = L S f , τ = L R = I ℓ p ( N ) S_{f,\tau}=LR=I_{\ell^p(\mathbb{N})} S f , τ = L R = I ℓ p ( N ) P f , τ = R L P_{f,\tau}=RL P f , τ = R L
( I ℓ p ( N ) − P f , τ ) ( x n ) n = 1 ∞ = ( x n ) n = 1 ∞ − R L ( x n ) n = 1 ∞ = ( x n ) n = 1 ∞ − ( 0 , x 2 , x 3 , … ) = ( x 1 , 0 , 0 , … ) , ∀ ( x n ) n = 1 ∞ ∈ ℓ p ( N ) \left(I_{\ell^p(\mathbb{N})}-P_{f,\tau}\right)(x_n)_{n=1}^{\infty}=(x_n)_{n=1}^{\infty}-RL(x_n)_{n=1}^{\infty}
=(x_n)_{n=1}^{\infty}-(0,x_2,x_3,\ldots)=(x_1,0,0,\ldots),\quad\forall(x_n)_{n=1}^{\infty}\in\ell^p(\mathbb{N}) ( I ℓ p ( N ) − P f , τ ) ( x n ) n = 1 ∞ = ( x n ) n = 1 ∞ − R L ( x n ) n = 1 ∞ = ( x n ) n = 1 ∞ − ( 0 , x 2 , x 3 , … ) = ( x 1 , 0 , 0 , … ) , ∀ ( x n ) n = 1 ∞ ∈ ℓ p ( N ) which says that ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N ) ) ≅ K (I_{\ell^p(\mathbb{N})}-P_{f,\tau})(\ell^p(\mathbb{N}))\cong\mathbb{K} ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N )) ≅ K Theorem 4.4.4 ,
X 1 = ℓ p ( N ) ⊕ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N ) ) ≅ ℓ p ( N ) ⊕ K ≅ ℓ p ( N ∪ { 0 } ) \mathcal{X}_1=\ell^p(\mathbb{N})\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})(\ell^p(\mathbb{N}))\cong\ell^p(\mathbb{N})\oplus\mathbb{K}\cong\ell^p(\mathbb{N}\cup\{0\}) X 1 = ℓ p ( N ) ⊕ ( I ℓ p ( N ) − P f , τ ) ( ℓ p ( N )) ≅ ℓ p ( N ) ⊕ K ≅ ℓ p ( N ∪ { 0 }) P : ℓ p ( N ∪ { 0 } ) ∋ ( x n ) n = 0 ∞ ↦ ( x n ) n = 1 ∞ ∈ ℓ p ( N ) , P:\ell^p(\mathbb{N}\cup\{0\})\ni(x_n)_{n=0}^{\infty}\mapsto(x_n)_{n=1}^{\infty}\in\ell^p(\mathbb{N}), P : ℓ p ( N ∪ { 0 }) ∋ ( x n ) n = 0 ∞ ↦ ( x n ) n = 1 ∞ ∈ ℓ p ( N ) , ω 1 = τ 1 ⊕ ( I ℓ p ( N ) − P f , τ ) τ 1 = L e 1 ⊕ ( I ℓ p ( N ) − P f , τ ) L e 1 = 0 ⊕ 0 , \omega_1=\tau_1\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\tau_1=Le_1\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})Le_1=0\oplus0, ω 1 = τ 1 ⊕ ( I ℓ p ( N ) − P f , τ ) τ 1 = L e 1 ⊕ ( I ℓ p ( N ) − P f , τ ) L e 1 = 0 ⊕ 0 , ω 2 = τ 2 ⊕ ( I ℓ p ( N ) − P f , τ ) τ 2 = L e n ⊕ ( I ℓ p ( N ) − P f , τ ) L e 2 = e 1 ⊕ ( I ℓ p ( N ) − P f , τ ) e 1 = e 1 ⊕ R L e 1 = e 1 ⊕ 0 , \omega_2=\tau_2\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\tau_2=Le_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})Le_2
=e_1\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})e_1=e_1\oplus RLe_1=e_1\oplus0, ω 2 = τ 2 ⊕ ( I ℓ p ( N ) − P f , τ ) τ 2 = L e n ⊕ ( I ℓ p ( N ) − P f , τ ) L e 2 = e 1 ⊕ ( I ℓ p ( N ) − P f , τ ) e 1 = e 1 ⊕ R L e 1 = e 1 ⊕ 0 , ω n = τ n ⊕ ( I ℓ p ( N ) − P f , τ ) τ n = L e n ⊕ ( I ℓ p ( N ) − P f , τ ) L e n = e n − 1 ⊕ ( I ℓ p ( N ) − P f , τ ) e n − 1 = e n − 1 ⊕ R L e n − 1 = e n − 1 ⊕ e n − 1 , ∀ n ≥ 3 , \omega_n=\tau_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})\tau_n=Le_n\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})Le_n
=e_{n-1}\oplus(I_{\ell^p(\mathbb{N})}-P_{f,\tau})e_{n-1}=e_{n-1}\oplus RLe_{n-1}=e_{n-1}\oplus e_{n-1},\quad\forall n\ge3, ω n = τ n ⊕ ( I ℓ p ( N ) − P f , τ ) τ n = L e n ⊕ ( I ℓ p ( N ) − P f , τ ) L e n = e n − 1 ⊕ ( I ℓ p ( N ) − P f , τ ) e n − 1 = e n − 1 ⊕ R L e n − 1 = e n − 1 ⊕ e n − 1 , ∀ n ≥ 3 , g n = f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) = ζ n R ⊕ ζ n R L = ζ n R ( I ℓ p ( N ) ⊕ L ) , ∀ n ∈ N g_n=f_n\oplus\zeta_n(I_{\ell^p(\mathbb{N})}-P_{f,\tau})=\zeta_nR\oplus\zeta_nRL=\zeta_nR(I_{\ell^p(\mathbb{N})}\oplus L),\quad\forall n\in\mathbb{N} g n = f n ⊕ ζ n ( I ℓ p ( N ) − P f , τ ) = ζ n R ⊕ ζ n R L = ζ n R ( I ℓ p ( N ) ⊕ L ) , ∀ n ∈ N and ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
NEW IDENTITY FOR PARSEVAL p-APPROXIMATE SCH-AUDER FRAMES
Certain classes of Banach spaces known as homogeneous semi-inner product spaces admit a kind of inner product and can be studied with certain similarities with Hilbert spaces. These spaces are introduced by Lumer [161 ] and studied extensively by Giles [119 ] . We now recall the fundamentals of semi-inner products. Let X \mathcal{X} X K \mathbb{K} K [ ⋅ , ⋅ ] : X × X → K [\cdot,\cdot]:\mathcal{X}\times\mathcal{X}\to\mathbb{K} [ ⋅ , ⋅ ] : X × X → K homogeneous semi-inner product if it satisfies the following.
(i) [ x , x ] > 0 [x,x]>0 [ x , x ] > 0 x ∈ X x\in\mathcal{X} x ∈ X x ≠ 0 x\ne0 x = 0
(ii) [ λ x , y ] = λ [ x , y ] [\lambda x,y]=\lambda[x,y] [ λ x , y ] = λ [ x , y ] x , y ∈ X x,y\in\mathcal{X} x , y ∈ X λ ∈ K \lambda\in\mathbb{K} λ ∈ K
(iii) [ x , λ y ] = λ ‾ [ x , y ] [x,\lambda y]=\overline{\lambda}[x,y] [ x , λ y ] = λ [ x , y ] x , y ∈ X x,y\in\mathcal{X} x , y ∈ X λ ∈ K \lambda\in\mathbb{K} λ ∈ K
(iv) [ x + y , z ] = [ x , z ] + [ y , z ] [x+y,z]=[x,z]+[y,z] [ x + y , z ] = [ x , z ] + [ y , z ] x , y , z ∈ X x,y,z\in\mathcal{X} x , y , z ∈ X
(v) ∣ [ x , y ] ∣ 2 ≤ [ x , x ] [ y , y ] |[x,y]|^2 \le[x,x][y,y] ∣ [ x , y ] ∣ 2 ≤ [ x , x ] [ y , y ] x , y ∈ X x,y \in\mathcal{X} x , y ∈ X
A homogeneous semi-inner product [ ⋅ , ⋅ ] [\cdot,\cdot] [ ⋅ , ⋅ ] norm which is defined as ∥ x ∥ : = [ x , x ] \lVert x\rVert:= \sqrt{[x,x]} ∥ x ∥ := [ x , x ] ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) 1 < p < ∞ 1 < p < \infty 1 < p < ∞ x = { x n } n x=\{x_n\}_n x = { x n } n y = { y n } n ∈ ℓ p ( N ) y=\{y_n\}_n \in\ell^p(\mathbb{N}) y = { y n } n ∈ ℓ p ( N )
[ x , y ] : = { ∑ n = 1 ∞ x n y n ‾ ∣ y n ∣ p − 2 ∥ y ∥ p p − 2 if y ≠ 0 , 0 if y = 0. [x,y] := \begin{cases}
\dfrac{\sum_{n=1}^{\infty}x_n\overline{y_n}|y_n|^{p-2}}{\lVert y\rVert_p^{p-2}} & \text{if } y \ne0,\\
0 & \text{if } y=0.
\end{cases} [ x , y ] := ⎩ ⎨ ⎧ ∥ y ∥ p p − 2 ∑ n = 1 ∞ x n y n ∣ y n ∣ p − 2 0 if y = 0 , if y = 0. For certain classes of Banach spaces we have Riesz representation theorem.
Theorem 4.5.1. (Giles [119 ] ) (Riesz representation theorem for Banach spaces ) Let X \mathcal{X} X X \mathcal{X} X f : X → K f:\mathcal{X}\to\mathbb{K} f : X → K y ∈ X y\in\mathcal{X} y ∈ X f ( x ) = [ x , y ] f(x)=[x,y] f ( x ) = [ x , y ] x ∈ X x\in\mathcal{X} x ∈ X
Theorem 4.5.1 leads to the notion of generalized adjoint whose existence is assured by the following theorem. Two state the result we need two definitions.
Definition 4.5.2. (cf. Giles [119 ] ) Let X \mathcal{X} X X \mathcal{X} X
Re ( [ x , y + λ x ] ) → Re ( [ x , y ] ) , for all real λ → 0 , ∀ x , y ∈ X such that ∥ x ∥ = ∥ y ∥ = 1. \operatorname{Re}([x,y+\lambda x]) \to\operatorname{Re}([x,y]), \text{ for all real } \lambda\to0,\ \forall x,y \in\mathcal{X} \text{ such that } \lVert x\rVert=\lVert y\rVert=1. Re ([ x , y + λ x ]) → Re ([ x , y ]) , for all real λ → 0 , ∀ x , y ∈ X such that ∥ x ∥ = ∥ y ∥ = 1. Definition 4.5.3. (Giles [119 ] ) A Banach space is said to be uniformly convex if given ε > 0 \varepsilon>0 ε > 0 δ > 0 \delta>0 δ > 0 x , y ∈ X x,y\in\mathcal{X} x , y ∈ X ∥ x ∥ = ∥ y ∥ = 1 \lVert x\rVert=\lVert y\rVert=1 ∥ x ∥ = ∥ y ∥ = 1 ∥ x − y ∥ > ε \lVert x-y\rVert>\varepsilon ∥ x − y ∥ > ε ∥ x + y ∥ ≤ 2 ( 1 − δ ) \lVert x+y\rVert\le2(1-\delta) ∥ x + y ∥ ≤ 2 ( 1 − δ )
Theorem 4.5.4. (Koehler [149 ] ) Let X \mathcal{X} X X \mathcal{X} X A : X → X A:\mathcal{X}\to\mathcal{X} A : X → X A † : X → X A^\dagger:\mathcal{X}\to\mathcal{X} A † : X → X generalized adjoint of A A A
[ A x , y ] = [ x , A † y ] , ∀ x , y ∈ X . [Ax,y]=[x,A^\dagger y], \qquad\forall x,y\in\mathcal{X}. [ A x , y ] = [ x , A † y ] , ∀ x , y ∈ X . Moreover, the following statements hold.
(i) ( λ A ) † = λ ‾ A † (\lambda A)^\dagger=\overline{\lambda}A^\dagger ( λ A ) † = λ A † λ ∈ K \lambda\in\mathbb{K} λ ∈ K
(ii) A † A^\dagger A † A ( X ) ‾ = X \overline{A(\mathcal{X})}=\mathcal{X} A ( X ) = X
(iii) If the norm of X \mathcal{X} X A † A^{\dagger} A †
Throughout this section we assume that X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X f n f_n f n ω n ∈ X \omega_n \in\mathcal{X} ω n ∈ X f n ( x ) = [ x , ω n ] f_n(x)=[x,\omega_n] f n ( x ) = [ x , ω n ] x ∈ X x\in\mathcal{X} x ∈ X
∑ n = 1 ∞ [ x , ( S ω , τ − 1 ) † ω n ] S ω , τ − 1 τ n = S ω , τ − 1 ( ∑ n = 1 ∞ [ S ω , τ − 1 x , ω n ] τ n ) = S ω , τ − 1 x , ∀ x ∈ X . \sum_{n=1}^{\infty}[x,(S_{\omega,\tau}^{-1})^{\dagger}\omega_n]S_{\omega,\tau}^{-1}\tau_n=S_{\omega,\tau}^{-1}\left(\sum_{n=1}^{\infty}[S_{\omega,\tau}^{-1}x,\omega_n]\tau_n\right)=S_{\omega,\tau}^{-1}x,\quad\forall x\in\mathcal{X}. n = 1 ∑ ∞ [ x , ( S ω , τ − 1 ) † ω n ] S ω , τ − 1 τ n = S ω , τ − 1 ( n = 1 ∑ ∞ [ S ω , τ − 1 x , ω n ] τ n ) = S ω , τ − 1 x , ∀ x ∈ X . Hence ( { ω ~ n : = ( S ω , τ − 1 ) † ω n } n , { τ ~ n : = S ω , τ − 1 τ n } n ) (\{\widetilde{\omega}_n:=(S_{\omega,\tau}^{-1})^{\dagger}\omega_n\}_n,\{\widetilde{\tau}_n:=S_{\omega,\tau}^{-1}\tau_n\}_n) ({ ω n := ( S ω , τ − 1 ) † ω n } n , { τ n := S ω , τ − 1 τ n } n ) X \mathcal{X} X ( { ω n } n , { τ n } n ) (\{\omega_n\}_n,\{\tau_n\}_n) ({ ω n } n , { τ n } n ) M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N S M : X ∋ x ↦ ∑ n ∈ M ⟨ x , ω n ⟩ τ n ∈ X S_{\mathbb{M}}:\mathcal{X}\ni x\mapsto\sum_{n\in\mathbb{M}}\langle x,\omega_n\rangle\tau_n\in\mathcal{X} S M : X ∋ x ↦ ∑ n ∈ M ⟨ x , ω n ⟩ τ n ∈ X S M S_{\mathbb{M}} S M S M S_{\mathbb{M}} S M [23 ] it is derived that if operators U , V : H → H U,V:\mathcal{H}\to\mathcal{H} U , V : H → H U + V = I H U+V=I_{\mathcal{H}} U + V = I H U − V = U 2 − V 2 U-V=U^2-V^2 U − V = U 2 − V 2
Lemma 4.5.5. If operators U , V : X → X U,V:\mathcal{X}\to\mathcal{X} U , V : X → X U + V = I X U+V=I_{\mathcal{X}} U + V = I X U − V = U 2 − V 2 U-V=U^2-V^2 U − V = U 2 − V 2
Proof. We follow the ideas in the proof of Proposition 2.2 in [23 ] :
U − V = U − ( I X − U ) = 2 U − I X = U 2 − ( I X − 2 U + U 2 ) = U 2 − ( I X − U ) 2 = U 2 − V 2 . \begin{aligned}
U-V&=U-(I_{\mathcal{X}}-U)=2U-I_{\mathcal{X}}=U^2-(I_{\mathcal{X}}-2U+U^2)\\
&=U^2-(I_{\mathcal{X}}-U)^2=U^2-V^2.
\end{aligned} U − V = U − ( I X − U ) = 2 U − I X = U 2 − ( I X − 2 U + U 2 ) = U 2 − ( I X − U ) 2 = U 2 − V 2 . We now have Banach space version of Theorem 1.2.47 .
Theorem 4.5.6. Let ( { ω n } n , { τ n } n ) (\{\omega_n\}_n,\{\tau_n\}_n) ({ ω n } n , { τ n } n ) X \mathcal{X} X M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N x ∈ X x\in\mathcal{X} x ∈ X
∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M x , ω ~ n ] [ τ ~ n , S M † x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M c x , ω ~ n ] [ τ ~ n , S M c † x ] . \sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}}x,\widetilde{\omega}_n][\widetilde{\tau}_n,S_{\mathbb{M}}^{\dagger}x]=\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}^{c}}x,\widetilde{\omega}_n][\widetilde{\tau}_n,S_{\mathbb{M}^{c}}^{\dagger}x]. n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M x , ω n ] [ τ n , S M † x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M c x , ω n ] [ τ n , S M c † x ] . Proof. For notational convenience, we denote S f , τ S_{f,\tau} S f , τ S S S S M + S M c = S S_{\mathbb{M}}+S_{\mathbb{M}^{c}}=S S M + S M c = S S − 1 S M + S − 1 S M c = I X S^{-1}S_{\mathbb{M}}+S^{-1}S_{\mathbb{M}^{c}}=I_{\mathcal{X}} S − 1 S M + S − 1 S M c = I X S − 1 S M − S − 1 S M c = ( S − 1 S M ) 2 − ( S − 1 S M c ) 2 = S − 1 S M S − 1 S M − S − 1 S M c S − 1 S M c S^{-1}S_{\mathbb{M}}-S^{-1}S_{\mathbb{M}^{c}}=(S^{-1}S_{\mathbb{M}})^2-(S^{-1}S_{\mathbb{M}^{c}})^2=S^{-1}S_{\mathbb{M}}S^{-1}S_{\mathbb{M}}-S^{-1}S_{\mathbb{M}^{c}}S^{-1}S_{\mathbb{M}^{c}} S − 1 S M − S − 1 S M c = ( S − 1 S M ) 2 − ( S − 1 S M c ) 2 = S − 1 S M S − 1 S M − S − 1 S M c S − 1 S M c
S − 1 S M − S − 1 S M S − 1 S M = S − 1 S M c − S − 1 S M c S − 1 S M c . S^{-1}S_{\mathbb{M}}-S^{-1}S_{\mathbb{M}}S^{-1}S_{\mathbb{M}}=S^{-1}S_{\mathbb{M}^{c}}-S^{-1}S_{\mathbb{M}^{c}}S^{-1}S_{\mathbb{M}^{c}}. S − 1 S M − S − 1 S M S − 1 S M = S − 1 S M c − S − 1 S M c S − 1 S M c . Therefore for all x , y ∈ X x,y\in\mathcal{X} x , y ∈ X
[ S − 1 S M x , y ] − [ S − 1 S M S − 1 S M x , y ] = [ S − 1 S M c x , y ] − [ S − 1 S M c S − 1 S M c x , y ] . [S^{-1}S_{\mathbb{M}}x,y]-[S^{-1}S_{\mathbb{M}}S^{-1}S_{\mathbb{M}}x,y]=[S^{-1}S_{\mathbb{M}^{c}}x,y]-[S^{-1}S_{\mathbb{M}^{c}}S^{-1}S_{\mathbb{M}^{c}}x,y]. [ S − 1 S M x , y ] − [ S − 1 S M S − 1 S M x , y ] = [ S − 1 S M c x , y ] − [ S − 1 S M c S − 1 S M c x , y ] . In particular, for all x ∈ X x \in\mathcal{X} x ∈ X
[ S − 1 S M x , S † x ] − [ S − 1 S M S − 1 S M x , S † x ] = [ S − 1 S M c x , S † x ] − [ S − 1 S M c S − 1 S M c x , S † x ] [S^{-1}S_{\mathbb{M}}x,S^{\dagger}x]-[S^{-1}S_{\mathbb{M}}S^{-1}S_{\mathbb{M}}x,S^{\dagger}x]=[S^{-1}S_{\mathbb{M}^{c}}x,S^{\dagger}x]-[S^{-1}S_{\mathbb{M}^{c}}S^{-1}S_{\mathbb{M}^{c}}x,S^{\dagger}x] [ S − 1 S M x , S † x ] − [ S − 1 S M S − 1 S M x , S † x ] = [ S − 1 S M c x , S † x ] − [ S − 1 S M c S − 1 S M c x , S † x ] which gives
[ S M x , x ] − [ S − 1 S M x , S M † x ] = [ S M c x , x ] − [ S − 1 S M c x , S M c † x ] , ∀ x ∈ X . (33) [S_{\mathbb{M}}x,x]-[S^{-1}S_{\mathbb{M}}x,S_{\mathbb{M}}^{\dagger}x]=[S_{\mathbb{M}^{c}}x,x]-[S^{-1}S_{\mathbb{M}^{c}}x,S_{\mathbb{M}^{c}}^{\dagger}x],\quad\forall x\in\mathcal{X}. \tag*{(33)} [ S M x , x ] − [ S − 1 S M x , S M † x ] = [ S M c x , x ] − [ S − 1 S M c x , S M c † x ] , ∀ x ∈ X . (33) Now note that
∑ n = 1 ∞ [ x , ω ~ n ] [ τ ~ n , y ] = ∑ n = 1 ∞ [ x , ( S − 1 ) † ω n ] [ S − 1 τ n , y ] = ∑ n = 1 ∞ [ S − 1 x , ω n ] [ S − 1 τ n , y ] = [ ∑ n = 1 ∞ [ S − 1 x , ω n ] S − 1 τ n , y ] = [ S − 1 ( ∑ n = 1 ∞ [ S − 1 x , ω n ] τ n ) , y ] = [ S − 1 x , y ] , ∀ x , y ∈ X . \begin{aligned}
\sum_{n=1}^{\infty}[x,\widetilde{\omega}_{n}][\widetilde{\tau}_{n},y]
&=\sum_{n=1}^{\infty}[x,(S^{-1})^{\dagger}\omega_{n}][S^{-1}\tau_{n},y]
=\sum_{n=1}^{\infty}[S^{-1}x,\omega_{n}][S^{-1}\tau_{n},y]\\
&=\left[\sum_{n=1}^{\infty}[S^{-1}x,\omega_{n}]S^{-1}\tau_{n},y\right]
=\left[S^{-1}\left(\sum_{n=1}^{\infty}[S^{-1}x,\omega_{n}]\tau_{n}\right),y\right]\\
&=[S^{-1}x,y],\quad\forall x,y\in\mathcal{X}.
\end{aligned} n = 1 ∑ ∞ [ x , ω n ] [ τ n , y ] = n = 1 ∑ ∞ [ x , ( S − 1 ) † ω n ] [ S − 1 τ n , y ] = n = 1 ∑ ∞ [ S − 1 x , ω n ] [ S − 1 τ n , y ] = [ n = 1 ∑ ∞ [ S − 1 x , ω n ] S − 1 τ n , y ] = [ S − 1 ( n = 1 ∑ ∞ [ S − 1 x , ω n ] τ n ) , y ] = [ S − 1 x , y ] , ∀ x , y ∈ X . Equation (4.5.1) now gives
∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M x , ω ~ n ] [ τ ~ n , S M † x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M c x , ω ~ n ] [ τ ~ n , S M c † x ] , ∀ x ∈ X . \sum_{n\in\mathbb{M}}[x,\omega_{n}][\tau_{n},x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}}x,\widetilde{\omega}_{n}][\widetilde{\tau}_{n},S_{\mathbb{M}}^{\dagger}x]=\sum_{n\in\mathbb{M}^{c}}[x,\omega_{n}][\tau_{n},x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}^{c}}x,\widetilde{\omega}_{n}][\widetilde{\tau}_{n},S_{\mathbb{M}^{c}}^{\dagger}x],\quad\forall x\in\mathcal{X}. n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M x , ω n ] [ τ n , S M † x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M c x , ω n ] [ τ n , S M c † x ] , ∀ x ∈ X . A look at Theorem 1.2.48 makes a guess of the following statement for Banach spaces. Let ( { ω n } n , { τ n } n ) (\{\omega_n\}_n,\{\tau_n\}_n) ({ ω n } n , { τ n } n ) X \mathcal{X} X M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N
∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∥ ∑ n ∈ M [ x , ω n ] τ n ∥ 2 = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∥ ∑ n ∈ M c [ x , ω n ] τ n ∥ 2 , ∀ x ∈ X . (34) \sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\left\|\sum_{n\in\mathbb{M}}[x,\omega_n]\tau_n\right\|^2=\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]-\left\|\sum_{n\in\mathbb{M}^{c}}[x,\omega_n]\tau_n\right\|^2,\quad\forall x\in\mathcal{X}. \tag*{(34)} n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n ∈ M ∑ [ x , ω n ] τ n 2 = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n ∈ M c ∑ [ x , ω n ] τ n 2 , ∀ x ∈ X . (34) However, the correct Banach space version of Theorem 1.2.48 is not Equation (34) but it is stated in the next theorem.
Theorem 4.5.7. (Parseval p-ASF identity ) Let ( { ω n } n , { τ n } n ) (\{\omega_n\}_n,\{\tau_n\}_n) ({ ω n } n , { τ n } n ) X \mathcal{X} X M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N
∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∑ n ∈ M ∑ k ∈ M [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∑ n ∈ M c ∑ k ∈ M c [ x , ω n ] [ τ n , ω k ] [ τ k , x ] , ∀ x ∈ X . \sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\sum_{n\in\mathbb{M}}\sum_{k\in\mathbb{M}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x]=\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]-\sum_{n\in\mathbb{M}^{c}}\sum_{k\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x],\quad\forall x\in\mathcal{X}. n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n ∈ M ∑ k ∈ M ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n ∈ M c ∑ k ∈ M c ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] , ∀ x ∈ X . Proof. Using Theorem 4.5.6 , for all x ∈ X x \in\mathcal{X} x ∈ X
∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∑ n ∈ M ∑ k ∈ M [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ ∑ n ∈ M [ x , ω n ] ∑ k ∈ M [ τ n , ω k ] τ k , x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ ∑ n ∈ M [ x , ω n ] S M τ n , x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ S M ( ∑ n ∈ M [ x , ω n ] τ n ) , x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ S M S M x , x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ S M x , S M † x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − [ ∑ n = 1 ∞ [ S M x , ω n ] τ n , S M † x ] = ∑ n ∈ M [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M x , ω n ] [ τ n , S M † x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∑ n = 1 ∞ [ S M c x , ω n ] [ τ n , S M c † x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] − ∑ n ∈ M c ∑ k ∈ M c [ x , ω n ] [ τ n , ω k ] [ τ k , x ] . \begin{aligned}
\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\sum_{n\in\mathbb{M}}\sum_{k\in\mathbb{M}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x]
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\left[\sum_{n\in\mathbb{M}}[x,\omega_n]\sum_{k\in\mathbb{M}}[\tau_n,\omega_k]\tau_k,x\right]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\left[\sum_{n\in\mathbb{M}}[x,\omega_n]S_{\mathbb{M}}\tau_n,x\right]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\left[S_{\mathbb{M}}\left(\sum_{n\in\mathbb{M}}[x,\omega_n]\tau_n\right),x\right]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-[S_{\mathbb{M}}S_{\mathbb{M}}x,x]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-[S_{\mathbb{M}}x,S_{\mathbb{M}}^{\dagger}x]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\left[\sum_{n=1}^{\infty}[S_{\mathbb{M}}x,\omega_n]\tau_n,S_{\mathbb{M}}^{\dagger}x\right]\\
&=\sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}}x,\omega_n][\tau_n,S_{\mathbb{M}}^{\dagger}x]\\
&=\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]-\sum_{n=1}^{\infty}[S_{\mathbb{M}^{c}}x,\omega_n][\tau_n,S_{\mathbb{M}^{c}}^{\dagger}x]\\
&=\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]-\sum_{n\in\mathbb{M}^{c}}\sum_{k\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x].
\end{aligned} n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n ∈ M ∑ k ∈ M ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ n ∈ M ∑ [ x , ω n ] k ∈ M ∑ [ τ n , ω k ] τ k , x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ n ∈ M ∑ [ x , ω n ] S M τ n , x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ S M ( n ∈ M ∑ [ x , ω n ] τ n ) , x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ S M S M x , x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ S M x , S M † x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − [ n = 1 ∑ ∞ [ S M x , ω n ] τ n , S M † x ] = n ∈ M ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M x , ω n ] [ τ n , S M † x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n = 1 ∑ ∞ [ S M c x , ω n ] [ τ n , S M c † x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] − n ∈ M c ∑ k ∈ M c ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] . In terms of S M S_{\mathbb{M}} S M S M 2 S_{\mathbb{M}}^{2} S M 2 Theorem 4.5.7 can be written as
S M − S M 2 = S M c − S M c 2 or S M + S M c 2 = S M c + S M 2 . (35) S_{\mathbb{M}}-S_{\mathbb{M}}^{2}=S_{\mathbb{M}^{c}}-S_{\mathbb{M}^{c}}^{2}
\quad\text{or}\quad
S_{\mathbb{M}}+S_{\mathbb{M}^{c}}^{2}=S_{\mathbb{M}^{c}}+S_{\mathbb{M}}^{2}. \tag*{(35)} S M − S M 2 = S M c − S M c 2 or S M + S M c 2 = S M c + S M 2 . (35) We now give an application of Theorem 4.5.7 . This is Banach space version of Theorem 1.2.49.
Theorem 4.5.8. Let ( { ω n } n , { τ n } n ) (\{\omega_n\}_n,\{\tau_n\}_n) ({ ω n } n , { τ n } n ) p p p X \mathcal{X} X M ⊆ N \mathbb{M}\subseteq\mathbb{N} M ⊆ N x ∈ X x\in\mathcal{X} x ∈ X [ ( S M − 1 2 I X ) 2 x , x ] ≥ 0 [(S_{\mathbb{M}}-\frac{1}{2}I_{\mathcal{X}})^2x,x]\geq0 [( S M − 2 1 I X ) 2 x , x ] ≥ 0
∑ n ∈ M [ x , ω n ] [ τ n , x ] + ∑ n ∈ M c ∑ k ∈ M c [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = ∑ n ∈ M c [ x , ω n ] [ τ n , x ] + ∑ n ∈ M ∑ k ∈ M [ x , ω n ] [ τ n , ω k ] [ τ k , x ] ≥ 3 4 ∥ x ∥ 2 , for that x . \sum_{n\in\mathbb{M}}[x,\omega_n][\tau_n,x]+\sum_{n\in\mathbb{M}^{c}}\sum_{k\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x]
=
\sum_{n\in\mathbb{M}^{c}}[x,\omega_n][\tau_n,x]+\sum_{n\in\mathbb{M}}\sum_{k\in\mathbb{M}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x]\geq\frac{3}{4}\lVert x\rVert^{2},\quad\text{for that }x. n ∈ M ∑ [ x , ω n ] [ τ n , x ] + n ∈ M c ∑ k ∈ M c ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = n ∈ M c ∑ [ x , ω n ] [ τ n , x ] + n ∈ M ∑ k ∈ M ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] ≥ 4 3 ∥ x ∥ 2 , for that x . :::`
Proof. We first compute
S M 2 + S M c 2 = S M 2 + ( I X − S M ) 2 = 2 S M 2 − 2 S M + I X = 2 ( S M − 1 2 I X ) 2 + 1 2 I X . \begin{aligned}
S_{\mathcal{M}}^{2}+S_{\mathcal{M}^{c}}^{2}=S_{\mathcal{M}}^{2}+(I_{\mathcal{X}}-S_{\mathcal{M}})^{2}=2S_{\mathcal{M}}^{2}-2S_{\mathcal{M}}+I_{\mathcal{X}} \\
&=2\left(S_{\mathcal{M}}-\frac{1}{2}I_{\mathcal{X}}\right)^{2}+\frac{1}{2}I_{\mathcal{X}}.
\end{aligned} S M 2 + S M c 2 = S M 2 + ( I X − S M ) 2 = 2 S M 2 − 2 S M + I X = 2 ( S M − 2 1 I X ) 2 + 2 1 I X . Hence if x ∈ X x\in\mathcal{X} x ∈ X [ ( S M − 1 2 I X ) 2 x , x ] ≥ 0 \left[\left(S_{\mathcal{M}}-\frac{1}{2}I_{\mathcal{X}}\right)^{2}x,x\right]\geq0 [ ( S M − 2 1 I X ) 2 x , x ] ≥ 0
[ ( S M 2 + S M c 2 ) x , x ] ≥ 1 2 ∥ x ∥ 2 . \left[\left(S_{\mathcal{M}}^{2}+S_{\mathcal{M}^{c}}^{2}\right)x,x\right]\geq\frac{1}{2}\lVert x\rVert^{2}. [ ( S M 2 + S M c 2 ) x , x ] ≥ 2 1 ∥ x ∥ 2 . Now using Equation (35) we get
2 ∑ n ∈ M [ x , ω n ] [ τ n , x ] + 2 ∑ n ∈ M c ∑ k ∈ M c [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = 2 [ S M x , x ] + 2 [ S M c 2 x , x ] = [ 2 ( S M + S M c 2 ) x , x ] = [ ( ( S M + S M c 2 ) + ( S M + S M c 2 ) ) x , x ] = [ ( ( S M + S M c 2 ) + ( S M c + S M 2 ) ) x , x ] = [ ( I X + S M c 2 + S M 2 ) x , x ] = ∥ x ∥ 2 + [ ( S M 2 + S M c 2 ) x , x ] ≥ 3 2 ∥ x ∥ 2 , ∀ x ∈ X . \begin{aligned}
2\sum_{n\in\mathcal{M}}[x,\omega_n][\tau_n,x]+2\sum_{n\in\mathcal{M}^{c}}\sum_{k\in\mathcal{M}^{c}}[x,\omega_n][\tau_n,\omega_k][\tau_k,x]=2[S_{\mathcal{M}}x,x]+2[S_{\mathcal{M}^{c}}^{2}x,x] \\
&=[2(S_{\mathcal{M}}+S_{\mathcal{M}^{c}}^{2})x,x]=[((S_{\mathcal{M}}+S_{\mathcal{M}^{c}}^{2})+(S_{\mathcal{M}}+S_{\mathcal{M}^{c}}^{2}))x,x] \\
&=[((S_{\mathcal{M}}+S_{\mathcal{M}^{c}}^{2})+(S_{\mathcal{M}^{c}}+S_{\mathcal{M}}^{2}))x,x]=[(I_{\mathcal{X}}+S_{\mathcal{M}^{c}}^{2}+S_{\mathcal{M}}^{2})x,x] \\
&=\lVert x\rVert^{2}+[(S_{\mathcal{M}}^{2}+S_{\mathcal{M}^{c}}^{2})x,x]\geq\frac{3}{2}\lVert x\rVert^{2},\quad\forall x\in\mathcal{X}.
\end{aligned} 2 n ∈ M ∑ [ x , ω n ] [ τ n , x ] + 2 n ∈ M c ∑ k ∈ M c ∑ [ x , ω n ] [ τ n , ω k ] [ τ k , x ] = 2 [ S M x , x ] + 2 [ S M c 2 x , x ] = [ 2 ( S M + S M c 2 ) x , x ] = [(( S M + S M c 2 ) + ( S M + S M c 2 )) x , x ] = [(( S M + S M c 2 ) + ( S M c + S M 2 )) x , x ] = [( I X + S M c 2 + S M 2 ) x , x ] = ∥ x ∥ 2 + [( S M 2 + S M c 2 ) x , x ] ≥ 2 3 ∥ x ∥ 2 , ∀ x ∈ X . PALEY-WIENER THEOREM FOR p-APPROXIMATE SCHAUDER FRAMES
In order to derive Paley-Wiener theorem for p-ASFs, we need a generalization of result of Hilding [132 ] .
Theorem 4.6.1. [48 , 54 ] (); van Eijndhoven 1996) (Casazza-Christensen-Kalton-van Eijndhoven perturbation ) Let X , Y \mathcal{X},\mathcal{Y} X , Y A : X → Y A:\mathcal{X}\to\mathcal{Y} A : X → Y B : X → Y B:\mathcal{X}\to\mathcal{Y} B : X → Y α , β ∈ [ 0 , 1 ) \alpha,\beta\in[0,1) α , β ∈ [ 0 , 1 )
∥ A x − B x ∥ ≤ α ∥ A x ∥ + β ∥ B x ∥ , ∀ x ∈ X , \lVert Ax-Bx\rVert\leq\alpha\lVert Ax\rVert+\beta\lVert Bx\rVert,\quad\forall x\in\mathcal{X}, ∥ A x − B x ∥ ≤ α ∥ A x ∥ + β ∥ B x ∥ , ∀ x ∈ X , then B B B
1 − α 1 + β ∥ A x ∥ ≤ ∥ B x ∥ ≤ 1 + α 1 − β ∥ A x ∥ , ∀ x ∈ X ; \frac{1-\alpha}{1+\beta}\lVert Ax\rVert\leq\lVert Bx\rVert\leq\frac{1+\alpha}{1-\beta}\lVert Ax\rVert,\quad\forall x\in\mathcal{X}; 1 + β 1 − α ∥ A x ∥ ≤ ∥ B x ∥ ≤ 1 − β 1 + α ∥ A x ∥ , ∀ x ∈ X ; 1 − β 1 + α 1 ∥ A ∥ ∥ y ∥ ≤ ∥ B − 1 y ∥ ≤ 1 + β 1 − α ∥ A − 1 ∥ ∥ y ∥ , ∀ y ∈ Y . \frac{1-\beta}{1+\alpha}\frac{1}{\lVert A\rVert}\lVert y\rVert\leq\lVert B^{-1}y\rVert\leq\frac{1+\beta}{1-\alpha}\lVert A^{-1}\rVert\lVert y\rVert,\quad\forall y\in\mathcal{Y}. 1 + α 1 − β ∥ A ∥ 1 ∥ y ∥ ≤ ∥ B − 1 y ∥ ≤ 1 − α 1 + β ∥ A − 1 ∥ ∥ y ∥ , ∀ y ∈ Y . In the sequel, the standard Schauder basis for ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { e n } n \{e_n\}_n { e n } n
Theorem 4.6.2. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|,\beta\}<1 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 m c n ω n ∥ , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . (36) \left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|\leq\alpha\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{m}c_n\omega_n\right\|,
\qquad\forall c_1,\ldots,c_m\in\mathbb{K},\ m=1,2,\ldots. \tag*{(36)} n = 1 ∑ m c n ( τ n − ω n ) ≤ α n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 + β n = 1 ∑ m c n ω n , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . (36) Then ( { f n } n , { ω n } n ) \left(\{f_n\}_n,\{\omega_n\}_n\right) ( { f n } n , { ω n } n ) p p p X \mathcal{X} X
1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) ( 1 + β ) ∥ S f , τ − 1 ∥ and ( 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β ) ∥ θ f ∥ . \frac{1-(\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|)}{(1+\beta)\|S_{f,\tau}^{-1}\|}
\qquad\text{and}\qquad
\left(\frac{1+\alpha}{1-\beta}\|\theta_\tau\|+\frac{\gamma}{1-\beta}\right)\|\theta_f\|. ( 1 + β ) ∥ S f , τ − 1 ∥ 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) and ( 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ ) ∥ θ f ∥. Proof. For m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , … c 1 , … , c m ∈ K c_1,\ldots,c_m\in\mathbb{K} c 1 , … , c m ∈ K
∥ ∑ n = 1 m c n ω n ∥ ≤ ∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ + ∥ ∑ n = 1 m c n τ n ∥ ≤ ( 1 + α ) ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 m c n ω n ∥ . \begin{aligned}
\left\|\sum_{n=1}^{m}c_n\omega_n\right\|
\leq\left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|+\left\|\sum_{n=1}^{m}c_n\tau_n\right\| \\
&\leq(1+\alpha)\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{m}c_n\omega_n\right\|.
\end{aligned} n = 1 ∑ m c n ω n ≤ n = 1 ∑ m c n ( τ n − ω n ) + n = 1 ∑ m c n τ n ≤ ( 1 + α ) n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 + β n = 1 ∑ m c n ω n . Hence
∥ ∑ n = 1 m c n ω n ∥ ≤ 1 + α 1 − β ∥ ∑ n = 1 m c n τ n ∥ + γ 1 − β ( ∑ n = 1 m ∣ c n ∣ p ) 1 p , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . \left\|\sum_{n=1}^{m}c_n\omega_n\right\|
\leq\frac{1+\alpha}{1-\beta}\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\frac{\gamma}{1-\beta}\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}},
\qquad\forall c_1,\ldots,c_m\in\mathbb{K},\ m=1,2,\ldots. n = 1 ∑ m c n ω n ≤ 1 − β 1 + α n = 1 ∑ m c n τ n + 1 − β γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . Therefore θ ω \theta_\omega θ ω
∥ θ ω ∥ ≤ 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β . \|\theta_\omega\|\leq\frac{1+\alpha}{1-\beta}\|\theta_\tau\|+\frac{\gamma}{1-\beta}. ∥ θ ω ∥ ≤ 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ . Now Equation (4.6.1) gives
∥ ∑ n = 1 ∞ c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 ∞ c n τ n ∥ + γ ( ∑ n = 1 ∞ ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 ∞ c n ω n ∥ , ∀ { c n } n ∈ ℓ p ( N ) . \left\|\sum_{n=1}^{\infty}c_n(\tau_n-\omega_n)\right\|\leq\alpha\left\|\sum_{n=1}^{\infty}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{\infty}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{\infty}c_n\omega_n\right\|,
\qquad\forall\{c_n\}_n\in\ell^p(\mathbb{N}). n = 1 ∑ ∞ c n ( τ n − ω n ) ≤ α n = 1 ∑ ∞ c n τ n + γ ( n = 1 ∑ ∞ ∣ c n ∣ p ) p 1 + β n = 1 ∑ ∞ c n ω n , ∀ { c n } n ∈ ℓ p ( N ) . That is,
∥ θ τ { c n } n − θ ω { c n } n ∥ ≤ α ∥ θ τ { c n } n ∥ + γ ( ∑ n = 1 ∞ ∣ c n ∣ p ) 1 p + β ∥ θ ω { c n } n ∥ , \|\theta_\tau\{c_n\}_n-\theta_\omega\{c_n\}_n\|\leq\alpha\|\theta_\tau\{c_n\}_n\|+\gamma\left(\sum_{n=1}^{\infty}|c_n|^p\right)^{\frac{1}{p}}+\beta\|\theta_\omega\{c_n\}_n\|, ∥ θ τ { c n } n − θ ω { c n } n ∥ ≤ α ∥ θ τ { c n } n ∥ + γ ( n = 1 ∑ ∞ ∣ c n ∣ p ) p 1 + β ∥ θ ω { c n } n ∥ , @page 113/200 starts_mid=0 ends_mid=1 In the sequel, the standard Schauder basis for ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { e n } n \{e_n\}_n { e n } n
Theorem 4.6.2. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|,\beta\}<1 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 m c n ω n ∥ , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . (37) \left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|\leq\alpha\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{m}c_n\omega_n\right\|,
\qquad\forall c_1,\ldots,c_m\in\mathbb{K},\ m=1,2,\ldots. \tag*{(37)} n = 1 ∑ m c n ( τ n − ω n ) ≤ α n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 + β n = 1 ∑ m c n ω n , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . (37) Then ( { f n } n , { ω n } n ) \left(\{f_n\}_n,\{\omega_n\}_n\right) ( { f n } n , { ω n } n ) p p p X \mathcal{X} X
1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) ( 1 + β ) ∥ S f , τ − 1 ∥ and ( 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β ) ∥ θ f ∥ . \frac{1-(\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|)}{(1+\beta)\|S_{f,\tau}^{-1}\|}
\qquad\text{and}\qquad
\left(\frac{1+\alpha}{1-\beta}\|\theta_\tau\|+\frac{\gamma}{1-\beta}\right)\|\theta_f\|. ( 1 + β ) ∥ S f , τ − 1 ∥ 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) and ( 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ ) ∥ θ f ∥. Proof. For m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , … c 1 , … , c m ∈ K c_1,\ldots,c_m\in\mathbb{K} c 1 , … , c m ∈ K
∥ ∑ n = 1 m c n ω n ∥ ≤ ∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ + ∥ ∑ n = 1 m c n τ n ∥ ≤ ( 1 + α ) ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 m c n ω n ∥ . \begin{aligned}
\left\|\sum_{n=1}^{m}c_n\omega_n\right\|
\leq\left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|+\left\|\sum_{n=1}^{m}c_n\tau_n\right\| \\
&\leq(1+\alpha)\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{m}c_n\omega_n\right\|.
\end{aligned} n = 1 ∑ m c n ω n ≤ n = 1 ∑ m c n ( τ n − ω n ) + n = 1 ∑ m c n τ n ≤ ( 1 + α ) n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 + β n = 1 ∑ m c n ω n . Hence
∥ ∑ n = 1 m c n ω n ∥ ≤ 1 + α 1 − β ∥ ∑ n = 1 m c n τ n ∥ + γ 1 − β ( ∑ n = 1 m ∣ c n ∣ p ) 1 p , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . \left\|\sum_{n=1}^{m}c_n\omega_n\right\|
\leq\frac{1+\alpha}{1-\beta}\left\|\sum_{n=1}^{m}c_n\tau_n\right\|+\frac{\gamma}{1-\beta}\left(\sum_{n=1}^{m}|c_n|^p\right)^{\frac{1}{p}},
\qquad\forall c_1,\ldots,c_m\in\mathbb{K},\ m=1,2,\ldots. n = 1 ∑ m c n ω n ≤ 1 − β 1 + α n = 1 ∑ m c n τ n + 1 − β γ ( n = 1 ∑ m ∣ c n ∣ p ) p 1 , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … . Therefore θ ω \theta_\omega θ ω
∥ θ ω ∥ ≤ 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β . \|\theta_\omega\|\leq\frac{1+\alpha}{1-\beta}\|\theta_\tau\|+\frac{\gamma}{1-\beta}. ∥ θ ω ∥ ≤ 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ . Now Equation (4.6.1) gives
∥ ∑ n = 1 ∞ c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 ∞ c n τ n ∥ + γ ( ∑ n = 1 ∞ ∣ c n ∣ p ) 1 p + β ∥ ∑ n = 1 ∞ c n ω n ∥ , ∀ { c n } n ∈ ℓ p ( N ) . \left\|\sum_{n=1}^{\infty}c_n(\tau_n-\omega_n)\right\|\leq\alpha\left\|\sum_{n=1}^{\infty}c_n\tau_n\right\|+\gamma\left(\sum_{n=1}^{\infty}|c_n|^p\right)^{\frac{1}{p}}+\beta\left\|\sum_{n=1}^{\infty}c_n\omega_n\right\|,
\qquad\forall\{c_n\}_n\in\ell^p(\mathbb{N}). n = 1 ∑ ∞ c n ( τ n − ω n ) ≤ α n = 1 ∑ ∞ c n τ n + γ ( n = 1 ∑ ∞ ∣ c n ∣ p ) p 1 + β n = 1 ∑ ∞ c n ω n , ∀ { c n } n ∈ ℓ p ( N ) . That is,
∥ θ τ { c n } n − θ ω { c n } n ∥ ≤ α ∥ θ τ { c n } n ∥ + γ ( ∑ n = 1 ∞ ∣ c n ∣ p ) 1 p + β ∥ θ ω { c n } n ∥ , \|\theta_\tau\{c_n\}_n-\theta_\omega\{c_n\}_n\|\leq\alpha\|\theta_\tau\{c_n\}_n\|+\gamma\left(\sum_{n=1}^{\infty}|c_n|^p\right)^{\frac{1}{p}}+\beta\|\theta_\omega\{c_n\}_n\|, ∥ θ τ { c n } n − θ ω { c n } n ∥ ≤ α ∥ θ τ { c n } n ∥ + γ ( n = 1 ∑ ∞ ∣ c n ∣ p ) p 1 + β ∥ θ ω { c n } n ∥ , ∀ { c n } n ∈ ℓ p ( N ) . (38) \forall\{c_n\}_n \in\ell^p(\mathbb{N}). \tag*{(38)} ∀ { c n } n ∈ ℓ p ( N ) . (38) By taking { c n } n = { f n ( S f , τ − 1 x ) } n = θ f S f , τ − 1 x \{c_n\}_n=\{f_n(S_{f,\tau}^{-1}x)\}_n=\theta_fS_{f,\tau}^{-1}x { c n } n = { f n ( S f , τ − 1 x ) } n = θ f S f , τ − 1 x
∥ θ τ θ f S f , τ − 1 x − θ ω θ f S f , τ − 1 x ∥ ≤ α ∥ θ τ θ f S f , τ − 1 x ∥ + γ ( ∑ n = 1 ∞ ∣ f n ( S f , τ − 1 x ) ∣ p ) 1 p + β ∥ θ ω θ f S f , τ − 1 x ∥ , \left\|\theta_\tau\theta_fS_{f,\tau}^{-1}x-\theta_\omega\theta_fS_{f,\tau}^{-1}x\right\|\leq\alpha\left\|\theta_\tau\theta_fS_{f,\tau}^{-1}x\right\|+\gamma\left(\sum_{n=1}^{\infty}\left|f_n(S_{f,\tau}^{-1}x)\right|^p\right)^{\frac{1}{p}}+\beta\left\|\theta_\omega\theta_fS_{f,\tau}^{-1}x\right\|, θ τ θ f S f , τ − 1 x − θ ω θ f S f , τ − 1 x ≤ α θ τ θ f S f , τ − 1 x + γ ( n = 1 ∑ ∞ f n ( S f , τ − 1 x ) p ) p 1 + β θ ω θ f S f , τ − 1 x , for all x ∈ X x\in\mathcal{X} x ∈ X
∥ x − S f , ω S f , τ − 1 x ∥ ≤ α ∥ x ∥ + γ ∥ θ f S f , τ − 1 x ∥ + β ∥ S f , ω S f , τ − 1 x ∥ ≤ ( α + γ ∥ θ f S f , τ − 1 ∥ ) ∥ x ∥ + β ∥ S f , ω S f , τ − 1 x ∥ , ∀ x ∈ X . \begin{aligned}
\left\|x-S_{f,\omega}S_{f,\tau}^{-1}x\right\|\leq\alpha\|x\|+\gamma\|\theta_fS_{f,\tau}^{-1}x\|+\beta\|S_{f,\omega}S_{f,\tau}^{-1}x\| \\
&\leq\left(\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|\right)\|x\|+\beta\|S_{f,\omega}S_{f,\tau}^{-1}x\|,\qquad\forall x\in\mathcal{X}.
\end{aligned} x − S f , ω S f , τ − 1 x ≤ α ∥ x ∥ + γ ∥ θ f S f , τ − 1 x ∥ + β ∥ S f , ω S f , τ − 1 x ∥ ≤ ( α + γ ∥ θ f S f , τ − 1 ∥ ) ∥ x ∥ + β ∥ S f , ω S f , τ − 1 x ∥ , ∀ x ∈ X . Since max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|,\beta\}<1 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1 S f , ω S f , τ − 1 S_{f,\omega}S_{f,\tau}^{-1} S f , ω S f , τ − 1
∥ ( S g , ω S f , τ − 1 ) − 1 ∥ ≤ 1 + β 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) . \left\|(S_{g,\omega}S_{f,\tau}^{-1})^{-1}\right\|\leq\frac{1+\beta}{1-(\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|)}. ( S g , ω S f , τ − 1 ) − 1 ≤ 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) 1 + β . Hence the operator S f , ω = ( S f , ω S f , τ − 1 ) S f , τ S_{f,\omega}=(S_{f,\omega}S_{f,\tau}^{-1})S_{f,\tau} S f , ω = ( S f , ω S f , τ − 1 ) S f , τ ( { f n } n , { ω n } n ) (\{f_n\}_n,\{\omega_n\}_n) ({ f n } n , { ω n } n ) X \mathcal{X} X
∥ S f , ω − 1 ∥ ≤ ∥ S f , τ − 1 ∥ ∥ S f , τ S f , ω − 1 ∥ ≤ ∥ S f , τ − 1 ∥ ( 1 + β ) 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) \|S_{f,\omega}^{-1}\|\leq\|S_{f,\tau}^{-1}\|\|S_{f,\tau}S_{f,\omega}^{-1}\|\leq\frac{\|S_{f,\tau}^{-1}\|(1+\beta)}{1-(\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|)} ∥ S f , ω − 1 ∥ ≤ ∥ S f , τ − 1 ∥∥ S f , τ S f , ω − 1 ∥ ≤ 1 − ( α + γ ∥ θ f S f , τ − 1 ∥ ) ∥ S f , τ − 1 ∥ ( 1 + β ) and
∥ S f , ω ∥ ≤ ∥ θ ω ∥ ∥ θ f ∥ ≤ ( 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β ) ∥ θ f ∥ . \|S_{f,\omega}\|\leq\|\theta_\omega\|\|\theta_f\|\leq\left(\frac{1+\alpha}{1-\beta}\|\theta_\tau\|+\frac{\gamma}{1-\beta}\right)\|\theta_f\|. ∥ S f , ω ∥ ≤ ∥ θ ω ∥∥ θ f ∥ ≤ ( 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ ) ∥ θ f ∥. Remark 4.6.3. Theorem 1.2.55 is a corollary of Theorem 4.6.2. In particular, Theorems 1.2.53 and 1.2.54 are corollaries of Theorem 4.6.2. Indeed, let { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
f n : H ∋ h ↦ f n ( h ) : = ⟨ h , τ n ⟩ ∈ K , ∀ n ∈ N . f_n:\mathcal{H}\ni h\mapsto f_n(h):=\langle h,\tau_n\rangle\in\mathbb{K},\qquad\forall n\in\mathbb{N}. f n : H ∋ h ↦ f n ( h ) := ⟨ h , τ n ⟩ ∈ K , ∀ n ∈ N . Then θ f = θ τ \theta_f=\theta_\tau θ f = θ τ ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) H \mathcal{H} H ( { f n } n , { ω n } n ) (\{f_n\}_n,\{\omega_n\}_n) ({ f n } n , { ω n } n ) H \mathcal{H} H { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H ( { f n } n , { ω n } n ) (\{f_n\}_n,\{\omega_n\}_n) ({ f n } n , { ω n } n ) H \mathcal{H} H θ ω \theta_\omega θ ω { ω n } n \{\omega_n\}_n { ω n } n H \mathcal{H} H
Theorem 4.6.4. (Christensen [62 ] ) A collection { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H
if the map
T : ℓ 2 ( N ) ∋ { c n } n ↦ ∑ n = 1 ∞ c n τ n ∈ H T:\ell^2(\mathbb{N})\ni\{c_n\}_n\mapsto\sum_{n=1}^{\infty}c_n\tau_n\in\mathcal{H} T : ℓ 2 ( N ) ∋ { c n } n ↦ n = 1 ∑ ∞ c n τ n ∈ H is a well-defined bounded linear surjective operator.
Corollary 4.6.5. Let q q q p p p ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X
λ : = ∑ n = 1 ∞ ∥ τ n − ω n ∥ q < 1 ∥ θ f S f , τ − 1 ∥ q . \lambda:=\sum_{n=1}^{\infty}\|\tau_n-\omega_n\|^q<\frac{1}{\|\theta_{f}S_{f,\tau}^{-1}\|^q}. λ := n = 1 ∑ ∞ ∥ τ n − ω n ∥ q < ∥ θ f S f , τ − 1 ∥ q 1 . Then ( { f n } n , { ω n } n ) (\{f_n\}_n,\{\omega_n\}_n) ({ f n } n , { ω n } n ) p p p X \mathcal{X} X
1 − λ 1 / p ∥ θ f S f , τ − 1 ∥ ∥ S f , τ − 1 ∥ \frac{1-\lambda^{1/p}\|\theta_fS_{f,\tau}^{-1}\|}{\|S_{f,\tau}^{-1}\|} ∥ S f , τ − 1 ∥ 1 − λ 1/ p ∥ θ f S f , τ − 1 ∥ and
∥ θ τ ∥ + λ 1 / p . \|\theta_\tau\|+\lambda^{1/p}. ∥ θ τ ∥ + λ 1/ p . Proof. Take α = 0 , β = 0 , γ = λ 1 / p \alpha=0,\beta=0,\gamma=\lambda^{1/p} α = 0 , β = 0 , γ = λ 1/ p max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_fS_{f,\tau}^{-1}\|,\beta\}<1 max { α + γ ∥ θ f S f , τ − 1 ∥ , β } < 1
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ ( ∑ n = 1 m ∥ τ n − ω n ∥ q ) 1 / q ( ∑ n = 1 m ∣ c n ∣ p ) 1 / p ≤ γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 / p , \left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|
\leq
\left(\sum_{n=1}^{m}\|\tau_n-\omega_n\|^q\right)^{1/q}
\left(\sum_{n=1}^{m}|c_n|^p\right)^{1/p}
\leq
\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{1/p}, n = 1 ∑ m c n ( τ n − ω n ) ≤ ( n = 1 ∑ m ∥ τ n − ω n ∥ q ) 1/ q ( n = 1 ∑ m ∣ c n ∣ p ) 1/ p ≤ γ ( n = 1 ∑ m ∣ c n ∣ p ) 1/ p , ∀ c 1 , … , c m ∈ K \forall c_1,\ldots,c_m\in\mathbb{K} ∀ c 1 , … , c m ∈ K m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , …
By using Theorem 4.6.2 we now get the result.
We next derive stability result which does not demand maximum condition on parameters α \alpha α γ \gamma γ
Theorem 4.6.6. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) p p p X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { g n } n \{g_n\}_n { g n } n X ∗ \mathcal{X}^* X ∗ r , s , t , α , β , γ ≥ 0 r,s,t,\alpha,\beta,\gamma\geq0 r , s , t , α , β , γ ≥ 0 max { β , s } < 1 \max\{\beta,s\}<1 max { β , s } < 1
∥ ∑ n = 1 m ( f n − g n ) ( x ) e n ∥ ≤ r ∥ ∑ n = 1 m f n ( x ) e n ∥ + t ∥ x ∥ + s ∥ ∑ n = 1 m g n ( x ) e n ∥ , \left\|\sum_{n=1}^{m}(f_n-g_n)(x)e_n\right\|
\leq
r\left\|\sum_{n=1}^{m}f_n(x)e_n\right\|+t\|x\|+s\left\|\sum_{n=1}^{m}g_n(x)e_n\right\|, n = 1 ∑ m ( f n − g n ) ( x ) e n ≤ r n = 1 ∑ m f n ( x ) e n + t ∥ x ∥ + s n = 1 ∑ m g n ( x ) e n , ∀ x ∈ X , m = 1 , 2 , … , \forall x\in\mathcal{X},m=1,2,\ldots, ∀ x ∈ X , m = 1 , 2 , … ,
∥ ∑ n = 1 m c n ( τ n − ω n ) ∥ ≤ α ∥ ∑ n = 1 m c n τ n ∥ + γ ( ∑ n = 1 m ∣ c n ∣ p ) 1 / p + β ∥ ∑ n = 1 m c n ω n ∥ , \left\|\sum_{n=1}^{m}c_n(\tau_n-\omega_n)\right\|
\leq
\alpha\left\|\sum_{n=1}^{m}c_n\tau_n\right\|
+\gamma\left(\sum_{n=1}^{m}|c_n|^p\right)^{1/p}
+\beta\left\|\sum_{n=1}^{m}c_n\omega_n\right\|, n = 1 ∑ m c n ( τ n − ω n ) ≤ α n = 1 ∑ m c n τ n + γ ( n = 1 ∑ m ∣ c n ∣ p ) 1/ p + β n = 1 ∑ m c n ω n , ∀ c 1 , … , c m ∈ K , m = 1 , 2 , … \forall c_1,\ldots,c_m\in\mathbb{K},m=1,2,\ldots ∀ c 1 , … , c m ∈ K , m = 1 , 2 , …
Assume that one of the following holds.
∑ n = 1 ∞ ( ∥ f n − g n ∥ ∥ S f , τ − 1 τ n ∥ + ∥ g n ∥ ∥ S f , τ − 1 ( τ n − ω n ) ∥ ) < 1 \sum_{n=1}^{\infty}\left(\lVert f_n-g_n\rVert\lVert S_{f,\tau}^{-1}\tau_n\rVert+\lVert g_n\rVert\lVert S_{f,\tau}^{-1}(\tau_n-\omega_n)\rVert\right)<1 ∑ n = 1 ∞ ( ∥ f n − g n ∥ ∥ S f , τ − 1 τ n ∥ + ∥ g n ∥ ∥ S f , τ − 1 ( τ n − ω n )∥ ) < 1
∑ n = 1 ∞ ( ∥ f n − g n ∥ ∥ S f , τ − 1 ω n ∥ + ∥ f n ∥ ∥ S f , τ − 1 ( τ n − ω n ) ∥ ) < 1 \sum_{n=1}^{\infty}\left(\lVert f_n-g_n\rVert\lVert S_{f,\tau}^{-1}\omega_n\rVert+\lVert f_n\rVert\lVert S_{f,\tau}^{-1}(\tau_n-\omega_n)\rVert\right)<1 ∑ n = 1 ∞ ( ∥ f n − g n ∥ ∥ S f , τ − 1 ω n ∥ + ∥ f n ∥ ∥ S f , τ − 1 ( τ n − ω n )∥ ) < 1
∑ n = 1 ∞ ( ∥ ( f n − g n ) S f , τ − 1 ∥ ∥ τ n ∥ + ∥ g n S f , τ − 1 ∥ ∥ τ n − ω n ∥ ) < 1 \sum_{n=1}^{\infty}\left(\lVert(f_n-g_n)S_{f,\tau}^{-1}\rVert\lVert\tau_n\rVert+\lVert g_nS_{f,\tau}^{-1}\rVert\lVert\tau_n-\omega_n\rVert\right)<1 ∑ n = 1 ∞ ( ∥( f n − g n ) S f , τ − 1 ∥ ∥ τ n ∥ + ∥ g n S f , τ − 1 ∥ ∥ τ n − ω n ∥ ) < 1
∑ n = 1 ∞ ( ∥ ( f n − g n ) S f , τ − 1 ∥ ∥ ω n ∥ + ∥ f n S f , τ − 1 ∥ ∥ τ n − ω n ∥ ) < 1 \sum_{n=1}^{\infty}\left(\lVert(f_n-g_n)S_{f,\tau}^{-1}\rVert\lVert\omega_n\rVert+\lVert f_nS_{f,\tau}^{-1}\rVert\lVert\tau_n-\omega_n\rVert\right)<1 ∑ n = 1 ∞ ( ∥( f n − g n ) S f , τ − 1 ∥ ∥ ω n ∥ + ∥ f n S f , τ − 1 ∥ ∥ τ n − ω n ∥ ) < 1
Then ( { g n } n , { ω n } n ) \left(\{g_n\}_n,\{\omega_n\}_n\right) ( { g n } n , { ω n } n ) p p p X \mathcal{X} X
( 1 + α 1 − β ∥ θ τ ∥ + γ 1 − β ) ( 1 + r 1 − s ∥ θ f ∥ + t 1 − s ) . \left(\frac{1+\alpha}{1-\beta}\lVert\theta_\tau\rVert+\frac{\gamma}{1-\beta}\right)\left(\frac{1+r}{1-s}\lVert\theta_f\rVert+\frac{t}{1-s}\right). ( 1 − β 1 + α ∥ θ τ ∥ + 1 − β γ ) ( 1 − s 1 + r ∥ θ f ∥ + 1 − s t ) . Proof. Following the initial lines in the proof of Theorem 4.6.2, we see that θ g \theta_g θ g θ ω \theta_\omega θ ω
Assume (i). Then
∥ x − ∑ n = 1 ∞ g n ( x ) S f , τ − 1 ω n ∥ = ∥ ∑ n = 1 ∞ f n ( x ) S f , τ − 1 τ n − ∑ n = 1 ∞ g n ( x ) S f , τ − 1 ω n ∥ ≤ ∑ n = 1 ∞ ∥ f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n ∥ ≤ ∑ n = 1 ∞ { ∥ f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 τ n ∥ + ∥ g n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n ∥ } = ∑ n = 1 ∞ { ∥ ( f n − g n ) ( x ) S f , τ − 1 τ n ∥ + ∥ g n ( x ) S f , τ − 1 ( τ n − ω n ) ∥ } ≤ ( ∑ n = 1 ∞ { ∥ f n − g n ∥ ∥ S f , τ − 1 τ n ∥ + ∥ g n ∥ ∥ S f , τ − 1 ( τ n − ω n ) ∥ } ) ∥ x ∥ . \begin{aligned}
\left\lVert x-\sum_{n=1}^{\infty}g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert
=
\left\lVert\sum_{n=1}^{\infty}f_n(x)S_{f,\tau}^{-1}\tau_n-\sum_{n=1}^{\infty}g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\\
&\leq\sum_{n=1}^{\infty}\left\lVert f_n(x)S_{f,\tau}^{-1}\tau_n-g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\\
&\leq\sum_{n=1}^{\infty}\left\{\left\lVert f_n(x)S_{f,\tau}^{-1}\tau_n-g_n(x)S_{f,\tau}^{-1}\tau_n\right\rVert+\left\lVert g_n(x)S_{f,\tau}^{-1}\tau_n-g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\right\} \\
&=\sum_{n=1}^{\infty}\left\{\left\lVert(f_n-g_n)(x)S_{f,\tau}^{-1}\tau_n\right\rVert+\left\lVert g_n(x)S_{f,\tau}^{-1}(\tau_n-\omega_n)\right\rVert\right\} \\
&\leq\left(\sum_{n=1}^{\infty}\left\{\lVert f_n-g_n\rVert\lVert S_{f,\tau}^{-1}\tau_n\rVert+\lVert g_n\rVert\lVert S_{f,\tau}^{-1}(\tau_n-\omega_n)\rVert\right\}\right)\lVert x\rVert.
\end{aligned} x − n = 1 ∑ ∞ g n ( x ) S f , τ − 1 ω n = n = 1 ∑ ∞ f n ( x ) S f , τ − 1 τ n − n = 1 ∑ ∞ g n ( x ) S f , τ − 1 ω n ≤ n = 1 ∑ ∞ f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n ≤ n = 1 ∑ ∞ { f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 τ n + g n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n } = n = 1 ∑ ∞ { ( f n − g n ) ( x ) S f , τ − 1 τ n + g n ( x ) S f , τ − 1 ( τ n − ω n ) } ≤ ( n = 1 ∑ ∞ { ∥ f n − g n ∥ ∥ S f , τ − 1 τ n ∥ + ∥ g n ∥ ∥ S f , τ − 1 ( τ n − ω n )∥ } ) ∥ x ∥ . Therefore the operator S f , τ − 1 S g , ω S_{f,\tau}^{-1}S_{g,\omega} S f , τ − 1 S g , ω
Assume (ii). Then
∥ x − ∑ n = 1 ∞ g n ( x ) S f , τ − 1 ω n ∥ = ∥ ∑ n = 1 ∞ f n ( x ) S f , τ − 1 τ n − ∑ n = 1 ∞ g n ( x ) S f , τ − 1 ω n ∥ ≤ ∑ n = 1 ∞ ∥ f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n ∥ ≤ ∑ n = 1 ∞ { ∥ f n ( x ) S f , τ − 1 τ n − f n ( x ) S f , τ − 1 ω n ∥ + ∥ f n ( x ) S f , τ − 1 ω n − g n ( x ) S f , τ − 1 ω n ∥ } = ∑ n = 1 ∞ { ∥ f n ( x ) S f , τ − 1 ( τ n − ω n ) ∥ + ∥ ( f n − g n ) ( x ) S f , τ − 1 ω n ∥ } ≤ ( ∑ n = 1 ∞ { ∥ f n ∥ ∥ S f , τ − 1 ( τ n − ω n ) ∥ + ∥ f n − g n ∥ ∥ S f , τ − 1 ω n ∥ } ) ∥ x ∥ . \begin{aligned}
\left\lVert x-\sum_{n=1}^{\infty}g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert
=
\left\lVert\sum_{n=1}^{\infty}f_n(x)S_{f,\tau}^{-1}\tau_n-\sum_{n=1}^{\infty}g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\\
&\leq\sum_{n=1}^{\infty}\left\lVert f_n(x)S_{f,\tau}^{-1}\tau_n-g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\\
&\leq\sum_{n=1}^{\infty}\left\{\left\lVert f_n(x)S_{f,\tau}^{-1}\tau_n-f_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert+\left\lVert f_n(x)S_{f,\tau}^{-1}\omega_n-g_n(x)S_{f,\tau}^{-1}\omega_n\right\rVert\right\} \\
&=\sum_{n=1}^{\infty}\left\{\left\lVert f_n(x)S_{f,\tau}^{-1}(\tau_n-\omega_n)\right\rVert+\left\lVert(f_n-g_n)(x)S_{f,\tau}^{-1}\omega_n\right\rVert\right\} \\
&\le\left(\sum_{n=1}^{\infty}\left\{\|f_n\|\|S_{f,\tau}^{-1}(\tau_n-\omega_n)\|+\|f_n-g_n\|\|S_{f,\tau}^{-1}\omega_n\|\right\}\right)\|x\|.
\end{aligned} x − n = 1 ∑ ∞ g n ( x ) S f , τ − 1 ω n = n = 1 ∑ ∞ f n ( x ) S f , τ − 1 τ n − n = 1 ∑ ∞ g n ( x ) S f , τ − 1 ω n ≤ n = 1 ∑ ∞ f n ( x ) S f , τ − 1 τ n − g n ( x ) S f , τ − 1 ω n ≤ n = 1 ∑ ∞ { f n ( x ) S f , τ − 1 τ n − f n ( x ) S f , τ − 1 ω n + f n ( x ) S f , τ − 1 ω n − g n ( x ) S f , τ − 1 ω n } = n = 1 ∑ ∞ { f n ( x ) S f , τ − 1 ( τ n − ω n ) + ( f n − g n ) ( x ) S f , τ − 1 ω n } ≤ ( n = 1 ∑ ∞ { ∥ f n ∥∥ S f , τ − 1 ( τ n − ω n ) ∥ + ∥ f n − g n ∥∥ S f , τ − 1 ω n ∥ } ) ∥ x ∥. Therefore the operator S f , τ − 1 S g , ω S_{f,\tau}^{-1}S_{g,\omega} S f , τ − 1 S g , ω
Assume (iii). Then
∥ x − ∑ n = 1 ∞ g n ( S f , τ − 1 x ) ω n ∥ = ∥ ∑ n = 1 ∞ f n ( S f , τ − 1 x ) τ n − ∑ n = 1 ∞ g n ( S f , τ − 1 x ) ω n ∥ ≤ ∑ n = 1 ∞ ∥ f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n ∥ ≤ ∑ n = 1 ∞ { ∥ f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) τ n ∥ + ∥ g n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n ∥ } = ∑ n = 1 ∞ { ∥ ( f n − g n ) ( S f , τ − 1 x ) τ n ∥ + ∥ g n ( S f , τ − 1 x ) ( τ n − ω n ) ∥ } ≤ ( ∑ n = 1 ∞ { ∥ ( f n − g n ) S f , τ − 1 ∥ ∥ τ n ∥ + ∥ g n S f , τ − 1 ∥ ∥ τ n − ω n ∥ } ) ∥ x ∥ . \begin{aligned}
\left\|x-\sum_{n=1}^{\infty}g_n(S_{f,\tau}^{-1}x)\omega_n\right\|=\left\|\sum_{n=1}^{\infty}f_n(S_{f,\tau}^{-1}x)\tau_n-\sum_{n=1}^{\infty}g_n(S_{f,\tau}^{-1}x)\omega_n\right\| \\
&\le\sum_{n=1}^{\infty}\left\|f_n(S_{f,\tau}^{-1}x)\tau_n-g_n(S_{f,\tau}^{-1}x)\omega_n\right\| \\
&\le\sum_{n=1}^{\infty}\left\{\left\|f_n(S_{f,\tau}^{-1}x)\tau_n-g_n(S_{f,\tau}^{-1}x)\tau_n\right\|+\left\|g_n(S_{f,\tau}^{-1}x)\tau_n-g_n(S_{f,\tau}^{-1}x)\omega_n\right\|\right\} \\
&=\sum_{n=1}^{\infty}\left\{\left\|(f_n-g_n)(S_{f,\tau}^{-1}x)\tau_n\right\|+\left\|g_n(S_{f,\tau}^{-1}x)(\tau_n-\omega_n)\right\|\right\} \\
&\le\left(\sum_{n=1}^{\infty}\left\{\|(f_n-g_n)S_{f,\tau}^{-1}\|\|\tau_n\|+\|g_nS_{f,\tau}^{-1}\|\|\tau_n-\omega_n\|\right\}\right)\|x\|.
\end{aligned} x − n = 1 ∑ ∞ g n ( S f , τ − 1 x ) ω n = n = 1 ∑ ∞ f n ( S f , τ − 1 x ) τ n − n = 1 ∑ ∞ g n ( S f , τ − 1 x ) ω n ≤ n = 1 ∑ ∞ f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n ≤ n = 1 ∑ ∞ { f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) τ n + g n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n } = n = 1 ∑ ∞ { ( f n − g n ) ( S f , τ − 1 x ) τ n + g n ( S f , τ − 1 x ) ( τ n − ω n ) } ≤ ( n = 1 ∑ ∞ { ∥ ( f n − g n ) S f , τ − 1 ∥∥ τ n ∥ + ∥ g n S f , τ − 1 ∥∥ τ n − ω n ∥ } ) ∥ x ∥. Therefore the operator S g , ω S f , τ − 1 S_{g,\omega}S_{f,\tau}^{-1} S g , ω S f , τ − 1
Assume (iv). Then
∥ x − ∑ n = 1 ∞ g n ( S f , τ − 1 x ) ω n ∥ = ∥ ∑ n = 1 ∞ f n ( S f , τ − 1 x ) τ n − ∑ n = 1 ∞ g n ( S f , τ − 1 x ) ω n ∥ ≤ ∑ n = 1 ∞ ∥ f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n ∥ ≤ ∑ n = 1 ∞ { ∥ f n ( S f , τ − 1 x ) τ n − f n ( S f , τ − 1 x ) ω n ∥ + ∥ f n ( S f , τ − 1 x ) ω n − g n ( S f , τ − 1 x ) ω n ∥ } = ∑ n = 1 ∞ { ∥ f n ( S f , τ − 1 x ) ( τ n − ω n ) ∥ + ∥ ( f n − g n ) ( S f , τ − 1 x ) ω n ∥ } ≤ ( ∑ n = 1 ∞ { ∥ f n S f , τ − 1 ∥ ∥ τ n − ω n ∥ + ∥ ( f n − g n ) S f , τ − 1 ∥ ∥ ω n ∥ } ) ∥ x ∥ . \begin{aligned}
\left\|x-\sum_{n=1}^{\infty}g_n(S_{f,\tau}^{-1}x)\omega_n\right\|=\left\|\sum_{n=1}^{\infty}f_n(S_{f,\tau}^{-1}x)\tau_n-\sum_{n=1}^{\infty}g_n(S_{f,\tau}^{-1}x)\omega_n\right\| \\
&\le\sum_{n=1}^{\infty}\left\|f_n(S_{f,\tau}^{-1}x)\tau_n-g_n(S_{f,\tau}^{-1}x)\omega_n\right\| \\
&\le\sum_{n=1}^{\infty}\left\{\left\|f_n(S_{f,\tau}^{-1}x)\tau_n-f_n(S_{f,\tau}^{-1}x)\omega_n\right\|+\left\|f_n(S_{f,\tau}^{-1}x)\omega_n-g_n(S_{f,\tau}^{-1}x)\omega_n\right\|\right\} \\
&=\sum_{n=1}^{\infty}\left\{\left\|f_n(S_{f,\tau}^{-1}x)(\tau_n-\omega_n)\right\|+\left\|(f_n-g_n)(S_{f,\tau}^{-1}x)\omega_n\right\|\right\} \\
&\le\left(\sum_{n=1}^{\infty}\left\{\|f_nS_{f,\tau}^{-1}\|\|\tau_n-\omega_n\|+\|(f_n-g_n)S_{f,\tau}^{-1}\|\|\omega_n\|\right\}\right)\|x\|.
\end{aligned} x − n = 1 ∑ ∞ g n ( S f , τ − 1 x ) ω n = n = 1 ∑ ∞ f n ( S f , τ − 1 x ) τ n − n = 1 ∑ ∞ g n ( S f , τ − 1 x ) ω n ≤ n = 1 ∑ ∞ f n ( S f , τ − 1 x ) τ n − g n ( S f , τ − 1 x ) ω n ≤ n = 1 ∑ ∞ { f n ( S f , τ − 1 x ) τ n − f n ( S f , τ − 1 x ) ω n + f n ( S f , τ − 1 x ) ω n − g n ( S f , τ − 1 x ) ω n } = n = 1 ∑ ∞ { f n ( S f , τ − 1 x ) ( τ n − ω n ) + ( f n − g n ) ( S f , τ − 1 x ) ω n } ≤ ( n = 1 ∑ ∞ { ∥ f n S f , τ − 1 ∥∥ τ n − ω n ∥ + ∥ ( f n − g n ) S f , τ − 1 ∥∥ ω n ∥ } ) ∥ x ∥. Therefore the operator S g , ω S f , τ − 1 S_{g,\omega}S_{f,\tau}^{-1} S g , ω S f , τ − 1
Hence in each of the assumptions we get that ( { g n } n , { ω n } n ) (\{g_n\}_n,\{\omega_n\}_n) ({ g n } n , { ω n } n ) X \mathcal{X} X
We end this chapter by deriving results on the expansion of sequences to approximate Schauder frames.
A routine Hilbert space argument shows that a sequence { τ n } n \{\tau_n\}_n { τ n } n H \mathcal{H} H S τ : H ∋ h ↦ ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H S_\tau:\mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}\langle h,\tau_n\rangle\tau_n\in\mathcal{H} S τ : H ∋ h ↦ ∑ n = 1 ∞ ⟨ h , τ n ⟩ τ n ∈ H { τ n } n \{\tau_n\}_n { τ n } n θ τ : H ∋ h ↦ θ τ h : = { ⟨ h , τ n ⟩ } n ∈ ℓ 2 ( N ) \theta_\tau:\mathcal{H}\ni h\mapsto\theta_\tau h:=\{\langle h,\tau_n\rangle\}_n\in\ell^2(\mathbb{N}) θ τ : H ∋ h ↦ θ τ h := {⟨ h , τ n ⟩ } n ∈ ℓ 2 ( N ) θ τ ∗ : ℓ 2 ( N ) ∋ { a n } n ↦ θ τ ∗ { a n } n : = ∑ n = 1 ∞ a n τ n ∈ H \theta_\tau^*:\ell^2(\mathbb{N})\ni\{a_n\}_n\mapsto\theta_\tau^*\{a_n\}_n:=\sum_{n=1}^{\infty}a_n\tau_n\in\mathcal{H} θ τ ∗ : ℓ 2 ( N ) ∋ { a n } n ↦ θ τ ∗ { a n } n := ∑ n = 1 ∞ a n τ n ∈ H [65 ] ). Now θ τ ∗ θ τ = S τ \theta_\tau^*\theta_\tau=S_\tau θ τ ∗ θ τ = S τ S τ S_\tau S τ S τ S_\tau S τ S τ S_\tau S τ b > 0 b>0 b > 0 ⟨ S τ h , h ⟩ ≤ b ∥ h ∥ 2 \langle S_\tau h,h\rangle\leq b\lVert h\rVert^2 ⟨ S τ h , h ⟩ ≤ b ∥ h ∥ 2 ∀ h ∈ H \forall h\in\mathcal{H} ∀ h ∈ H S τ S_\tau S τ { τ n } n \{\tau_n\}_n { τ n } n
Definition 4.6.7. Let { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) weak reconstruction sequence or approximate Bessel sequence (ABS) for X \mathcal{X} X S f , τ : X ∋ x ↦ S f , τ x : = ∑ n = 1 ∞ f n ( x ) τ n ∈ X S_{f,\tau}:\mathcal{X}\ni x\mapsto S_{f,\tau}x:=\sum_{n=1}^{\infty}f_n(x)\tau_n\in\mathcal{X} S f , τ : X ∋ x ↦ S f , τ x := ∑ n = 1 ∞ f n ( x ) τ n ∈ X
We next recall the reconstruction property of Banach spaces.
Definition 4.6.8. (Casazza and Christensen [49 ] ) A Banach space X \mathcal{X} X { τ n } n \{\tau_n\}_n { τ n } n X \mathcal{X} X { f n } n \{f_n\}_n { f n } n X ∗ \mathcal{X}^* X ∗ x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X x=\sum_{n=1}^{\infty}f_n(x)\tau_n,\forall x\in\mathcal{X} x = ∑ n = 1 ∞ f n ( x ) τ n , ∀ x ∈ X
Using approximation property of Banach spaces (cf. Casazza [47 ] ), Casazza and Christensen proved the following result.
Theorem 4.6.9. (Casazza and Christensen [49 ] ) There exists a Banach space X \mathcal{X} X X \mathcal{X} X
Now we have the following characterization. This is a result which is in contrast with Theorem 1.2.59 .
Theorem 4.6.10. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
(i) ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
(ii) X \mathcal{X} X
Proof. (i) ⇒ \Rightarrow ⇒ { ω n } n \{\omega_n\}_n { ω n } n X \mathcal{X} X { g n } n \{g_n\}_n { g n } n X ∗ \mathcal{X}^* X ∗ ( { f n } n ∪ { g n } n , { τ n } n ∪ { ω n } n ) (\{f_n\}_n\cup\{g_n\}_n,\{\tau_n\}_n\cup\{\omega_n\}_n) ({ f n } n ∪ { g n } n , { τ n } n ∪ { ω n } n ) X \mathcal{X} X S ( f , g ) , ( τ , ω ) S_{(f,g),(\tau,\omega)} S ( f , g ) , ( τ , ω ) ( { f n } n ∪ { g n } n , { τ n } n ∪ { ω n } n ) (\{f_n\}_n\cup\{g_n\}_n,\{\tau_n\}_n\cup\{\omega_n\}_n) ({ f n } n ∪ { g n } n , { τ n } n ∪ { ω n } n )
x = S ( f , g ) , ( τ , ω ) − 1 S ( f , g ) , ( τ , ω ) x = S ( f , g ) , ( τ , ω ) − 1 ( ∑ n = 1 ∞ f n ( x ) τ n + ∑ n = 1 ∞ g n ( x ) ω n ) = ∑ n = 1 ∞ f n ( x ) S ( f , g ) , ( τ , ω ) − 1 τ n + ∑ n = 1 ∞ g n ( x ) S ( f , g ) , ( τ , ω ) − 1 ω n , ∀ x ∈ X \begin{aligned}
x=S_{(f,g),(\tau,\omega)}^{-1}S_{(f,g),(\tau,\omega)}x=S_{(f,g),(\tau,\omega)}^{-1}\left(\sum_{n=1}^{\infty}f_n(x)\tau_n+\sum_{n=1}^{\infty}g_n(x)\omega_n\right) \\
&= \sum_{n=1}^{\infty} f_n(x) S_{(f,g),(\tau,\omega)}^{-1}\tau_n + \sum_{n=1}^{\infty} g_n(x) S_{(f,g),(\tau,\omega)}^{-1}\omega_n,\quad\forall x \in\mathcal{X}
\end{aligned} x = S ( f , g ) , ( τ , ω ) − 1 S ( f , g ) , ( τ , ω ) x = S ( f , g ) , ( τ , ω ) − 1 ( n = 1 ∑ ∞ f n ( x ) τ n + n = 1 ∑ ∞ g n ( x ) ω n ) = n = 1 ∑ ∞ f n ( x ) S ( f , g ) , ( τ , ω ) − 1 τ n + n = 1 ∑ ∞ g n ( x ) S ( f , g ) , ( τ , ω ) − 1 ω n , ∀ x ∈ X which shows that X \mathcal{X} X
(ii) ⇒ \Rightarrow ⇒ { ω n } n \{\omega_n\}_n { ω n } n X \mathcal{X} X { g n } n \{g_n\}_n { g n } n X ∗ \mathcal{X}^* X ∗ x = ∑ n = 1 ∞ g n ( x ) ω n x = \sum_{n=1}^{\infty} g_n(x)\omega_n x = ∑ n = 1 ∞ g n ( x ) ω n ∀ x ∈ X \forall x \in\mathcal{X} ∀ x ∈ X h n : = g n h_n := g_n h n := g n ρ n : = ( I X − S f , τ ) ω n \rho_n := (I_{\mathcal{X}} - S_{f,\tau})\omega_n ρ n := ( I X − S f , τ ) ω n n ∈ N n \in\mathbb{N} n ∈ N
∑ n = 1 ∞ f n ( x ) τ n + ∑ n = 1 ∞ h n ( x ) ρ n = ∑ n = 1 ∞ f n ( x ) τ n + ∑ n = 1 ∞ g n ( x ) ( I X − S f , τ ) ω n = S f , τ x + ( I X − S f , τ ) ( ∑ n = 1 ∞ g n ( x ) ω n ) = S f , τ x + ( I X − S f , τ ) x = x , ∀ x ∈ X . \begin{aligned}
\sum_{n=1}^{\infty} f_n(x)\tau_n + \sum_{n=1}^{\infty} h_n(x)\rho_n
&= \sum_{n=1}^{\infty} f_n(x)\tau_n + \sum_{n=1}^{\infty} g_n(x)(I_{\mathcal{X}} - S_{f,\tau})\omega_n \\
&= S_{f,\tau}x + (I_{\mathcal{X}} - S_{f,\tau})\left(\sum_{n=1}^{\infty} g_n(x)\omega_n\right) \\
&= S_{f,\tau}x + (I_{\mathcal{X}} - S_{f,\tau})x = x,\quad\forall x \in\mathcal{X}.
\end{aligned} n = 1 ∑ ∞ f n ( x ) τ n + n = 1 ∑ ∞ h n ( x ) ρ n = n = 1 ∑ ∞ f n ( x ) τ n + n = 1 ∑ ∞ g n ( x ) ( I X − S f , τ ) ω n = S f , τ x + ( I X − S f , τ ) ( n = 1 ∑ ∞ g n ( x ) ω n ) = S f , τ x + ( I X − S f , τ ) x = x , ∀ x ∈ X . Therefore ( { f n } n ∪ { h n } n , { τ n } n ∪ { ρ n } n ) (\{f_n\}_n \cup\{h_n\}_n,\{\tau_n\}_n \cup\{\rho_n\}_n) ({ f n } n ∪ { h n } n , { τ n } n ∪ { ρ n } n ) X \mathcal{X} X
We now show that there are infinitely many ways to expand a weak reconstruction sequence into an ASF.
Corollary 4.6.11. There exists a Banach space X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
Proof. From Theorem 4.6.9, there exists a Banach space X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
Following corollary is an easy consequence of Theorem 4.6.10.
Corollary 4.6.12. Let ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X X \mathcal{X} X ( { f n } n , { τ n } n ) (\{f_n\}_n,\{\tau_n\}_n) ({ f n } n , { τ n } n ) X \mathcal{X} X
Note that Theorem 4.6.10 may not add countably many elements to a weak reconstruction sequence to get an ASF. In the following example we show that it adds just one element to a weak reconstruction sequence and yields an ASF.
Example 4.6.13. Let p ∈ [ 1 , ∞ ) p \in[1,\infty) p ∈ [ 1 , ∞ ) { e n } n \{e_n\}_n { e n } n ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) { ζ n } n \{\zeta_n\}_n { ζ n } n { e n } n \{e_n\}_n { e n } n
R : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( 0 , x 1 , x 2 , … ) ∈ ℓ p ( N ) , R:\ell^p(\mathbb{N}) \ni(x_n)_{n=1}^{\infty} \mapsto(0,x_1,x_2,\ldots) \in\ell^p(\mathbb{N}), R : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( 0 , x 1 , x 2 , … ) ∈ ℓ p ( N ) , L : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( x 2 , x 3 , x 4 , … ) ∈ ℓ p ( N ) . L:\ell^{p}(\mathbb{N}) \ni(x_n)_{n=1}^{\infty} \mapsto(x_2,x_3,x_4,\ldots) \in\ell^{p}(\mathbb{N}). L : ℓ p ( N ) ∋ ( x n ) n = 1 ∞ ↦ ( x 2 , x 3 , x 4 , … ) ∈ ℓ p ( N ) . Clearly ( { f n : = ζ n L } n , { τ n : = R e n } n ) \left(\{f_n:=\zeta_nL\}_n,\{\tau_n:=Re_n\}_n\right) ( { f n := ζ n L } n , { τ n := R e n } n ) ℓ p ( N ) \ell^{p}(\mathbb{N}) ℓ p ( N ) S f , τ = R L S_{f,\tau}=RL S f , τ = R L
( I ℓ p ( N ) − S f , τ ) e 1 = e 1 − R L e 1 = e 1 − 0 = e 1 , (I_{\ell^{p}(\mathbb{N})}-S_{f,\tau})e_1=e_1-RLe_1=e_1-0=e_1, ( I ℓ p ( N ) − S f , τ ) e 1 = e 1 − R L e 1 = e 1 − 0 = e 1 , ( I ℓ p ( N ) − S f , τ ) e n = e n − R L e n = e n − R e n − 1 = e n − e n = 0 , ∀ n ≥ 2. (I_{\ell^{p}(\mathbb{N})}-S_{f,\tau})e_n=e_n-RLe_n=e_n-Re_{n-1}=e_n-e_n=0,\quad\forall n\geq2. ( I ℓ p ( N ) − S f , τ ) e n = e n − R L e n = e n − R e n − 1 = e n − e n = 0 , ∀ n ≥ 2. Let g n : = ζ n g_n:=\zeta_n g n := ζ n ω n : = e n \omega_n:=e_n ω n := e n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N ( { f n } n ∪ { h 1 } , { τ n } n ∪ { ρ 1 } ) \left(\{f_n\}_n\cup\{h_1\},\{\tau_n\}_n\cup\{\rho_1\}\right) ( { f n } n ∪ { h 1 } , { τ n } n ∪ { ρ 1 } ) ℓ p ( N ) \ell^{p}(\mathbb{N}) ℓ p ( N )
It may be possible to expand a weak reconstruction sequence to a tight ASF by adding finitely many elements Hilbert space. In this case, we can estimate the number of elements added to a tight ASF. This is given in the following theorem which can be compared with Theorem 1.2.60.
Theorem 4.6.14. Let ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X ( { f n } n ∪ { g k } k = 1 N , { τ n } n ∪ { ω k } k = 1 N ) \left(\{f_n\}_n\cup\{g_k\}_{k=1}^{N},\{\tau_n\}_n\cup\{\omega_k\}_{k=1}^{N}\right) ( { f n } n ∪ { g k } k = 1 N , { τ n } n ∪ { ω k } k = 1 N ) λ \lambda λ X \mathcal{X} X
N ≥ dim ( λ I X − S f , τ ) ( X ) . (39) N\geq\dim(\lambda I_{\mathcal{X}}-S_{f,\tau})(\mathcal{X}). \tag*{(39)} N ≥ dim ( λ I X − S f , τ ) ( X ) . (39) Further, the Inequality (4.6.3) can not be improved.
Proof. Let S ( f , g ) , ( τ , ω ) S_{(f,g),(\tau,\omega)} S ( f , g ) , ( τ , ω ) ( { f n } n ∪ { g k } k = 1 N , { τ n } n ∪ { ω k } k = 1 N ) \left(\{f_n\}_n\cup\{g_k\}_{k=1}^{N},\{\tau_n\}_n\cup\{\omega_k\}_{k=1}^{N}\right) ( { f n } n ∪ { g k } k = 1 N , { τ n } n ∪ { ω k } k = 1 N ) S g , ω ( x ) : = ∑ k = 1 N g k ( x ) ω k S_{g,\omega}(x):=\sum_{k=1}^{N}g_k(x)\omega_k S g , ω ( x ) := ∑ k = 1 N g k ( x ) ω k ∀ x ∈ X \forall x\in\mathcal{X} ∀ x ∈ X
λ x = S ( f , g ) , ( τ , ω ) x = ∑ n = 1 ∞ f n ( x ) τ n + ∑ k = 1 N g k ( x ) ω k = S f , τ x + S g , ω x , ∀ x ∈ X . \lambda x=S_{(f,g),(\tau,\omega)}x=\sum_{n=1}^{\infty}f_n(x)\tau_n+\sum_{k=1}^{N}g_k(x)\omega_k=S_{f,\tau}x+S_{g,\omega}x,\quad\forall x\in\mathcal{X}. λ x = S ( f , g ) , ( τ , ω ) x = n = 1 ∑ ∞ f n ( x ) τ n + k = 1 ∑ N g k ( x ) ω k = S f , τ x + S g , ω x , ∀ x ∈ X . Therefore
N ≥ dim S g , ω ( X ) = dim ( λ I X − S f , τ ) ( X ) . N\geq\dim S_{g,\omega}(\mathcal{X})=\dim(\lambda I_{\mathcal{X}}-S_{f,\tau})(\mathcal{X}). N ≥ dim S g , ω ( X ) = dim ( λ I X − S f , τ ) ( X ) . Example 4.6.13 says that inequality in Theorem 4.6.14 can not be improved.
We now state the definition of a p-weak reconstruction sequence and give a extension theorem for p-weak reconstruction sequences.
Definition 4.6.15. Let p ∈ [ 1 , ∞ ) p\in[1,\infty) p ∈ [ 1 , ∞ ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) X \mathcal{X} X p-weak reconstruction sequence or p-approximate Bessel sequence (p-ABS) for X \mathcal{X} X θ f : X ∋ x ↦ θ f x : = { f n ( x ) } n ∈ ℓ p ( N ) \theta_f:\mathcal{X}\ni x\mapsto\theta_fx:=\{f_n(x)\}_n\in\ell^{p}(\mathbb{N}) θ f : X ∋ x ↦ θ f x := { f n ( x ) } n ∈ ℓ p ( N ) θ τ : ℓ p ( N ) ∋ { a n } n ↦ θ τ { a n } n : = ∑ n = 1 ∞ a n τ n ∈ X \theta_\tau:\ell^{p}(\mathbb{N})\ni\{a_n\}_n\mapsto\theta_\tau\{a_n\}_n:=\sum_{n=1}^{\infty}a_n\tau_n\in\mathcal{X} θ τ : ℓ p ( N ) ∋ { a n } n ↦ θ τ { a n } n := ∑ n = 1 ∞ a n τ n ∈ X
Theorem 4.6.16. Let p ∈ [ 1 , ∞ ) p \in[1,\infty) p ∈ [ 1 , ∞ ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N ) ( { f n } n , { τ n } n ) \left(\{f_n\}_n,\{\tau_n\}_n\right) ( { f n } n , { τ n } n ) p p p
Proof. Let { e n } n \{e_n\}_n { e n } n { ζ n } n \{\zeta_n\}_n { ζ n } n h n : = ζ n h_n:=\zeta_n h n := ζ n ρ n : = ( I ℓ p ( N ) − S f , τ ) e n \rho_n:=\left(I_{\ell^p(\mathbb{N})}-S_{f,\tau}\right)e_n ρ n := ( I ℓ p ( N ) − S f , τ ) e n n ∈ N n\in\mathbb{N} n ∈ N ( { f n } n ∪ { h n } n , { τ n } n ∪ { ρ n } n ) \left(\{f_n\}_n\cup\{h_n\}_n,\{\tau_n\}_n\cup\{\rho_n\}_n\right) ( { f n } n ∪ { h n } n , { τ n } n ∪ { ρ n } n ) p p p ℓ p ( N ) \ell^p(\mathbb{N}) ℓ p ( N )
WEAK OPERATOR-VALUED FRAMES
BASIC PROPERTIES
Let H , H 0 \mathcal{H},\mathcal{H}_0 H , H 0 B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) H \mathcal{H} H H 0 \mathcal{H}_0 H 0 ∑ n = 1 ∞ Ψ n ∗ A n \sum_{n=1}^{\infty} \Psi_n^*A_n ∑ n = 1 ∞ Ψ n ∗ A n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
Definition 5.1.1. Let { A n } n \{A_n\}_n { A n } n { Ψ n } n \{\Psi_n\}_n { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) weak operator-valued frame (weak OVF) in B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(Operator-valued frame operator) S A , Ψ : = ∑ n = 1 ∞ Ψ n ∗ A n \text{(Operator-valued frame operator)}\qquad S_{A,\Psi}:=\sum_{n=1}^{\infty}\Psi_n^*A_n (Operator-valued frame operator) S A , Ψ := n = 1 ∑ ∞ Ψ n ∗ A n converges in the strong-operator topology on B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) S A , Ψ = I H S_{A,\Psi}=I_{\mathcal{H}} S A , Ψ = I H
We now give various examples of weak OVFs.
Example 5.1.2. (i) By taking Ψ n : = A n \Psi_n:=A_n Ψ n := A n n ∈ N n\in\mathbb{N} n ∈ N operator-valued frame is a weak OVF. In particular, a G G G
(ii) Let ( { τ n } n , { f n } n ) (\{\tau_n\}_n,\{f_n\}_n) ({ τ n } n , { f n } n ) pseudo-frame for H \mathcal{H} H Ψ n : = f n \Psi_n:=f_n Ψ n := f n A n h : = ⟨ h , τ n ⟩ A_nh:=\langle h,\tau_n\rangle A n h := ⟨ h , τ n ⟩ n ∈ N n\in\mathbb{N} n ∈ N h ∈ H h\in\mathcal{H} h ∈ H ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) B ( H , K ) \mathcal{B}(\mathcal{H},\mathbb{K}) B ( H , K ) frames for subspaces, fusion frames, outer frames, oblique frames and quasi-projectors are all weak OVFs.
(iii) Let C ∈ B ( H ) C\in\mathcal{B}(\mathcal{H}) C ∈ B ( H ) { τ n } n \{\tau_n\}_n { τ n } n C C C H \mathcal{H} H [27 ] ). If we define Ψ n h : = ⟨ h , C τ n ⟩ \Psi_nh:=\langle h,C\tau_n\rangle Ψ n h := ⟨ h , C τ n ⟩ A n h : = ⟨ h , τ n ⟩ A_nh:=\langle h,\tau_n\rangle A n h := ⟨ h , τ n ⟩ n ∈ N n\in\mathbb{N} n ∈ N h ∈ H h \in\mathcal{H} h ∈ H ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , K ) \mathcal{B}(\mathcal{H},\mathcal{K}) B ( H , K ) weighted frame is a weak OVF.
(iv) Let ( { τ n } n , { f n } n ) \left(\{\tau_n\}_n,\{f_n\}_n\right) ( { τ n } n , { f n } n ) approximate Schauder frame for H \mathcal{H} H [111 ] ). Note that it is possible for Hilbert spaces to have approximate Schauder frames which are not frames. If we define Ψ n : = f n \Psi_n := f_n Ψ n := f n A n h : = ⟨ h , τ n ⟩ A_nh := \langle h,\tau_n\rangle A n h := ⟨ h , τ n ⟩ n ∈ N n \in\mathbb{N} n ∈ N h ∈ H h \in\mathcal{H} h ∈ H ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , K ) \mathcal{B}(\mathcal{H},\mathcal{K}) B ( H , K ) atomic decompositions ([53 ] ), framings ([53 ] ), cb-frames ([159 ] ) and Schauder frames ([51 ] ) for Hilbert spaces are all weak OVFs.
(v) Let { τ n } n \{\tau_n\}_n { τ n } n signed frame for H \mathcal{H} H { σ n } n \{\sigma_n\}_n { σ n } n Ψ n : = h : = ⟨ h , σ n τ n ⟩ \Psi_n := h := \langle h,\sigma_n\tau_n\rangle Ψ n := h := ⟨ h , σ n τ n ⟩ A n h : = ⟨ h , τ n ⟩ A_nh := \langle h,\tau_n\rangle A n h := ⟨ h , τ n ⟩ n ∈ N n \in\mathbb{N} n ∈ N h ∈ H h \in\mathcal{H} h ∈ H ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , K ) \mathcal{B}(\mathcal{H},\mathcal{K}) B ( H , K )
Unlike in the case of OVFs, the frame operator S A , Ψ S_{A,\Psi} S A , Ψ S A , Ψ S_{A,\Psi} S A , Ψ a , b > 0 a,b>0 a , b > 0
a ∥ h ∥ ≤ ∥ S A , Ψ h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . a\lVert h\rVert\leq\lVert S_{A,\Psi}h\rVert\leq b\lVert h\rVert,\quad\forall h\in\mathcal{H}. a ∥ h ∥ ≤ ∥ S A , Ψ h ∥ ≤ b ∥ h ∥ , ∀ h ∈ H . We call such a , b a,b a , b
optimal lower frame bound = ∥ S A , Ψ − 1 ∥ − 1 , \text{optimal lower frame bound} = \lVert S_{A,\Psi}^{-1}\rVert^{-1}, optimal lower frame bound = ∥ S A , Ψ − 1 ∥ − 1 , optimal upper frame bound = ∥ S A , Ψ ∥ . \text{optimal upper frame bound} = \lVert S_{A,\Psi}\rVert. optimal upper frame bound = ∥ S A , Ψ ∥ . We now define the notion of dual weak OVFs.
Definition 5.1.3. A weak OVF ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
∑ n = 1 ∞ Ψ n ∗ B n = ∑ n = 1 ∞ Φ n ∗ A n = I H . \sum_{n=1}^{\infty}\Psi_n^{*}B_n=\sum_{n=1}^{\infty}\Phi_n^{*}A_n=I_{\mathcal{H}}. n = 1 ∑ ∞ Ψ n ∗ B n = n = 1 ∑ ∞ Φ n ∗ A n = I H . Note that dual always exists for a given weak OVF. In fact, a direct calculation shows that
( { A ~ n : = A n S A , Ψ − 1 } n , { Ψ ~ n : = Ψ n ( S A , Ψ − 1 ) ∗ } n ) \left(\{\widetilde{A}_n:=A_nS_{A,\Psi}^{-1}\}_n,\{\widetilde{\Psi}_n:=\Psi_n(S_{A,\Psi}^{-1})^{*}\}_n\right) ( { A n := A n S A , Ψ − 1 } n , { Ψ n := Ψ n ( S A , Ψ − 1 ) ∗ } n ) is a weak OVF and is a dual for ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) canonical dual for ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
Proposition 5.1.4. Let ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) h ∈ H h \in\mathcal{H} h ∈ H h = ∑ n = 1 ∞ A n ∗ y n = ∑ n = 1 ∞ Ψ n ∗ z n h = \sum_{n=1}^{\infty} A_n^*y_n = \sum_{n=1}^{\infty}\Psi_n^*z_n h = ∑ n = 1 ∞ A n ∗ y n = ∑ n = 1 ∞ Ψ n ∗ z n { y n } n , { z n } n \{y_n\}_n,\{z_n\}_n { y n } n , { z n } n H 0 \mathcal{H}_0 H 0
∑ n = 1 ∞ ⟨ y n , z n ⟩ = ∑ n = 1 ∞ ⟨ Ψ ~ n h , A ~ n h ⟩ + ∑ n = 1 ∞ ⟨ y n − Ψ ~ n h , z n − A ~ n h ⟩ . \sum_{n=1}^{\infty}\langle y_n,z_n\rangle= \sum_{n=1}^{\infty}\langle\widetilde{\Psi}_n h,\widetilde{A}_n h\rangle+\sum_{n=1}^{\infty}\langle y_n-\widetilde{\Psi}_n h,z_n-\widetilde{A}_n h\rangle. n = 1 ∑ ∞ ⟨ y n , z n ⟩ = n = 1 ∑ ∞ ⟨ Ψ n h , A n h ⟩ + n = 1 ∑ ∞ ⟨ y n − Ψ n h , z n − A n h ⟩ . Proof. We start from the right side and see
∑ n = 1 ∞ ⟨ Ψ ~ n h , A ~ n h ⟩ + ∑ n = 1 ∞ ⟨ y n , z n ⟩ − ∑ n = 1 ∞ ⟨ y n , A ~ n h ⟩ − ∑ n = 1 ∞ ⟨ Ψ ~ n h , z n ⟩ + ∑ n = 1 ∞ ⟨ Ψ ~ n h , A ~ n h ⟩ = 2 ∑ n = 1 ∞ ⟨ Ψ ~ n h , A ~ n h ⟩ + ∑ n = 1 ∞ ⟨ y n , z n ⟩ − ∑ n = 1 ∞ ⟨ y n , A n S A , Ψ − 1 h ⟩ − ∑ n = 1 ∞ ⟨ Ψ n ( S A , Ψ − 1 ) ∗ h , z n ⟩ = 2 ⟨ ∑ n = 1 ∞ ( S A , Ψ − 1 ) ∗ A n ∗ Ψ n ( S A , Ψ − 1 ) ∗ h , h ⟩ + ∑ n = 1 ∞ ⟨ y n , z n ⟩ − ⟨ ∑ n = 1 ∞ A n ∗ y n , S A , Ψ − 1 h ⟩ − ⟨ ( S A , Ψ − 1 ) ∗ h , ∑ n = 1 ∞ Ψ n ∗ z n ⟩ = 2 ⟨ ( S A , Ψ − 1 ) ∗ h , h ⟩ + ∑ n = 1 ∞ ⟨ y n , z n ⟩ − ⟨ h , S A , Ψ − 1 h ⟩ − ⟨ ( S A , Ψ − 1 ) ∗ h , h ⟩ \begin{aligned}
&\sum_{n=1}^{\infty}\langle\widetilde{\Psi}_n h,\widetilde{A}_n h\rangle+\sum_{n=1}^{\infty}\langle y_n,z_n\rangle-\sum_{n=1}^{\infty}\langle y_n,\widetilde{A}_n h\rangle-\sum_{n=1}^{\infty}\langle\widetilde{\Psi}_n h,z_n\rangle+\sum_{n=1}^{\infty}\langle\widetilde{\Psi}_n h,\widetilde{A}_n h\rangle\\
&=2\sum_{n=1}^{\infty}\langle\widetilde{\Psi}_n h,\widetilde{A}_n h\rangle+\sum_{n=1}^{\infty}\langle y_n,z_n\rangle-\sum_{n=1}^{\infty}\langle y_n,A_nS_{A,\Psi}^{-1}h\rangle-\sum_{n=1}^{\infty}\langle\Psi_n(S_{A,\Psi}^{-1})^*h,z_n\rangle\\
&=2\left\langle\sum_{n=1}^{\infty}(S_{A,\Psi}^{-1})^*A_n^*\Psi_n(S_{A,\Psi}^{-1})^*h,h\right\rangle+\sum_{n=1}^{\infty}\langle y_n,z_n\rangle-\left\langle\sum_{n=1}^{\infty}A_n^*y_n,S_{A,\Psi}^{-1}h\right\rangle\\
&\qquad-\left\langle(S_{A,\Psi}^{-1})^*h,\sum_{n=1}^{\infty}\Psi_n^*z_n\right\rangle\\
&=2\langle(S_{A,\Psi}^{-1})^*h,h\rangle+\sum_{n=1}^{\infty}\langle y_n,z_n\rangle-\langle h,S_{A,\Psi}^{-1}h\rangle-\langle(S_{A,\Psi}^{-1})^*h,h\rangle
\end{aligned} n = 1 ∑ ∞ ⟨ Ψ n h , A n h ⟩ + n = 1 ∑ ∞ ⟨ y n , z n ⟩ − n = 1 ∑ ∞ ⟨ y n , A n h ⟩ − n = 1 ∑ ∞ ⟨ Ψ n h , z n ⟩ + n = 1 ∑ ∞ ⟨ Ψ n h , A n h ⟩ = 2 n = 1 ∑ ∞ ⟨ Ψ n h , A n h ⟩ + n = 1 ∑ ∞ ⟨ y n , z n ⟩ − n = 1 ∑ ∞ ⟨ y n , A n S A , Ψ − 1 h ⟩ − n = 1 ∑ ∞ ⟨ Ψ n ( S A , Ψ − 1 ) ∗ h , z n ⟩ = 2 ⟨ n = 1 ∑ ∞ ( S A , Ψ − 1 ) ∗ A n ∗ Ψ n ( S A , Ψ − 1 ) ∗ h , h ⟩ + n = 1 ∑ ∞ ⟨ y n , z n ⟩ − ⟨ n = 1 ∑ ∞ A n ∗ y n , S A , Ψ − 1 h ⟩ − ⟨ ( S A , Ψ − 1 ) ∗ h , n = 1 ∑ ∞ Ψ n ∗ z n ⟩ = 2 ⟨( S A , Ψ − 1 ) ∗ h , h ⟩ + n = 1 ∑ ∞ ⟨ y n , z n ⟩ − ⟨ h , S A , Ψ − 1 h ⟩ − ⟨( S A , Ψ − 1 ) ∗ h , h ⟩ which gives the left side.
Theorem 5.1.5. Let ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) a a a b b b
(i) The canonical dual weak OVF for the canonical dual weak OVF for ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
(ii) 1 b , 1 a \frac{1}{b},\frac{1}{a} b 1 , a 1 ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
(iii) If a , b a,b a , b ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) 1 b , 1 a \frac{1}{b},\frac{1}{a} b 1 , a 1
Proof. Since (ii) and (iii) follow from the property of invertible operators on Banach spaces, we have to argue for (i): frame operator for ( { A n S A , Ψ − 1 } n , { Ψ n ( S A , Ψ − 1 ) ∗ } n ) (\{A_nS_{A,\Psi}^{-1}\}_n,\{\Psi_n(S_{A,\Psi}^{-1})^*\}_n) ({ A n S A , Ψ − 1 } n , { Ψ n ( S A , Ψ − 1 ) ∗ } n )
∑ n = 1 ∞ ( Ψ n ( S A , Ψ − 1 ) ∗ ) ∗ ( A n S A , Ψ − 1 ) = S A , Ψ − 1 ( ∑ n = 1 ∞ Ψ n ∗ A n ) S A , Ψ − 1 = S A , Ψ − 1 S A , Ψ S A , Ψ − 1 = S A , Ψ − 1 . \sum_{n=1}^{\infty}\bigl(\Psi_n(S_{A,\Psi}^{-1})^*\bigr)^*(A_nS_{A,\Psi}^{-1})=S_{A,\Psi}^{-1}\left(\sum_{n=1}^{\infty}\Psi_n^*A_n\right)S_{A,\Psi}^{-1}=S_{A,\Psi}^{-1}S_{A,\Psi}S_{A,\Psi}^{-1}=S_{A,\Psi}^{-1}. n = 1 ∑ ∞ ( Ψ n ( S A , Ψ − 1 ) ∗ ) ∗ ( A n S A , Ψ − 1 ) = S A , Ψ − 1 ( n = 1 ∑ ∞ Ψ n ∗ A n ) S A , Ψ − 1 = S A , Ψ − 1 S A , Ψ S A , Ψ − 1 = S A , Ψ − 1 . Therefore, its canonical dual is ( { ( A n S A , Ψ − 1 ) S A , Ψ } n , { ( Ψ n ( S A , Ψ − 1 ) ∗ ) S A , Ψ ∗ } n ) (\{(A_nS_{A,\Psi}^{-1})S_{A,\Psi}\}_n,\{(\Psi_n(S_{A,\Psi}^{-1})^*)S_{A,\Psi}^*\}_n) ({( A n S A , Ψ − 1 ) S A , Ψ } n , {( Ψ n ( S A , Ψ − 1 ) ∗ ) S A , Ψ ∗ } n )
For the further study of weak OVFs, we impose some conditions so that the frame operator splits.
Definition 5.1.6. A weak OVF ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) factorable if both the maps (called analysis operator )
θ A : H ∋ h ↦ θ A h : = ∑ n = 1 ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 \theta_A:\mathcal{H}\ni h\mapsto\theta_Ah:=\sum_{n=1}^{\infty}L_nA_nh\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 θ A : H ∋ h ↦ θ A h := n = 1 ∑ ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 θ Ψ : H ∋ h ↦ θ Ψ h : = ∑ n = 1 ∞ L n Ψ n h ∈ ℓ 2 ( N ) ⊗ H 0 \theta_\Psi:\mathcal{H}\ni h\mapsto\theta_\Psi h:=\sum_{n=1}^{\infty}L_n\Psi_nh\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 θ Ψ : H ∋ h ↦ θ Ψ h := n = 1 ∑ ∞ L n Ψ n h ∈ ℓ 2 ( N ) ⊗ H 0 are well-defined bounded linear operators.
We next give an example which shows that a weak OVF need not be factorable.
Example 5.1.7. On C \mathbb{C} C A n x : = x n A_nx:=\frac{x}{\sqrt{n}} A n x := n x ∀ x ∈ C , ∀ n ∈ N \forall x\in\mathbb{C},\forall n\in\mathbb{N} ∀ x ∈ C , ∀ n ∈ N Ψ 1 x : = x , Ψ n x : = 0 \Psi_1x:=x,\Psi_nx:=0 Ψ 1 x := x , Ψ n x := 0 ∀ x ∈ C , ∀ n ∈ N ∖ { 1 } \forall x\in\mathbb{C},\forall n\in\mathbb{N}\setminus\{1\} ∀ x ∈ C , ∀ n ∈ N ∖ { 1 } ∑ n = 1 ∞ Ψ n ∗ A n x \sum_{n=1}^{\infty}\Psi_n^*A_nx ∑ n = 1 ∞ Ψ n ∗ A n x ∑ n = 1 ∞ L n A n x \sum_{n=1}^{\infty}L_nA_nx ∑ n = 1 ∞ L n A n x
∥ ∑ n = 1 m L n A n 1 ∥ 2 = ∑ n = 1 m ∥ A n 1 ∥ 2 = ∑ n = 1 m 1 n → ∞ as m → ∞ . \left\|\sum_{n=1}^{m}L_nA_n1\right\|^2=\sum_{n=1}^{m}\|A_n1\|^2=\sum_{n=1}^{m}\frac{1}{n}\to\infty\quad\text{as}\quad m\to\infty. n = 1 ∑ m L n A n 1 2 = n = 1 ∑ m ∥ A n 1 ∥ 2 = n = 1 ∑ m n 1 → ∞ as m → ∞. Equation (1.6.1) gives the following theorem easily.
Theorem 5.1.8. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(i) Analysis operator
θ A : H ∋ h ↦ θ A h : = ∑ n = 1 ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 \theta_A:\mathcal{H}\ni h\mapsto\theta_Ah:=\sum_{n=1}^{\infty}L_nA_nh\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 θ A : H ∋ h ↦ θ A h := n = 1 ∑ ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 is a well-defined bounded linear injective operator.
(ii) Synthesis operator
θ Ψ ∗ : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ ∑ n = 1 ∞ Ψ n ∗ L n ∗ z ∈ H \theta_\Psi^*:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\ni z\mapsto\sum_{n=1}^{\infty}\Psi_n^*L_n^*z\in\mathcal{H} θ Ψ ∗ : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ n = 1 ∑ ∞ Ψ n ∗ L n ∗ z ∈ H is a well-defined bounded linear surjective operator.
(iii) Frame operator factors as S A , Ψ = θ Ψ ∗ θ A S_{A,\Psi}=\theta_\Psi^*\theta_A S A , Ψ = θ Ψ ∗ θ A
(iv) P A , Ψ : = θ A S A , Ψ − 1 θ Ψ ∗ : ℓ 2 ( N ) ⊗ H 0 → ℓ 2 ( N ) ⊗ H 0 P_{A,\Psi}:=\theta_AS_{A,\Psi}^{-1}\theta_\Psi^*:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 P A , Ψ := θ A S A , Ψ − 1 θ Ψ ∗ : ℓ 2 ( N ) ⊗ H 0 → ℓ 2 ( N ) ⊗ H 0 θ A ( H ) \theta_A(\mathcal{H}) θ A ( H )
We next define the notions of Riesz and orthonormal factorable weak OVFs.
Definition 5.1.9. A factorable weak OVF ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) P A , Ψ = I ℓ 2 ( N ) ⊗ H 0 P_{A,\Psi}=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} P A , Ψ = I ℓ 2 ( N ) ⊗ H 0 θ Ψ ∗ θ A = I H \theta_\Psi^*\theta_A=I_{\mathcal{H}} θ Ψ ∗ θ A = I H θ A θ Ψ ∗ = I ℓ 2 ( N ) ⊗ H 0 \theta_A\theta_\Psi^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} θ A θ Ψ ∗ = I ℓ 2 ( N ) ⊗ H 0 orthonormal OVF .
Proposition 5.1.10. A factorable weak OVF ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) A n Ψ m ∗ = δ n , m I H 0 A_n\Psi_m^*=\delta_{n,m}I_{\mathcal{H}_0} A n Ψ m ∗ = δ n , m I H 0 ∀ n , m ∈ N \forall n,m\in\mathbb{N} ∀ n , m ∈ N
Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) θ A θ Ψ ∗ = I ℓ 2 ( N ) ⊗ H 0 \theta_A\theta_\Psi^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} θ A θ Ψ ∗ = I ℓ 2 ( N ) ⊗ H 0
e m ⊗ y = θ A θ Ψ ∗ ( e m ⊗ y ) = ∑ n = 1 ∞ L n A n ( ∑ k = 1 ∞ Ψ k ∗ L k ∗ ( e m ⊗ y ) ) = ∑ n = 1 ∞ L n A n Ψ m ∗ y = ∑ n = 1 ∞ ( e n ⊗ A n Ψ m ∗ y ) = e m ⊗ ( A m Ψ m ∗ y ) + ∑ n = 1 n ≠ m ∞ ( e n ⊗ A n Ψ m ∗ y ) , ∀ m ∈ N , y ∈ H 0 . \begin{aligned}
e_m\otimes y=\theta_A\theta_\Psi^*(e_m\otimes y)=\sum_{n=1}^{\infty}L_nA_n\left(\sum_{k=1}^{\infty}\Psi_k^*L_k^*(e_m\otimes y)\right) \\
&=\sum_{n=1}^{\infty}L_nA_n\Psi_m^*y=\sum_{n=1}^{\infty}(e_n\otimes A_n\Psi_m^*y) \\
&=e_m\otimes(A_m\Psi_m^*y)+\sum_{\substack{n=1\\n\ne m}}^{\infty}(e_n\otimes A_n\Psi_m^*y),\quad\forall m\in\mathbb{N},\quad y\in\mathcal{H}_0.
\end{aligned} e m ⊗ y = θ A θ Ψ ∗ ( e m ⊗ y ) = n = 1 ∑ ∞ L n A n ( k = 1 ∑ ∞ Ψ k ∗ L k ∗ ( e m ⊗ y ) ) = n = 1 ∑ ∞ L n A n Ψ m ∗ y = n = 1 ∑ ∞ ( e n ⊗ A n Ψ m ∗ y ) = e m ⊗ ( A m Ψ m ∗ y ) + n = 1 n = m ∑ ∞ ( e n ⊗ A n Ψ m ∗ y ) , ∀ m ∈ N , y ∈ H 0 . We then have A n Ψ m ∗ y = δ n , m y , ∀ y ∈ H 0 A_n\Psi_m^*y=\delta_{n,m}y,\forall y\in\mathcal{H}_0 A n Ψ m ∗ y = δ n , m y , ∀ y ∈ H 0
( ⇐ ) (\Leftarrow) ( ⇐ ) θ A θ Ψ ∗ = ∑ n = 1 ∞ L n A n ( ∑ k = 1 ∞ Ψ k ∗ L k ∗ ) = ∑ n = 1 ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 \theta_A\theta_\Psi^*=\sum_{n=1}^{\infty}L_nA_n\left(\sum_{k=1}^{\infty}\Psi_k^*L_k^*\right)=\sum_{n=1}^{\infty}L_nL_n^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} θ A θ Ψ ∗ = ∑ n = 1 ∞ L n A n ( ∑ k = 1 ∞ Ψ k ∗ L k ∗ ) = ∑ n = 1 ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0
:::
We now derive a dilation result for factorable weak OVFs. First we need a lemma for this.
Lemma 5.1.11. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) θ A \theta_A θ A
Proof. Let { h n } n = 1 ∞ \{h_n\}_{n=1}^{\infty} { h n } n = 1 ∞ H \mathcal{H} H { θ A h n } n = 1 ∞ \{\theta_Ah_n\}_{n=1}^{\infty} { θ A h n } n = 1 ∞ y ∈ H 0 y\in\mathcal{H}_0 y ∈ H 0 S A , Ψ h n → θ Ψ ∗ y S_{A,\Psi}h_n\to\theta_\Psi^*y S A , Ψ h n → θ Ψ ∗ y n → ∞ n\to\infty n → ∞ h n → S A , Ψ − 1 θ Ψ ∗ y h_n\to S_{A,\Psi}^{-1}\theta_\Psi^*y h n → S A , Ψ − 1 θ Ψ ∗ y n → ∞ n\to\infty n → ∞ θ A \theta_A θ A θ A h n → θ A S A , Ψ − 1 θ Ψ ∗ y \theta_Ah_n\to\theta_AS_{A,\Psi}^{-1}\theta_\Psi^*y θ A h n → θ A S A , Ψ − 1 θ Ψ ∗ y n → ∞ n\to\infty n → ∞ y = θ A ( S A , Ψ − 1 θ Ψ ∗ y ) y=\theta_A(S_{A,\Psi}^{-1}\theta_\Psi^*y) y = θ A ( S A , Ψ − 1 θ Ψ ∗ y )
Theorem 5.1.12. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) θ A ( H ) = θ Ψ ( H ) \theta_A(\mathcal{H})=\theta_\Psi(\mathcal{H}) θ A ( H ) = θ Ψ ( H ) P A , Ψ P_{A,\Psi} P A , Ψ H 1 \mathcal{H}_1 H 1 H \mathcal{H} H B n , Φ n : H 1 → H 0 , ∀ n B_n,\Phi_n:\mathcal{H}_1\to\mathcal{H}_0,\forall n B n , Φ n : H 1 → H 0 , ∀ n ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H 1 , H 0 ) \mathcal{B}(\mathcal{H}_1,\mathcal{H}_0) B ( H 1 , H 0 ) B n ∣ H = A n , Φ n ∣ H = Ψ n , ∀ n ∈ N B_n|_{\mathcal{H}}=A_n,\Phi_n|_{\mathcal{H}}=\Psi_n,\forall n\in\mathbb{N} B n ∣ H = A n , Φ n ∣ H = Ψ n , ∀ n ∈ N
Proof. We first see that P A , Ψ P_{A,\Psi} P A , Ψ ℓ 2 ( N ) ⊗ H 0 \ell^2(\mathbb{N})\otimes\mathcal{H}_0 ℓ 2 ( N ) ⊗ H 0 θ A ( H ) = θ Ψ ( H ) \theta_A(\mathcal{H})=\theta_\Psi(\mathcal{H}) θ A ( H ) = θ Ψ ( H ) H 1 : = H ⊕ θ A ( H ) ⊥ \mathcal{H}_1:=\mathcal{H}\oplus\theta_A(\mathcal{H})^\perp H 1 := H ⊕ θ A ( H ) ⊥ H 1 \mathcal{H}_1 H 1 H ∋ h ↦ h ⊕ 0 ∈ H 1 \mathcal{H}\ni h\mapsto h\oplus0\in\mathcal{H}_1 H ∋ h ↦ h ⊕ 0 ∈ H 1 P A , Ψ ⊥ : = I ℓ 2 ( N ) ⊗ H 0 − P A , Ψ P_{A,\Psi}^\perp:=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-P_{A,\Psi} P A , Ψ ⊥ := I ℓ 2 ( N ) ⊗ H 0 − P A , Ψ
B n : H 1 ∋ h ⊕ g ↦ A n h + L n ∗ P A , Ψ ⊥ g ∈ H 0 , B_n:\mathcal{H}_1\ni h\oplus g\mapsto A_nh+L_n^*P_{A,\Psi}^\perp g\in\mathcal{H}_0, B n : H 1 ∋ h ⊕ g ↦ A n h + L n ∗ P A , Ψ ⊥ g ∈ H 0 , Φ n : H 1 ∋ h ⊕ g ↦ Ψ n h + L n ∗ P A , Ψ ⊥ g ∈ H 0 , ∀ n ∈ N . \Phi_n:\mathcal{H}_1\ni h\oplus g\mapsto\Psi_nh+L_n^*P_{A,\Psi}^{\perp}g\in\mathcal{H}_0,\quad\forall n\in\mathbb{N}. Φ n : H 1 ∋ h ⊕ g ↦ Ψ n h + L n ∗ P A , Ψ ⊥ g ∈ H 0 , ∀ n ∈ N . Then clearly B n ∣ H = A n B_n|_{\mathcal{H}}=A_n B n ∣ H = A n Φ n ∣ H = Ψ n \Phi_n|_{\mathcal{H}}=\Psi_n Φ n ∣ H = Ψ n ∀ n ∈ N \forall n\in\mathbb{N} ∀ n ∈ N
θ B ( h ⊕ g ) = ∑ n = 1 ∞ L n A n h + ∑ n = 1 ∞ L n L n ∗ P A , Ψ ⊥ g = θ A h + P A , Ψ ⊥ g , ∀ h ⊕ g ∈ H 1 . \theta_B(h\oplus g)=\sum_{n=1}^{\infty}L_nA_nh+\sum_{n=1}^{\infty}L_nL_n^*P_{A,\Psi}^{\perp}g=\theta_Ah+P_{A,\Psi}^{\perp}g,\quad\forall h\oplus g\in\mathcal{H}_1. θ B ( h ⊕ g ) = n = 1 ∑ ∞ L n A n h + n = 1 ∑ ∞ L n L n ∗ P A , Ψ ⊥ g = θ A h + P A , Ψ ⊥ g , ∀ h ⊕ g ∈ H 1 . Similarly θ Φ ( h ⊕ g ) = θ Ψ h + P A , Ψ ⊥ g , ∀ h ⊕ g ∈ H 1 \theta_\Phi(h\oplus g)=\theta_\Psi h+P_{A,\Psi}^{\perp}g,\forall h\oplus g\in\mathcal{H}_1 θ Φ ( h ⊕ g ) = θ Ψ h + P A , Ψ ⊥ g , ∀ h ⊕ g ∈ H 1
⟨ θ B ∗ z , h ⊕ g ⟩ = ⟨ z , θ B ( h ⊕ g ) ⟩ = ⟨ θ A ∗ z , h ⟩ + ⟨ P A , Ψ ⊥ z , g ⟩ = ⟨ θ A ∗ z ⊕ P A , Ψ ⊥ z , h ⊕ g ⟩ , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 , ∀ h ⊕ g ∈ H 1 . \begin{aligned}
\langle\theta_{B}^{*}z,h\oplus g\rangle&=\langle z,\theta_B(h\oplus g)\rangle=\langle\theta_A^{*}z,h\rangle+\langle P_{A,\Psi}^{\perp}z,g\rangle\\
&=\langle\theta_A^{*}z\oplus P_{A,\Psi}^{\perp}z,h\oplus g\rangle,\quad\forall z\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0,\forall h\oplus g\in\mathcal{H}_1.
\end{aligned} ⟨ θ B ∗ z , h ⊕ g ⟩ = ⟨ z , θ B ( h ⊕ g )⟩ = ⟨ θ A ∗ z , h ⟩ + ⟨ P A , Ψ ⊥ z , g ⟩ = ⟨ θ A ∗ z ⊕ P A , Ψ ⊥ z , h ⊕ g ⟩ , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 , ∀ h ⊕ g ∈ H 1 . Hence θ B ∗ z = θ A ∗ z ⊕ P A , Ψ ⊥ z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 \theta_B^{*}z=\theta_A^{*}z\oplus P_{A,\Psi}^{\perp}z,\forall z\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 θ B ∗ z = θ A ∗ z ⊕ P A , Ψ ⊥ z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 θ Φ ∗ z = θ Ψ ∗ z ⊕ P A , Ψ ⊥ z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 \theta_\Phi^{*}z=\theta_\Psi^{*}z\oplus P_{A,\Psi}^{\perp}z,\ \forall z\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 θ Φ ∗ z = θ Ψ ∗ z ⊕ P A , Ψ ⊥ z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 θ A ( H ) = θ Ψ ( H ) \theta_A(\mathcal{H})=\theta_\Psi(\mathcal{H}) θ A ( H ) = θ Ψ ( H ) θ Ψ ∗ P A , Ψ ⊥ = 0 = P A , Ψ ⊥ θ A \theta_\Psi^{*}P_{A,\Psi}^{\perp}=0=P_{A,\Psi}^{\perp}\theta_A θ Ψ ∗ P A , Ψ ⊥ = 0 = P A , Ψ ⊥ θ A
S B , Φ ( h ⊕ g ) = θ Φ ∗ ( θ A h + P A , Ψ ⊥ g ) = θ Ψ ∗ ( θ A h + P A , Ψ ⊥ g ) ⊕ P A , Ψ ⊥ ( θ A h + P A , Ψ ⊥ g ) = ( S A , Ψ h + 0 ) ⊕ ( 0 + P A , Ψ ⊥ g ) = S A , Ψ h ⊕ P A , Ψ ⊥ g = I H h ⊕ I θ A ( H ) ⊥ g , ∀ h ⊕ g ∈ H 1 . \begin{aligned}
S_{B,\Phi}(h\oplus g)&=\theta_\Phi^{*}(\theta_Ah+P_{A,\Psi}^{\perp}g)\\
&=\theta_\Psi^{*}(\theta_Ah+P_{A,\Psi}^{\perp}g)\oplus P_{A,\Psi}^{\perp}(\theta_Ah+P_{A,\Psi}^{\perp}g)\\
&=(S_{A,\Psi}h+0)\oplus(0+P_{A,\Psi}^{\perp}g)=S_{A,\Psi}h\oplus P_{A,\Psi}^{\perp}g\\
&=I_{\mathcal{H}}h\oplus I_{\theta_A(\mathcal{H})^\perp}g,\quad\forall h\oplus g\in\mathcal{H}_1.
\end{aligned} S B , Φ ( h ⊕ g ) = θ Φ ∗ ( θ A h + P A , Ψ ⊥ g ) = θ Ψ ∗ ( θ A h + P A , Ψ ⊥ g ) ⊕ P A , Ψ ⊥ ( θ A h + P A , Ψ ⊥ g ) = ( S A , Ψ h + 0 ) ⊕ ( 0 + P A , Ψ ⊥ g ) = S A , Ψ h ⊕ P A , Ψ ⊥ g = I H h ⊕ I θ A ( H ) ⊥ g , ∀ h ⊕ g ∈ H 1 . Hence ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H 1 , H 0 ) \mathcal{B}(\mathcal{H}_1,\mathcal{H}_0) B ( H 1 , H 0 )
P B , Φ z = θ B S B , Φ − 1 θ Φ ∗ z = θ B θ Φ ∗ z = θ B ( θ Ψ ∗ z ⊕ P A , Ψ ⊥ z ) = θ A ( θ Ψ ∗ z ) + P A , Ψ ⊥ ( P A , Ψ ⊥ z ) = P A , Ψ z + P A , Ψ ⊥ z = P A , Ψ z + ( ( I ℓ 2 ( N ) ⊗ I H 0 ) − P A , Ψ ) z = ( I ℓ 2 ( N ) ⊗ I H 0 ) z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 . \begin{aligned}
P_{B,\Phi}z&=\theta_BS_{B,\Phi}^{-1}\theta_\Phi^{*}z=\theta_B\theta_\Phi^{*}z=\theta_B(\theta_\Psi^{*}z\oplus P_{A,\Psi}^{\perp}z)\\
&=\theta_A(\theta_\Psi^{*}z)+P_{A,\Psi}^{\perp}(P_{A,\Psi}^{\perp}z)=P_{A,\Psi}z+P_{A,\Psi}^{\perp}z\\
&=P_{A,\Psi}z+\left((I_{\ell^2(\mathbb{N})}\otimes I_{\mathcal{H}_0})-P_{A,\Psi}\right)z=(I_{\ell^2(\mathbb{N})}\otimes I_{\mathcal{H}_0})z,\quad\forall z\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0.
\end{aligned} P B , Φ z = θ B S B , Φ − 1 θ Φ ∗ z = θ B θ Φ ∗ z = θ B ( θ Ψ ∗ z ⊕ P A , Ψ ⊥ z ) = θ A ( θ Ψ ∗ z ) + P A , Ψ ⊥ ( P A , Ψ ⊥ z ) = P A , Ψ z + P A , Ψ ⊥ z = P A , Ψ z + ( ( I ℓ 2 ( N ) ⊗ I H 0 ) − P A , Ψ ) z = ( I ℓ 2 ( N ) ⊗ I H 0 ) z , ∀ z ∈ ℓ 2 ( N ) ⊗ H 0 . Therefore ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H 1 , H 0 ) \mathcal{B}(\mathcal{H}_1,\mathcal{H}_0) B ( H 1 , H 0 ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H 1 , H 0 ) \mathcal{B}(\mathcal{H}_1,\mathcal{H}_0) B ( H 1 , H 0 )
Theorem 5.1.13. A pair ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , A_n=L_n^*U,\quad\Psi_n=L_n^*V,\quad\forall n\in\mathbb{N}, A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , where U , V : H → ℓ 2 ( N ) ⊗ H 0 U,V:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U , V : H → ℓ 2 ( N ) ⊗ H 0 V ∗ U V^*U V ∗ U
Proof. ( ⇐ ) (\Leftarrow) ( ⇐ ) θ A \theta_A θ A θ Ψ \theta_\Psi θ Ψ h ∈ H h\in\mathcal{H} h ∈ H (13) , we have
S A , Ψ = ∑ n = 1 ∞ ( L n ∗ V ) ∗ L n ∗ U h = V ∗ ( ∑ n = 1 ∞ L n L n ∗ ) U h = V ∗ U h . (40) S_{A,\Psi}=\sum_{n=1}^{\infty}(L_n^*V)^*L_n^*Uh=V^*\left(\sum_{n=1}^{\infty}L_nL_n^*\right)Uh=V^*Uh. \tag*{(40)} S A , Ψ = n = 1 ∑ ∞ ( L n ∗ V ) ∗ L n ∗ U h = V ∗ ( n = 1 ∑ ∞ L n L n ∗ ) U h = V ∗ U h . (40) Hence S A , Ψ S_{A,\Psi} S A , Ψ
( ⇒ ) (\Rightarrow) ( ⇒ ) U : = ∑ n = 1 ∞ L n A n U:=\sum_{n=1}^{\infty}L_nA_n U := ∑ n = 1 ∞ L n A n V : = ∑ n = 1 ∞ L n Ψ n V:=\sum_{n=1}^{\infty}L_n\Psi_n V := ∑ n = 1 ∞ L n Ψ n
L n ∗ U = L n ∗ ( ∑ k = 1 ∞ L k A k ) = ∑ k = 1 ∞ L n ∗ L k A k = A n , L_n^*U=L_n^*\left(\sum_{k=1}^{\infty}L_kA_k\right)=\sum_{k=1}^{\infty}L_n^*L_kA_k=A_n, L n ∗ U = L n ∗ ( k = 1 ∑ ∞ L k A k ) = k = 1 ∑ ∞ L n ∗ L k A k = A n , L n ∗ V = L n ∗ ( ∑ k = 1 ∞ L k Ψ k ) = ∑ k = 1 ∞ L n ∗ L k Ψ k = Ψ n , ∀ n ∈ N L_n^*V=L_n^*\left(\sum_{k=1}^{\infty}L_k\Psi_k\right)=\sum_{k=1}^{\infty}L_n^*L_k\Psi_k=\Psi_n,\qquad\forall n\in\mathbb{N} L n ∗ V = L n ∗ ( k = 1 ∑ ∞ L k Ψ k ) = k = 1 ∑ ∞ L n ∗ L k Ψ k = Ψ n , ∀ n ∈ N and
V ∗ U = ( ∑ n = 1 ∞ Ψ n ∗ L n ∗ ) ( ∑ k = 1 ∞ L k A k ) = ∑ n = 1 ∞ Ψ n ∗ A n = S A , Ψ V^*U=\left(\sum_{n=1}^{\infty}\Psi_n^*L_n^*\right)\left(\sum_{k=1}^{\infty}L_kA_k\right)=\sum_{n=1}^{\infty}\Psi_n^*A_n=S_{A,\Psi} V ∗ U = ( n = 1 ∑ ∞ Ψ n ∗ L n ∗ ) ( k = 1 ∑ ∞ L k A k ) = n = 1 ∑ ∞ Ψ n ∗ A n = S A , Ψ which is bounded invertible.
Corollary 5.1.14. A pair { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , A_n=L_n^*U,\qquad\Psi_n=L_n^*V,\qquad\forall n\in\mathbb{N}, A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , where U , V : H → ℓ 2 ( N ) ⊗ H 0 U,V:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U , V : H → ℓ 2 ( N ) ⊗ H 0 V ∗ U V^*U V ∗ U U ( V ∗ U ) − 1 V ∗ = I ℓ 2 ( N ) ⊗ H 0 U(V^*U)^{-1}V^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} U ( V ∗ U ) − 1 V ∗ = I ℓ 2 ( N ) ⊗ H 0
Proof. ( ⇐ ) (\Leftarrow) ( ⇐ ) P A , Ψ = U ( V ∗ U ) − 1 V ∗ = I ℓ 2 ( N ) ⊗ H 0 P_{A,\Psi}=U(V^*U)^{-1}V^*=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} P A , Ψ = U ( V ∗ U ) − 1 V ∗ = I ℓ 2 ( N ) ⊗ H 0
( ⇒ ) (\Rightarrow) ( ⇒ ) U U U V V V Theorem 5.1.13 . Then U ( V ∗ U ) − 1 V ∗ = P A , Ψ = I ℓ 2 ( N ) ⊗ H 0 U(V^*U)^{-1}V^*=P_{A,\Psi}=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} U ( V ∗ U ) − 1 V ∗ = P A , Ψ = I ℓ 2 ( N ) ⊗ H 0
Corollary 5.1.15. A pair { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , A_n=L_n^*U,\qquad\Psi_n=L_n^*V,\qquad\forall n\in\mathbb{N}, A n = L n ∗ U , Ψ n = L n ∗ V , ∀ n ∈ N , where U , V : H → ℓ 2 ( N ) ⊗ H 0 U,V:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U , V : H → ℓ 2 ( N ) ⊗ H 0 V ∗ U V^*U V ∗ U V ∗ U = I H V^*U=I_{\mathcal{H}} V ∗ U = I H I ℓ 2 ( N ) ⊗ H 0 = U V ∗ I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}=UV^* I ℓ 2 ( N ) ⊗ H 0 = U V ∗
Proof. We use Corollary 5.1.14.
( ⇐ ) (\Leftarrow) ( ⇐ ) S A , Ψ = V ∗ U = I H , P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = θ A θ Ψ ∗ = θ F U V ∗ θ F ∗ = I ℓ 2 ( N ) ⊗ H 0 S_{A,\Psi}=V^{*}U=I_{\mathcal{H}}, P_{A,\Psi}=\theta_A S_{A,\Psi}^{-1}\theta_{\Psi}^{*}=\theta_A\theta_{\Psi}^{*}=\theta_FUV^{*}\theta_F^{*}=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0} S A , Ψ = V ∗ U = I H , P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = θ A θ Ψ ∗ = θ F U V ∗ θ F ∗ = I ℓ 2 ( N ) ⊗ H 0
( ⇒ ) (\Rightarrow) ( ⇒ ) V ∗ U = S A , Ψ = I H V^{*}U=S_{A,\Psi}=I_{\mathcal{H}} V ∗ U = S A , Ψ = I H
U V ∗ = ( ∑ n = 1 ∞ L n ∗ A n ) ( ∑ k = 1 ∞ Ψ k ∗ L k ) = ∑ n = 1 ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . UV^{*}=\left(\sum_{n=1}^{\infty}L_n^{*}A_n\right)\left(\sum_{k=1}^{\infty}\Psi_k^{*}L_k\right)=\sum_{n=1}^{\infty}L_nL_n^{*}=I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}. U V ∗ = ( n = 1 ∑ ∞ L n ∗ A n ) ( k = 1 ∑ ∞ Ψ k ∗ L k ) = n = 1 ∑ ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . Theorem 5.1.16. Let { F n } n \{F_n\}_n { F n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = F n U , Ψ n = F n V , ∀ n ∈ N , A_n=F_nU,\qquad\Psi_n=F_nV,\qquad\forall n\in\mathbb{N}, A n = F n U , Ψ n = F n V , ∀ n ∈ N , where U , V : H → H U,V:\mathcal{H}\to\mathcal{H} U , V : H → H V ∗ U V^{*}U V ∗ U
Proof. ( ⇐ ) (\Leftarrow) ( ⇐ ) ∑ n = 1 ∞ L n ( F n U ) = ( ∑ n = 1 ∞ L n F n ) U \sum_{n=1}^{\infty}L_n(F_nU)=\left(\sum_{n=1}^{\infty}L_nF_n\right)U ∑ n = 1 ∞ L n ( F n U ) = ( ∑ n = 1 ∞ L n F n ) U ∑ n = 1 ∞ L n ( F n V ) = ( ∑ n = 1 ∞ L n F n ) V \sum_{n=1}^{\infty}L_n(F_nV)=\left(\sum_{n=1}^{\infty}L_nF_n\right)V ∑ n = 1 ∞ L n ( F n V ) = ( ∑ n = 1 ∞ L n F n ) V ( { F n U } n , { F n V } n ) (\{F_nU\}_n,\{F_nV\}_n) ({ F n U } n , { F n V } n )
∑ n = 1 ∞ ( F n V ) ∗ ( F n U ) = V ∗ U \sum_{n=1}^{\infty}(F_nV)^{*}(F_nU)=V^{*}U n = 1 ∑ ∞ ( F n V ) ∗ ( F n U ) = V ∗ U shows that it is a factorable weak OVF.
( ⇒ ) (\Rightarrow) ( ⇒ ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) ∑ n = 1 ∞ F n ∗ A n \sum_{n=1}^{\infty}F_n^{*}A_n ∑ n = 1 ∞ F n ∗ A n ∑ n = 1 ∞ F n ∗ Ψ n \sum_{n=1}^{\infty}F_n^{*}\Psi_n ∑ n = 1 ∞ F n ∗ Ψ n h ∈ H h\in\mathcal{H} h ∈ H
∥ ∑ n = 1 m F n ∗ A n h ∥ 2 = ⟨ ∑ n = 1 m F n ∗ A n h , ∑ k = 1 m F k ∗ A k h ⟩ = ∑ n = 1 m ⟨ A n h , F n ( ∑ k = 1 m F k ∗ A k h ) ⟩ = ∑ n = 1 m ∥ A n h ∥ 2 . \left\|\sum_{n=1}^{m}F_n^{*}A_nh\right\|^2
=\left\langle\sum_{n=1}^{m}F_n^{*}A_nh,\sum_{k=1}^{m}F_k^{*}A_kh\right\rangle
=\sum_{n=1}^{m}\left\langle A_nh,F_n\left(\sum_{k=1}^{m}F_k^{*}A_kh\right)\right\rangle
=\sum_{n=1}^{m}\|A_nh\|^2. n = 1 ∑ m F n ∗ A n h 2 = ⟨ n = 1 ∑ m F n ∗ A n h , k = 1 ∑ m F k ∗ A k h ⟩ = n = 1 ∑ m ⟨ A n h , F n ( k = 1 ∑ m F k ∗ A k h ) ⟩ = n = 1 ∑ m ∥ A n h ∥ 2 . which converges to ∥ θ A h ∥ 2 = ∥ ∑ n = 1 ∞ L n A n h ∥ 2 = ∑ n = 1 ∞ ∥ A n h ∥ 2 \|\theta_Ah\|^2=\|\sum_{n=1}^{\infty}L_nA_nh\|^2=\sum_{n=1}^{\infty}\|A_nh\|^2 ∥ θ A h ∥ 2 = ∥ ∑ n = 1 ∞ L n A n h ∥ 2 = ∑ n = 1 ∞ ∥ A n h ∥ 2 U : = ∑ n = 1 ∞ F n ∗ A n U:=\sum_{n=1}^{\infty}F_n^{*}A_n U := ∑ n = 1 ∞ F n ∗ A n V : = ∑ n = 1 ∞ F n ∗ Ψ n V:=\sum_{n=1}^{\infty}F_n^{*}\Psi_n V := ∑ n = 1 ∞ F n ∗ Ψ n F n U = A n , F n V = Ψ n , ∀ n ∈ N F_nU=A_n,F_nV=\Psi_n,\forall n\in\mathbb{N} F n U = A n , F n V = Ψ n , ∀ n ∈ N
V ∗ U = ( ∑ n = 1 ∞ Ψ n ∗ F n ) ( ∑ k = 1 ∞ F k ∗ A k ) = ∑ n = 1 ∞ Ψ n ∗ A n = S A , Ψ V^{*}U=\left(\sum_{n=1}^{\infty}\Psi_n^{*}F_n\right)\left(\sum_{k=1}^{\infty}F_k^{*}A_k\right)=\sum_{n=1}^{\infty}\Psi_n^{*}A_n=S_{A,\Psi} V ∗ U = ( n = 1 ∑ ∞ Ψ n ∗ F n ) ( k = 1 ∑ ∞ F k ∗ A k ) = n = 1 ∑ ∞ Ψ n ∗ A n = S A , Ψ which is bounded invertible.
Corollary 5.1.17. Let { F n } n \{F_n\}_n { F n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = F n U , Ψ n = F n V , ∀ n ∈ N , A_n=F_nU,\qquad\Psi_n=F_nV,\qquad\forall n\in\mathbb{N}, A n = F n U , Ψ n = F n V , ∀ n ∈ N , where U , V : H → H U,V:\mathcal{H}\to\mathcal{H} U , V : H → H V ∗ U V^*U V ∗ U U ( V ∗ U ) − 1 V ∗ = I H U(V^*U)^{-1}V^*=I_{\mathcal{H}} U ( V ∗ U ) − 1 V ∗ = I H
(ii) an orthonormal factorable weak OVF in B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
A n = F n U , Ψ n = F n V , ∀ n ∈ N , A_n=F_nU,\qquad\Psi_n=F_nV,\qquad\forall n\in\mathbb{N}, A n = F n U , Ψ n = F n V , ∀ n ∈ N , where U , V : H → H U,V:\mathcal{H}\to\mathcal{H} U , V : H → H V ∗ U V^*U V ∗ U V ∗ U = I H = U V ∗ V^*U=I_{\mathcal{H}}=UV^* V ∗ U = I H = U V ∗
Proof. (i) ( ⇐ ) (\Leftarrow) ( ⇐ )
P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = ( ∑ n = 1 ∞ L n F n U ) ( V ∗ U ) − 1 ( ∑ k = 1 ∞ V ∗ F k ∗ L k ∗ ) = θ F U ( V ∗ U ) − 1 V ∗ θ F ∗ = θ F I H θ F ∗ = ∑ n = 1 ∞ L n F n ( ∑ k = 1 ∞ F k ∗ L k ∗ ) = ∑ n = 1 ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . \begin{aligned}
P_{A,\Psi}=\theta_AS_{A,\Psi}^{-1}\theta_{\Psi}^{*}
=\left(\sum_{n=1}^{\infty}L_nF_nU\right)(V^*U)^{-1}\left(\sum_{k=1}^{\infty}V^*F_k^*L_k^*\right) \\
&=\theta_FU(V^*U)^{-1}V^*\theta_F^*
=\theta_FI_{\mathcal{H}}\theta_F^*
=\sum_{n=1}^{\infty}L_nF_n\left(\sum_{k=1}^{\infty}F_k^*L_k^*\right) \\
&=\sum_{n=1}^{\infty}L_nL_n^*
=I_{\ell_2(\mathbb{N})\otimes\mathcal{H}_0}.
\end{aligned} P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = ( n = 1 ∑ ∞ L n F n U ) ( V ∗ U ) − 1 ( k = 1 ∑ ∞ V ∗ F k ∗ L k ∗ ) = θ F U ( V ∗ U ) − 1 V ∗ θ F ∗ = θ F I H θ F ∗ = n = 1 ∑ ∞ L n F n ( k = 1 ∑ ∞ F k ∗ L k ∗ ) = n = 1 ∑ ∞ L n L n ∗ = I ℓ 2 ( N ) ⊗ H 0 . ( ⇒ ) (\Rightarrow) ( ⇒ ) U U U V V V
U ( V ∗ U ) − 1 V ∗ = ( ∑ k = 1 ∞ F k ∗ A k ) S A , Ψ − 1 ( ∑ n = 1 ∞ Ψ n ∗ F n ) = ( ∑ r = 1 ∞ F r ∗ L r ∗ ) ( ∑ k = 1 ∞ L k A k ) S A , Ψ − 1 ( ∑ n = 1 ∞ Ψ n ∗ L n ∗ ) ( ∑ m = 1 ∞ L m F m ) = ( ∑ r = 1 ∞ F r ∗ L r ∗ ) θ A S A , Ψ − 1 θ Ψ ∗ ( ∑ m = 1 ∞ L m F m ) = ( ∑ r = 1 ∞ F r ∗ L r ∗ ) P A , Ψ ( ∑ m = 1 ∞ L m F m ) = ( ∑ r = 1 ∞ F r ∗ L r ∗ ) ( I ℓ 2 ( N ) ⊗ H 0 ) ( ∑ m = 1 ∞ L m F m ) = ( ∑ r = 1 ∞ F r ∗ L r ∗ ) ( ∑ m = 1 ∞ L m F m ) = ∑ r = 1 ∞ F r ∗ F r = I H . \begin{aligned}
U(V^*U)^{-1}V^*
=\left(\sum_{k=1}^{\infty}F_k^*A_k\right)S_{A,\Psi}^{-1}\left(\sum_{n=1}^{\infty}\Psi_n^*F_n\right) \\
&=\left(\sum_{r=1}^{\infty}F_r^*L_r^*\right)\left(\sum_{k=1}^{\infty}L_kA_k\right)S_{A,\Psi}^{-1}\left(\sum_{n=1}^{\infty}\Psi_n^*L_n^*\right)\left(\sum_{m=1}^{\infty}L_mF_m\right) \\
&=\left(\sum_{r=1}^{\infty}F_r^*L_r^*\right)\theta_AS_{A,\Psi}^{-1}\theta_\Psi^*\left(\sum_{m=1}^{\infty}L_mF_m\right) \\
&=\left(\sum_{r=1}^{\infty}F_r^*L_r^*\right)P_{A,\Psi}\left(\sum_{m=1}^{\infty}L_mF_m\right) \\
&=\left(\sum_{r=1}^{\infty}F_r^*L_r^*\right)\left(I_{\ell_2(\mathbb{N})\otimes\mathcal{H}_0}\right)\left(\sum_{m=1}^{\infty}L_mF_m\right) \\
&=\left(\sum_{r=1}^{\infty}F_r^*L_r^*\right)\left(\sum_{m=1}^{\infty}L_mF_m\right)
=\sum_{r=1}^{\infty}F_r^*F_r=I_{\mathcal{H}}.
\end{aligned} U ( V ∗ U ) − 1 V ∗ = ( k = 1 ∑ ∞ F k ∗ A k ) S A , Ψ − 1 ( n = 1 ∑ ∞ Ψ n ∗ F n ) = ( r = 1 ∑ ∞ F r ∗ L r ∗ ) ( k = 1 ∑ ∞ L k A k ) S A , Ψ − 1 ( n = 1 ∑ ∞ Ψ n ∗ L n ∗ ) ( m = 1 ∑ ∞ L m F m ) = ( r = 1 ∑ ∞ F r ∗ L r ∗ ) θ A S A , Ψ − 1 θ Ψ ∗ ( m = 1 ∑ ∞ L m F m ) = ( r = 1 ∑ ∞ F r ∗ L r ∗ ) P A , Ψ ( m = 1 ∑ ∞ L m F m ) = ( r = 1 ∑ ∞ F r ∗ L r ∗ ) ( I ℓ 2 ( N ) ⊗ H 0 ) ( m = 1 ∑ ∞ L m F m ) = ( r = 1 ∑ ∞ F r ∗ L r ∗ ) ( m = 1 ∑ ∞ L m F m ) = r = 1 ∑ ∞ F r ∗ F r = I H . (ii) We use (i).
( ⇐ ) S A , Ψ = V ∗ U = I H , P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = θ A θ Ψ ∗ = θ F U V ∗ θ F ∗ = θ F I H θ F ∗ = I ℓ 2 ( N ) ⊗ H 0 . (\Leftarrow)\ S_{A,\Psi}=V^{*}U=I_{\mathcal{H}},\ P_{A,\Psi}=\theta_A S_{A,\Psi}^{-1}\theta_{\Psi}^{*}=\theta_A\theta_{\Psi}^{*}=\theta_FUV^{*}\theta_F^{*}=\theta_FI_{\mathcal{H}}\theta_F^{*}=I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0}. ( ⇐ ) S A , Ψ = V ∗ U = I H , P A , Ψ = θ A S A , Ψ − 1 θ Ψ ∗ = θ A θ Ψ ∗ = θ F U V ∗ θ F ∗ = θ F I H θ F ∗ = I ℓ 2 ( N ) ⊗ H 0 . ( ⇒ ) V ∗ U = S A , Ψ = I H and using Proposition 5.1.10, (\Rightarrow)\ V^{*}U=S_{A,\Psi}=I_{\mathcal{H}}\text{ and using Proposition 5.1.10,} ( ⇒ ) V ∗ U = S A , Ψ = I H and using Proposition 5.1.10, U V ∗ = ( ∑ n = 1 ∞ F n ∗ A n ) ( ∑ k = 1 ∞ Ψ k ∗ F k ) = ∑ n = 1 ∞ F n ∗ F n = I H . UV^{*}=\left(\sum_{n=1}^{\infty}F_n^{*}A_n\right)\left(\sum_{k=1}^{\infty}\Psi_k^{*}F_k\right)=\sum_{n=1}^{\infty}F_n^{*}F_n=I_{\mathcal{H}}. U V ∗ = ( n = 1 ∑ ∞ F n ∗ A n ) ( k = 1 ∑ ∞ Ψ k ∗ F k ) = n = 1 ∑ ∞ F n ∗ F n = I H . We next derive another characterization which is free from natural numbers.
Theorem 5.1.18. Let { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
(i) if and only if
U : ℓ 2 ( N ) ⊗ H 0 ∋ y ↦ ∑ n = 1 ∞ A n ∗ L n ∗ y ∈ H , and V : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ ∑ n = 1 ∞ Ψ n ∗ L n ∗ z ∈ H U:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0\ni y\mapsto\sum_{n=1}^{\infty}A_n^{*}L_n^{*}y\in\mathcal{H},\quad\text{and}\quad V:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0\ni z\mapsto\sum_{n=1}^{\infty}\Psi_n^{*}L_n^{*}z\in\mathcal{H} U : ℓ 2 ( N ) ⊗ H 0 ∋ y ↦ n = 1 ∑ ∞ A n ∗ L n ∗ y ∈ H , and V : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ n = 1 ∑ ∞ Ψ n ∗ L n ∗ z ∈ H are well-defined bounded linear operators such that V U ∗ VU^{*} V U ∗
(ii) if and only if
U : ℓ 2 ( N ) ⊗ H 0 ∋ y ↦ ∑ n = 1 ∞ A n ∗ L n ∗ y ∈ H , and S : H ∋ g ↦ ∑ n = 1 ∞ L n Ψ n g ∈ ℓ 2 ( N ) ⊗ H 0 U:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0\ni y\mapsto\sum_{n=1}^{\infty}A_n^{*}L_n^{*}y\in\mathcal{H},\quad\text{and}\quad S:\mathcal{H}\ni g\mapsto\sum_{n=1}^{\infty}L_n\Psi_n g\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 U : ℓ 2 ( N ) ⊗ H 0 ∋ y ↦ n = 1 ∑ ∞ A n ∗ L n ∗ y ∈ H , and S : H ∋ g ↦ n = 1 ∑ ∞ L n Ψ n g ∈ ℓ 2 ( N ) ⊗ H 0 are well-defined bounded linear operators such that S ∗ U ∗ S^{*}U^{*} S ∗ U ∗
(iii) if and only if
R : H ∋ h ↦ ∑ n = 1 ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 , and V : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ ∑ n = 1 ∞ Ψ n ∗ L n ∗ z ∈ H R:\mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}L_nA_nh\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0,\quad\text{and}\quad V:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0\ni z\mapsto\sum_{n=1}^{\infty}\Psi_n^{*}L_n^{*}z\in\mathcal{H} R : H ∋ h ↦ n = 1 ∑ ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 , and V : ℓ 2 ( N ) ⊗ H 0 ∋ z ↦ n = 1 ∑ ∞ Ψ n ∗ L n ∗ z ∈ H are well-defined bounded linear operators such that V R VR V R
(iv) if and only if
R : H ∋ h ↦ ∑ n = 1 ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 , and S : H ∋ g ↦ ∑ n = 1 ∞ L n Ψ n g ∈ ℓ 2 ( N ) ⊗ H 0 R:\mathcal{H}\ni h\mapsto\sum_{n=1}^{\infty}L_nA_nh\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0,\quad\text{and}\quad S:\mathcal{H}\ni g\mapsto\sum_{n=1}^{\infty}L_n\Psi_n g\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 R : H ∋ h ↦ n = 1 ∑ ∞ L n A n h ∈ ℓ 2 ( N ) ⊗ H 0 , and S : H ∋ g ↦ n = 1 ∑ ∞ L n Ψ n g ∈ ℓ 2 ( N ) ⊗ H 0 are well-defined bounded linear operators such that S ∗ R S^{*}R S ∗ R
Proof. We prove (i) and others are similar.
( ⇒ ) (\Rightarrow) ( ⇒ ) U = θ A ∗ U=\theta_A^* U = θ A ∗ V = θ Φ ∗ V=\theta_\Phi^* V = θ Φ ∗ V U ∗ = θ Φ ∗ θ A = S A , Ψ VU^*=\theta_\Phi^*\theta_A=S_{A,\Psi} V U ∗ = θ Φ ∗ θ A = S A , Ψ
( ⇐ ) (\Leftarrow) ( ⇐ ) θ A = U ∗ \theta_A=U^* θ A = U ∗ θ Ψ = V ∗ \theta_\Psi=V^* θ Ψ = V ∗ S A , Ψ = θ Ψ ∗ θ A = V U ∗ S_{A,\Psi}=\theta_\Psi^*\theta_A=VU^* S A , Ψ = θ Ψ ∗ θ A = V U ∗
Now we try to characterize all dual OVFs.
Lemma 5.1.19. Let { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { B n } n , { Φ n } n \{B_n\}_n,\{\Phi_n\}_n { B n } n , { Φ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n
B n = L n ∗ U , Φ n = L n ∗ V ∗ , ∀ n ∈ N B_n=L_n^*U,\qquad\Phi_n=L_n^*V^*,\qquad\forall n\in\mathbb{N} B n = L n ∗ U , Φ n = L n ∗ V ∗ , ∀ n ∈ N where U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U : H → ℓ 2 ( N ) ⊗ H 0 θ Ψ ∗ \theta_\Psi^* θ Ψ ∗ V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H θ A \theta_A θ A V U VU V U
Proof. ( ⇐ ) (\Leftarrow) ( ⇐ ) { B n } n , { Φ n } n \{B_n\}_n,\{\Phi_n\}_n { B n } n , { Φ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { B n } n , { Φ n } n \{B_n\}_n,\{\Phi_n\}_n { B n } n , { Φ n } n θ Φ ∗ θ A = V ∗ θ A = I H \theta_\Phi^*\theta_A=V^*\theta_A=I_{\mathcal{H}} θ Φ ∗ θ A = V ∗ θ A = I H θ Ψ ∗ θ B = θ Ψ ∗ U = I H \theta_\Psi^*\theta_B=\theta_\Psi^*U=I_{\mathcal{H}} θ Ψ ∗ θ B = θ Ψ ∗ U = I H
( ⇒ ) (\Rightarrow) ( ⇒ ) { B n } n , { Φ n } n \{B_n\}_n,\{\Phi_n\}_n { B n } n , { Φ n } n { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n θ Φ ∗ θ B = I H = θ Φ ∗ θ A \theta_\Phi^*\theta_B=I_{\mathcal{H}}=\theta_\Phi^*\theta_A θ Φ ∗ θ B = I H = θ Φ ∗ θ A U : = θ B , V : = θ Φ ∗ U:=\theta_B,V:=\theta_\Phi^* U := θ B , V := θ Φ ∗ U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U : H → ℓ 2 ( N ) ⊗ H 0 θ Ψ ∗ \theta_\Psi^* θ Ψ ∗ V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H θ A \theta_A θ A V U = θ Φ ∗ θ B = S B , Φ VU=\theta_\Phi^*\theta_B=S_{B,\Phi} V U = θ Φ ∗ θ B = S B , Φ
L n ∗ U = L n ∗ ( ∑ k = 1 ∞ L k B k ) = B n , L n ∗ V ∗ = L n ∗ ( ∑ k = 1 ∞ L k Φ k ) = Φ n , ∀ n ∈ N . L_n^*U=L_n^*\left(\sum_{k=1}^{\infty}L_kB_k\right)=B_n,\qquad L_n^*V^*=L_n^*\left(\sum_{k=1}^{\infty}L_k\Phi_k\right)=\Phi_n,\qquad\forall n\in\mathbb{N}. L n ∗ U = L n ∗ ( k = 1 ∑ ∞ L k B k ) = B n , L n ∗ V ∗ = L n ∗ ( k = 1 ∑ ∞ L k Φ k ) = Φ n , ∀ n ∈ N . Lemma 5.1.20. Let { A n } n , { Ψ n } n \{A_n\}_n,\{\Psi_n\}_n { A n } n , { Ψ n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
(i) R : H → ℓ 2 ( N ) ⊗ H 0 R:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 R : H → ℓ 2 ( N ) ⊗ H 0 θ Ψ ∗ \theta_\Psi^* θ Ψ ∗
R = θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U , R=\theta_AS_{A,\Psi}^{-1}+(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_AS_{A,\Psi}^{-1}\theta_\Psi^*)U, R = θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U , where U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U : H → ℓ 2 ( N ) ⊗ H 0
(ii) L : ℓ 2 ( N ) ⊗ H 0 → H L:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\mathcal{H} L : ℓ 2 ( N ) ⊗ H 0 → H θ A \theta_A θ A
L = S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) , L=S_{A,\Psi}^{-1}\theta_\Psi^*+V(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_AS_{A,\Psi}^{-1}\theta_\Psi^*), L = S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) , where V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H
Proof. (i) (⇐ \Leftarrow ⇐ U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0} U : H → ℓ 2 ( N ) ⊗ H 0
θ Ψ ∗ ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = I H + θ Ψ ∗ U − θ Ψ ∗ U = I H . \theta_{\Psi}^{*}\left(\theta_{A}S_{A,\Psi}^{-1}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})U\right)=I_{\mathcal{H}}+\theta_{\Psi}^{*}U-\theta_{\Psi}^{*}U=I_{\mathcal{H}}. θ Ψ ∗ ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = I H + θ Ψ ∗ U − θ Ψ ∗ U = I H . Therefore θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U \theta_{A}S_{A,\Psi}^{-1}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})U θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U θ Ψ ∗ \theta_{\Psi}^{*} θ Ψ ∗
(⇒ \Rightarrow ⇒ R : H → ℓ 2 ( N ) ⊗ H 0 R:\mathcal{H}\to\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0} R : H → ℓ 2 ( N ) ⊗ H 0 θ Ψ ∗ \theta_{\Psi}^{*} θ Ψ ∗ U : = R U:=R U := R
θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U = θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) R = θ A S A , Ψ − 1 + R − θ A S A , Ψ − 1 = R . \theta_{A}S_{A,\Psi}^{-1}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})U
=\theta_{A}S_{A,\Psi}^{-1}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})R
=\theta_{A}S_{A,\Psi}^{-1}+R-\theta_{A}S_{A,\Psi}^{-1}=R. θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U = θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) R = θ A S A , Ψ − 1 + R − θ A S A , Ψ − 1 = R . (ii) (⇐ \Leftarrow ⇐ V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H
( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) ) θ A = I H + V θ A − V θ A I H = I H . (S_{A,\Psi}^{-1}\theta_{\Psi}^{*}+V(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*}))\theta_{A}
=I_{\mathcal{H}}+V\theta_{A}-V\theta_{A}I_{\mathcal{H}}=I_{\mathcal{H}}. ( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ )) θ A = I H + V θ A − V θ A I H = I H . Therefore S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) S_{A,\Psi}^{-1}\theta_{\Psi}^{*}+V(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*}) S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) θ A \theta_{A} θ A
(⇒ \Rightarrow ⇒ L : ℓ 2 ( N ) ⊗ H 0 → H L:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}\to\mathcal{H} L : ℓ 2 ( N ) ⊗ H 0 → H θ A \theta_{A} θ A V : = L V:=L V := L
S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) = S A , Ψ − 1 θ Ψ ∗ + L ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) = S A , Ψ − 1 θ Ψ ∗ + L − I H S A , Ψ − 1 θ Ψ ∗ = L . S_{A,\Psi}^{-1}\theta_{\Psi}^{*}+V(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})
=S_{A,\Psi}^{-1}\theta_{\Psi}^{*}+L(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})
=S_{A,\Psi}^{-1}\theta_{\Psi}^{*}+L-I_{\mathcal{H}}S_{A,\Psi}^{-1}\theta_{\Psi}^{*}=L. S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) = S A , Ψ − 1 θ Ψ ∗ + L ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) = S A , Ψ − 1 θ Ψ ∗ + L − I H S A , Ψ − 1 θ Ψ ∗ = L . Theorem 5.1.21. Let ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { B n } n , { Φ n } n ) (\{B_n\}_n,\{\Phi_n\}_n) ({ B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
B n = A n S A , Ψ − 1 + L n ∗ U − A n S A , Ψ − 1 θ Ψ ∗ U , B_n=A_nS_{A,\Psi}^{-1}+L_n^{*}U-A_nS_{A,\Psi}^{-1}\theta_{\Psi}^{*}U, B n = A n S A , Ψ − 1 + L n ∗ U − A n S A , Ψ − 1 θ Ψ ∗ U , Φ n = Ψ n ( S A , Ψ − 1 ) ∗ + L n ∗ V ∗ − Ψ n ( S A , Ψ − 1 ) ∗ θ A ∗ V ∗ , ∀ n ∈ N \Phi_n=\Psi_n(S_{A,\Psi}^{-1})^{*}+L_n^{*}V^{*}-\Psi_n(S_{A,\Psi}^{-1})^{*}\theta_{A}^{*}V^{*},\qquad\forall n\in\mathbb{N} Φ n = Ψ n ( S A , Ψ − 1 ) ∗ + L n ∗ V ∗ − Ψ n ( S A , Ψ − 1 ) ∗ θ A ∗ V ∗ , ∀ n ∈ N such that the operator
S A , Ψ − 1 + V U − V θ A S A , Ψ − 1 θ Ψ ∗ U S_{A,\Psi}^{-1}+VU-V\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*}U S A , Ψ − 1 + V U − V θ A S A , Ψ − 1 θ Ψ ∗ U is bounded invertible, where U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0} U : H → ℓ 2 ( N ) ⊗ H 0 V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H
Proof. Lemmas 5.1.19 and 5.1.20 give the characterization of dual weak OVF as
B n = L n ∗ ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = A n S A , Ψ − 1 + L n ∗ U − A n S A , Ψ − 1 θ Ψ ∗ U , \begin{aligned}
B_n=L_n^{*}(\theta_{A}S_{A,\Psi}^{-1}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{A}S_{A,\Psi}^{-1}\theta_{\Psi}^{*})U) \\
&=A_nS_{A,\Psi}^{-1}+L_n^{*}U-A_nS_{A,\Psi}^{-1}\theta_{\Psi}^{*}U,
\end{aligned} B n = L n ∗ ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = A n S A , Ψ − 1 + L n ∗ U − A n S A , Ψ − 1 θ Ψ ∗ U , Φ n = L n ∗ ( θ Ψ ( S A , Ψ − 1 ) ∗ + ( I ℓ 2 ( N ) ⊗ H 0 − θ Ψ ( S A , Ψ − 1 ) ∗ θ A ∗ ) V ∗ ) = Ψ n ( S A , Ψ − 1 ) ∗ + L n ∗ V ∗ − Ψ n ( S A , Ψ − 1 ) ∗ θ A ∗ V ∗ , ∀ n ∈ N \begin{aligned}
\Phi_n=L_n^{*}(\theta_{\Psi}(S_{A,\Psi}^{-1})^{*}+(I_{\ell^{2}(\mathbb{N})\otimes\mathcal{H}_{0}}-\theta_{\Psi}(S_{A,\Psi}^{-1})^{*}\theta_{A}^{*})V^{*}) \\
&=\Psi_n(S_{A,\Psi}^{-1})^{*}+L_n^{*}V^{*}-\Psi_n(S_{A,\Psi}^{-1})^{*}\theta_{A}^{*}V^{*},\qquad\forall n\in\mathbb{N}
\end{aligned} Φ n = L n ∗ ( θ Ψ ( S A , Ψ − 1 ) ∗ + ( I ℓ 2 ( N ) ⊗ H 0 − θ Ψ ( S A , Ψ − 1 ) ∗ θ A ∗ ) V ∗ ) = Ψ n ( S A , Ψ − 1 ) ∗ + L n ∗ V ∗ − Ψ n ( S A , Ψ − 1 ) ∗ θ A ∗ V ∗ , ∀ n ∈ N such that the operator
( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) ) ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) \left(S_{A,\Psi}^{-1}\theta_\Psi^*+V\left(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_A S_{A,\Psi}^{-1}\theta_\Psi^*\right)\right)\left(\theta_A S_{A,\Psi}^{-1}+\left(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_A S_{A,\Psi}^{-1}\theta_\Psi^*\right)U\right) ( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) ) ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) is bounded invertible, where U : H → ℓ 2 ( N ) ⊗ H 0 U:\mathcal{H}\to\ell^2(\mathbb{N})\otimes\mathcal{H}_0 U : H → ℓ 2 ( N ) ⊗ H 0 V : ℓ 2 ( N ) ⊗ H 0 → H V:\ell^2(\mathbb{N})\otimes\mathcal{H}_0\to\mathcal{H} V : ℓ 2 ( N ) ⊗ H 0 → H
( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) ) ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = S A , Ψ − 1 + V U − V θ A S A , Ψ − 1 θ Ψ ∗ U . \left(S_{A,\Psi}^{-1}\theta_\Psi^*+V\left(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_A S_{A,\Psi}^{-1}\theta_\Psi^*\right)\right)\left(\theta_A S_{A,\Psi}^{-1}+\left(I_{\ell^2(\mathbb{N})\otimes\mathcal{H}_0}-\theta_A S_{A,\Psi}^{-1}\theta_\Psi^*\right)U\right)
=
S_{A,\Psi}^{-1}+VU-V\theta_A S_{A,\Psi}^{-1}\theta_\Psi^*U. ( S A , Ψ − 1 θ Ψ ∗ + V ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) ) ( θ A S A , Ψ − 1 + ( I ℓ 2 ( N ) ⊗ H 0 − θ A S A , Ψ − 1 θ Ψ ∗ ) U ) = S A , Ψ − 1 + V U − V θ A S A , Ψ − 1 θ Ψ ∗ U . We now define the orthogonality for weak OVFs.
Definition 5.1.22. A weak OVF ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
∑ n = 1 ∞ Ψ n ∗ B n = ∑ n = 1 ∞ Φ n ∗ A n = 0. \sum_{n=1}^{\infty}\Psi_n^*B_n=\sum_{n=1}^{\infty}\Phi_n^*A_n=0. n = 1 ∑ ∞ Ψ n ∗ B n = n = 1 ∑ ∞ Φ n ∗ A n = 0. Remarkable property of orthogonal frames is that we can interpolate as well as we can take direct sum of them to get new frames. These are illustrated in the following two results.
Proposition 5.1.23. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) C , D , E , F ∈ B ( H ) C,D,E,F\in\mathcal{B}(\mathcal{H}) C , D , E , F ∈ B ( H ) C ∗ E + D ∗ F = I H C^*E+D^*F=I_{\mathcal{H}} C ∗ E + D ∗ F = I H
( { A n C + B n D } n , { Ψ n E + Φ n F } n ) \left(\{A_nC+B_nD\}_n,\{\Psi_nE+\Phi_nF\}_n\right) ( { A n C + B n D } n , { Ψ n E + Φ n F } n ) is a Parseval weak OVF in B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) c , d , e , f c,d,e,f c , d , e , f c ‾ e + d ‾ f = 1 \overline{c}e+\overline{d}f=1 c e + d f = 1 ( { c A n + d B n } n , { e Ψ n + f Φ n } n ) \left(\{cA_n+dB_n\}_n,\{e\Psi_n+f\Phi_n\}_n\right) ( { c A n + d B n } n , { e Ψ n + f Φ n } n )
Proof. We use the definition of frame operator and get
S A C + B D , Ψ E + Φ F = ∑ n = 1 ∞ ( Ψ n E + Φ n F ) ∗ ( A n C + B n D ) = E ∗ S A , Ψ C + E ∗ ( ∑ n = 1 ∞ Ψ n ∗ B n ) D + F ∗ ( ∑ n = 1 ∞ Φ n ∗ A n ) C + F ∗ S B , Φ D = E ∗ I H C + E ∗ 0 D + F ∗ 0 C + F ∗ I H D = I H . \begin{aligned}
S_{AC+BD,\Psi E+\Phi F}=\sum_{n=1}^{\infty}(\Psi_nE+\Phi_nF)^*(A_nC+B_nD) \\
&=E^*S_{A,\Psi}C+E^*\left(\sum_{n=1}^{\infty}\Psi_n^*B_n\right)D+F^*\left(\sum_{n=1}^{\infty}\Phi_n^*A_n\right)C+F^*S_{B,\Phi}D \\
&=E^*I_{\mathcal{H}}C+E^*0D+F^*0C+F^*I_{\mathcal{H}}D=I_{\mathcal{H}}.
\end{aligned} S A C + B D , Ψ E + Φ F = n = 1 ∑ ∞ ( Ψ n E + Φ n F ) ∗ ( A n C + B n D ) = E ∗ S A , Ψ C + E ∗ ( n = 1 ∑ ∞ Ψ n ∗ B n ) D + F ∗ ( n = 1 ∑ ∞ Φ n ∗ A n ) C + F ∗ S B , Φ D = E ∗ I H C + E ∗ 0 D + F ∗ 0 C + F ∗ I H D = I H . Proposition 5.1.24. If ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n ⊕ B n } n , { Ψ n ⊕ Φ n } n ) \left(\{A_n\oplus B_n\}_n,\{\Psi_n\oplus\Phi_n\}_n\right) ( { A n ⊕ B n } n , { Ψ n ⊕ Φ n } n ) B ( H ⊕ H , H 0 ) \mathcal{B}(\mathcal{H}\oplus\mathcal{H},\mathcal{H}_0) B ( H ⊕ H , H 0 ) ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) ( { A n ⊕ B n } n , { Ψ n ⊕ Φ n } n ) \left(\{A_n\oplus B_n\}_n,\{\Psi_n\oplus\Phi_n\}_n\right) ( { A n ⊕ B n } n , { Ψ n ⊕ Φ n } n )
Proof. Let h ⊕ g ∈ H ⊕ H h\oplus g\in\mathcal{H}\oplus\mathcal{H} h ⊕ g ∈ H ⊕ H
S A ⊕ B , Ψ ⊕ Φ ( h ⊕ g ) = ∑ n = 1 ∞ ( Ψ n ⊕ Φ n ) ∗ ( A n ⊕ B n ) ( h ⊕ g ) = ∑ n = 1 ∞ ( Ψ n ⊕ Φ n ) ∗ ( A n h + B n g ) = ∑ n = 1 ∞ ( Ψ n ∗ ( A n h + B n g ) ⊕ Φ n ∗ ( A n h + B n g ) ) = ( ∑ n = 1 ∞ Ψ n ∗ A n h + ∑ n = 1 ∞ Ψ n ∗ B n g ) ⊕ ( ∑ n = 1 ∞ Φ n ∗ A n h + ∑ n = 1 ∞ Φ n ∗ B n g ) = ( S A , Ψ h + 0 ) ⊕ ( 0 + S B , Φ g ) = ( S A , Ψ ⊕ S B , Φ ) ( h ⊕ g ) . \begin{aligned}
S_{A\oplus B,\Psi\oplus\Phi}(h\oplus g)
&=\sum_{n=1}^{\infty}(\Psi_n\oplus\Phi_n)^*(A_n\oplus B_n)(h\oplus g) \\
&=\sum_{n=1}^{\infty}(\Psi_n\oplus\Phi_n)^*(A_nh+B_ng) \\
&=\sum_{n=1}^{\infty}\left(\Psi_n^*(A_nh+B_ng)\oplus\Phi_n^*(A_nh+B_ng)\right) \\
&=\left(\sum_{n=1}^{\infty}\Psi_n^*A_nh+\sum_{n=1}^{\infty}\Psi_n^*B_ng\right)
\oplus\left(\sum_{n=1}^{\infty}\Phi_n^*A_nh+\sum_{n=1}^{\infty}\Phi_n^*B_ng\right) \\
&=(S_{A,\Psi}h+0)\oplus(0+S_{B,\Phi}g)=(S_{A,\Psi}\oplus S_{B,\Phi})(h\oplus g).
\end{aligned} S A ⊕ B , Ψ ⊕ Φ ( h ⊕ g ) = n = 1 ∑ ∞ ( Ψ n ⊕ Φ n ) ∗ ( A n ⊕ B n ) ( h ⊕ g ) = n = 1 ∑ ∞ ( Ψ n ⊕ Φ n ) ∗ ( A n h + B n g ) = n = 1 ∑ ∞ ( Ψ n ∗ ( A n h + B n g ) ⊕ Φ n ∗ ( A n h + B n g ) ) = ( n = 1 ∑ ∞ Ψ n ∗ A n h + n = 1 ∑ ∞ Ψ n ∗ B n g ) ⊕ ( n = 1 ∑ ∞ Φ n ∗ A n h + n = 1 ∑ ∞ Φ n ∗ B n g ) = ( S A , Ψ h + 0 ) ⊕ ( 0 + S B , Φ g ) = ( S A , Ψ ⊕ S B , Φ ) ( h ⊕ g ) . EQUIVALENCE OF WEAK OPERATOR-VALUED FRAMES
Definition 1.6.19 introduced similarity for OVFs. Here is the similar notion for factorable weak OVFs.
Definition 5.2.1. A factorable weak OVF ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal{B}(\mathcal{H}) R A , B , R Ψ , Φ ∈ B ( H )
B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N . B_n=A_nR_{A,B},\qquad\Phi_n=\Psi_nR_{\Psi,\Phi},\qquad\forall n\in\mathbb{N}. B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N . Since R A , B R_{A,B} R A , B R Ψ , Φ R_{\Psi,\Phi} R Ψ , Φ
{ ( { A n } n , { Ψ n } n ) : ( { A n } n , { Ψ n } n ) is a factorable weak OVF } . \left\{\left(\{A_n\}_n,\{\Psi_n\}_n\right):\left(\{A_n\}_n,\{\Psi_n\}_n\right)\text{ is a factorable weak OVF}\right\}. { ( { A n } n , { Ψ n } n ) : ( { A n } n , { Ψ n } n ) is a factorable weak OVF } . Similar frames have nice property that knowing analysis, synthesis and frame operators of one give that of another.
Lemma 5.2.2. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N B_n=A_nR_{A,B},\Phi_n=\Psi_nR_{\Psi,\Phi},\forall n\in\mathbb{N} B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal{B}(\mathcal{H}) R A , B , R Ψ , Φ ∈ B ( H )
(i) θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ \theta_B=\theta_A R_{A,B},\theta_\Phi=\theta_\Psi R_{\Psi,\Phi} θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ
(ii) S B , Φ = R Ψ , Φ ∗ S A , Ψ R A , B S_{B,\Phi}=R_{\Psi,\Phi}^{*}S_{A,\Psi}R_{A,B} S B , Φ = R Ψ , Φ ∗ S A , Ψ R A , B
(iii) P B , Φ = P A , Ψ P_{B,\Phi}=P_{A,\Psi} P B , Φ = P A , Ψ
Proof. θ B = ∑ n = 1 ∞ L n B n = ∑ n = 1 ∞ L n A n R A , B = θ A R A , B \theta_B=\sum_{n=1}^{\infty}L_nB_n=\sum_{n=1}^{\infty}L_nA_nR_{A,B}=\theta_AR_{A,B} θ B = ∑ n = 1 ∞ L n B n = ∑ n = 1 ∞ L n A n R A , B = θ A R A , B θ Φ = θ Ψ R Ψ , Φ \theta_\Phi=\theta_\Psi R_{\Psi,\Phi} θ Φ = θ Ψ R Ψ , Φ θ B \theta_B θ B θ Φ \theta_\Phi θ Φ S B , Φ = ∑ n = 1 ∞ Φ n ∗ B n = ∑ n = 1 ∞ ( Ψ n R Ψ , Φ ) ∗ ( A n R A , B ) = R Ψ , Φ ∗ ( ∑ n = 1 ∞ Ψ n ∗ A n ) R A , B = R Ψ , Φ ∗ S A , Ψ R A , B S_{B,\Phi}=\sum_{n=1}^{\infty}\Phi_n^{*}B_n=\sum_{n=1}^{\infty}(\Psi_nR_{\Psi,\Phi})^{*}(A_nR_{A,B})=R_{\Psi,\Phi}^{*}(\sum_{n=1}^{\infty}\Psi_n^{*}A_n)R_{A,B}=R_{\Psi,\Phi}^{*}S_{A,\Psi}R_{A,B} S B , Φ = ∑ n = 1 ∞ Φ n ∗ B n = ∑ n = 1 ∞ ( Ψ n R Ψ , Φ ) ∗ ( A n R A , B ) = R Ψ , Φ ∗ ( ∑ n = 1 ∞ Ψ n ∗ A n ) R A , B = R Ψ , Φ ∗ S A , Ψ R A , B
P B , Φ = θ B S B , Φ − 1 θ Φ ∗ = ( θ A R A , B ) ( R Ψ , Φ ∗ S A , Ψ R A , B ) − 1 ( θ Ψ R Ψ , Φ ) ∗ = P A , Ψ . P_{B,\Phi}=\theta_BS_{B,\Phi}^{-1}\theta_\Phi^{*}=(\theta_AR_{A,B})(R_{\Psi,\Phi}^{*}S_{A,\Psi}R_{A,B})^{-1}(\theta_\Psi R_{\Psi,\Phi})^{*}=P_{A,\Psi}. P B , Φ = θ B S B , Φ − 1 θ Φ ∗ = ( θ A R A , B ) ( R Ψ , Φ ∗ S A , Ψ R A , B ) − 1 ( θ Ψ R Ψ , Φ ) ∗ = P A , Ψ . We now classify similarity using operators.
Theorem 5.2.3. For two factorable weak OVFs ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n )
(i) B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N B_n=A_nR_{A,B},\Phi_n=\Psi_nR_{\Psi,\Phi},\forall n\in\mathbb{N} B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal{B}(\mathcal{H}) R A , B , R Ψ , Φ ∈ B ( H )
(ii) θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ \theta_B=\theta_AR_{A,B},\theta_\Phi=\theta_\Psi R_{\Psi,\Phi} θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal{B}(\mathcal{H}) R A , B , R Ψ , Φ ∈ B ( H )
(iii) P B , Φ = P A , Ψ P_{B,\Phi}=P_{A,\Psi} P B , Φ = P A , Ψ
If one of the above conditions is satisfied, then invertible operators in (i) and (ii) are unique and are given R A , B = S A , Ψ − 1 θ Ψ ∗ θ B R_{A,B}=S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_B R A , B = S A , Ψ − 1 θ Ψ ∗ θ B R Ψ , Φ = ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ R_{\Psi,\Phi}=(S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Phi R Ψ , Φ = ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) ( { B n } n , { Φ n } n ) \left(\{B_n\}_n,\{\Phi_n\}_n\right) ( { B n } n , { Φ n } n ) R Ψ , Φ ∗ R A , B = I H R_{\Psi,\Phi}^{*}R_{A,B}=I_{\mathcal{H}} R Ψ , Φ ∗ R A , B = I H R A , B R Ψ , Φ ∗ = I H R_{A,B}R_{\Psi,\Phi}^{*}=I_{\mathcal{H}} R A , B R Ψ , Φ ∗ = I H
Proof. The implications (i) ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ B n = L n ∗ θ B = L n ∗ θ A R A , B = A n R A , B ′ B_n=L_n^{*}\theta_B=L_n^{*}\theta_AR_{A,B}=A_nR_{A,B}^{\prime} B n = L n ∗ θ B = L n ∗ θ A R A , B = A n R A , B ′ Φ n \Phi_n Φ n θ B = P B , Φ θ B \theta_B=P_{B,\Phi}\theta_B θ B = P B , Φ θ B θ Φ = P B , Φ ∗ θ Φ \theta_\Phi=P_{B,\Phi}^{*}\theta_\Phi θ Φ = P B , Φ ∗ θ Φ θ B = P A , Ψ θ B = θ A ( S A , Ψ − 1 θ Ψ ∗ θ B ) \theta_B=P_{A,\Psi}\theta_B=\theta_A(S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_B) θ B = P A , Ψ θ B = θ A ( S A , Ψ − 1 θ Ψ ∗ θ B ) θ Φ = P A , Ψ ∗ θ Φ = ( θ A S A , Ψ − 1 θ Ψ ∗ ) ∗ θ Φ = θ Ψ ( ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) \theta_\Phi=P_{A,\Psi}^{*}\theta_\Phi=(\theta_AS_{A,\Psi}^{-1}\theta_\Psi^{*})^{*}\theta_\Phi=\theta_\Psi((S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Phi) θ Φ = P A , Ψ ∗ θ Φ = ( θ A S A , Ψ − 1 θ Ψ ∗ ) ∗ θ Φ = θ Ψ (( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) S A , Ψ − 1 θ Ψ ∗ θ B S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_B S A , Ψ − 1 θ Ψ ∗ θ B ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ (S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Phi ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ
( S A , Ψ − 1 θ Ψ ∗ θ B ) ( S B , Φ − 1 θ Φ ∗ θ A ) = S A , Ψ − 1 θ Ψ ∗ P B , Φ θ A = S A , Ψ − 1 θ Ψ ∗ P A , Ψ θ A = S A , Ψ − 1 θ Ψ ∗ θ A = I H , (S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_B)(S_{B,\Phi}^{-1}\theta_\Phi^{*}\theta_A)=S_{A,\Psi}^{-1}\theta_\Psi^{*}P_{B,\Phi}\theta_A=S_{A,\Psi}^{-1}\theta_\Psi^{*}P_{A,\Psi}\theta_A=S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_A=I_{\mathcal{H}}, ( S A , Ψ − 1 θ Ψ ∗ θ B ) ( S B , Φ − 1 θ Φ ∗ θ A ) = S A , Ψ − 1 θ Ψ ∗ P B , Φ θ A = S A , Ψ − 1 θ Ψ ∗ P A , Ψ θ A = S A , Ψ − 1 θ Ψ ∗ θ A = I H , ( S B , Φ − 1 θ Φ ∗ θ A ) ( S A , Ψ − 1 θ Ψ ∗ θ B ) = S B , Φ − 1 θ Φ ∗ P A , Ψ θ B = S B , Φ − 1 θ Φ ∗ P B , Φ θ B = S B , Φ − 1 θ Φ ∗ θ B = I H (S_{B,\Phi}^{-1}\theta_\Phi^{*}\theta_A)(S_{A,\Psi}^{-1}\theta_\Psi^{*}\theta_B)=S_{B,\Phi}^{-1}\theta_\Phi^{*}P_{A,\Psi}\theta_B=S_{B,\Phi}^{-1}\theta_\Phi^{*}P_{B,\Phi}\theta_B=S_{B,\Phi}^{-1}\theta_\Phi^{*}\theta_B=I_{\mathcal{H}} ( S B , Φ − 1 θ Φ ∗ θ A ) ( S A , Ψ − 1 θ Ψ ∗ θ B ) = S B , Φ − 1 θ Φ ∗ P A , Ψ θ B = S B , Φ − 1 θ Φ ∗ P B , Φ θ B = S B , Φ − 1 θ Φ ∗ θ B = I H and
( ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) ( ( S B , Φ − 1 ) ∗ θ B ∗ θ Ψ ) = ( ( S A , Ψ − 1 ) ∗ θ A ∗ P B , Φ ∗ θ Ψ = ( ( S A , Ψ − 1 ) ∗ θ A ∗ P A , Ψ ∗ θ Ψ = ( S A , Ψ − 1 ) ∗ θ A ∗ θ Ψ = I H , \begin{aligned}
((S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Phi)((S_{B,\Phi}^{-1})^{*}\theta_B^{*}\theta_\Psi)=((S_{A,\Psi}^{-1})^{*}\theta_A^{*}P_{B,\Phi}^{*}\theta_\Psi=((S_{A,\Psi}^{-1})^{*}\theta_A^{*}P_{A,\Psi}^{*}\theta_\Psi\\
&= (S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Psi=I_{\mathcal H},
\end{aligned} (( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) (( S B , Φ − 1 ) ∗ θ B ∗ θ Ψ ) = (( S A , Ψ − 1 ) ∗ θ A ∗ P B , Φ ∗ θ Ψ = (( S A , Ψ − 1 ) ∗ θ A ∗ P A , Ψ ∗ θ Ψ = ( S A , Ψ − 1 ) ∗ θ A ∗ θ Ψ = I H , ( ( S B , Φ − 1 ) ∗ θ B ∗ θ Ψ ) ( ( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) = ( S B , Φ − 1 ) ∗ θ B ∗ P A , Ψ θ Φ = ( S B , Φ − 1 ) ∗ θ B ∗ P B , Φ ∗ θ Φ = ( S B , Φ − 1 ) ∗ θ B ∗ θ Φ = I H . \begin{aligned}
((S_{B,\Phi}^{-1})^{*}\theta_B^{*}\theta_\Psi)((S_{A,\Psi}^{-1})^{*}\theta_A^{*}\theta_\Phi)=(S_{B,\Phi}^{-1})^{*}\theta_B^{*}P_{A,\Psi}\theta_\Phi=(S_{B,\Phi}^{-1})^{*}\theta_B^{*}P_{B,\Phi}^{*}\theta_\Phi\\
&=(S_{B,\Phi}^{-1})^{*}\theta_B^{*}\theta_\Phi=I_{\mathcal H}.
\end{aligned} (( S B , Φ − 1 ) ∗ θ B ∗ θ Ψ ) (( S A , Ψ − 1 ) ∗ θ A ∗ θ Φ ) = ( S B , Φ − 1 ) ∗ θ B ∗ P A , Ψ θ Φ = ( S B , Φ − 1 ) ∗ θ B ∗ P B , Φ ∗ θ Φ = ( S B , Φ − 1 ) ∗ θ B ∗ θ Φ = I H . Let R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal B(\mathcal H) R A , B , R Ψ , Φ ∈ B ( H ) R A , B R_{A,B} R A , B R Ψ , Φ R_{\Psi,\Phi} R Ψ , Φ B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N B_n=A_nR_{A,B},\Phi_n=\Psi_nR_{\Psi,\Phi},\forall n\in\mathbb N B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ ⟹ θ B ∗ θ B = θ A ∗ θ A R A , B = S A , Ψ R A , B , θ Φ ∗ θ Φ = θ Ψ ∗ θ Ψ R Ψ , Φ = S A , Ψ ∗ R Ψ , Φ \theta_B=\theta_AR_{A,B},\theta_\Phi=\theta_\Psi R_{\Psi,\Phi}\Longrightarrow\theta_B^{*}\theta_B=\theta_A^{*}\theta_AR_{A,B}=S_{A,\Psi}R_{A,B},\theta_\Phi^{*}\theta_\Phi=\theta_\Psi^{*}\theta_\Psi R_{\Psi,\Phi}=S_{A,\Psi}^{*}R_{\Psi,\Phi} θ B = θ A R A , B , θ Φ = θ Ψ R Ψ , Φ ⟹ θ B ∗ θ B = θ A ∗ θ A R A , B = S A , Ψ R A , B , θ Φ ∗ θ Φ = θ Ψ ∗ θ Ψ R Ψ , Φ = S A , Ψ ∗ R Ψ , Φ R A , B R_{A,B} R A , B R Ψ , Φ R_{\Psi,\Phi} R Ψ , Φ S B , Φ = R Ψ , Φ ∗ S A , Ψ R A , B S_{B,\Phi}=R_{\Psi,\Phi}^{*}S_{A,\Psi}R_{A,B} S B , Φ = R Ψ , Φ ∗ S A , Ψ R A , B
Corollary 5.2.4. For any given factorable weak OVF ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) the canonical dual of ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) is the only dual factorable weak OVF that is similar to ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
Proof. Let ( { B n } n , { Φ n } n ) (\{B_n\}_n,\{\Phi_n\}_n) ({ B n } n , { Φ n } n ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) θ B ∗ θ Ψ = I H = θ Ψ ∗ θ A \theta_B^{*}\theta_\Psi=I_{\mathcal H}=\theta_\Psi^{*}\theta_A θ B ∗ θ Ψ = I H = θ Ψ ∗ θ A R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal B(\mathcal H) R A , B , R Ψ , Φ ∈ B ( H ) B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N B_n=A_nR_{A,B},\Phi_n=\Psi_nR_{\Psi,\Phi},\forall n\in\mathbb N B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N R A , B = S A , Ψ − 1 θ A ∗ θ B , R Ψ , Φ = S A , Ψ − 1 θ A ∗ θ Φ R_{A,B}=S_{A,\Psi}^{-1}\theta_A^{*}\theta_B,R_{\Psi,\Phi}=S_{A,\Psi}^{-1}\theta_A^{*}\theta_\Phi R A , B = S A , Ψ − 1 θ A ∗ θ B , R Ψ , Φ = S A , Ψ − 1 θ A ∗ θ Φ R A , B = S A , Ψ − 1 I H = S A , Ψ − 1 , R Ψ , Φ = ( S A , Ψ − 1 ) ∗ I H = ( S A , Ψ − 1 ) ∗ R_{A,B}=S_{A,\Psi}^{-1}I_{\mathcal H}=S_{A,\Psi}^{-1},R_{\Psi,\Phi}=(S_{A,\Psi}^{-1})^{*}I_{\mathcal H}=(S_{A,\Psi}^{-1})^{*} R A , B = S A , Ψ − 1 I H = S A , Ψ − 1 , R Ψ , Φ = ( S A , Ψ − 1 ) ∗ I H = ( S A , Ψ − 1 ) ∗ ( { B n } n , { Φ n } n ) (\{B_n\}_n,\{\Phi_n\}_n) ({ B n } n , { Φ n } n ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
Corollary 5.2.5. Two similar factorable weak OVF cannot be orthogonal.
Proof. Let a factorable weak OVF ( { B n } n , { Φ n } n ) (\{B_n\}_n,\{\Phi_n\}_n) ({ B n } n , { Φ n } n ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) R A , B , R Ψ , Φ ∈ B ( H ) R_{A,B},R_{\Psi,\Phi}\in\mathcal B(\mathcal H) R A , B , R Ψ , Φ ∈ B ( H ) B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N B_n=A_nR_{A,B},\Phi_n=\Psi_nR_{\Psi,\Phi},\forall n\in\mathbb N B n = A n R A , B , Φ n = Ψ n R Ψ , Φ , ∀ n ∈ N R A , B ∗ R_{A,B}^{*} R A , B ∗ S A , Ψ ∗ S_{A,\Psi}^{*} S A , Ψ ∗
θ B ∗ θ Ψ = ( θ A R A , B ) ∗ θ Ψ = R A , B ∗ θ A ∗ θ Ψ = R A , B ∗ S A , Ψ ∗ ≠ 0. \theta_B^{*}\theta_\Psi=(\theta_AR_{A,B})^{*}\theta_\Psi=R_{A,B}^{*}\theta_A^{*}\theta_\Psi=R_{A,B}^{*}S_{A,\Psi}^{*}\ne0. θ B ∗ θ Ψ = ( θ A R A , B ) ∗ θ Ψ = R A , B ∗ θ A ∗ θ Ψ = R A , B ∗ S A , Ψ ∗ = 0. For every factorable weak OVF ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n ) ( { A n S A , Ψ − 1 } n , { Ψ n } n ) (\{A_nS_{A,\Psi}^{-1}\}_n,\{\Psi_n\}_n) ({ A n S A , Ψ − 1 } n , { Ψ n } n ) ( { A n } n , { Ψ n ( S A , Ψ − 1 ) ∗ } n ) (\{A_n\}_n,\{\Psi_n(S_{A,\Psi}^{-1})^{*}\}_n) ({ A n } n , { Ψ n ( S A , Ψ − 1 ) ∗ } n ) ( { A n } n , { Ψ n } n ) (\{A_n\}_n,\{\Psi_n\}_n) ({ A n } n , { Ψ n } n )
WEAK OPERATOR-VALUED FRAMES GENERATED BY GROUPS AND GROUP LIKE UNITARY SYSTEMS
In this section G G G π \pi π G G G G G G e e e
Definition 5.3.1. Let π \pi π G G G H \mathcal{H} H A A A B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) factorable operator frame generator (resp. a Parseval frame generator ) w.r.t. an operator Ψ \Psi Ψ B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { A g : = A π g − 1 } g ∈ G , { Ψ g : = Ψ π g − 1 } g ∈ G \{A_g := A\pi_{g^{-1}}\}_{g\in G},\{\Psi_g := \Psi\pi_{g^{-1}}\}_{g\in G} { A g := A π g − 1 } g ∈ G , { Ψ g := Ψ π g − 1 } g ∈ G B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( A , Ψ ) (A,\Psi) ( A , Ψ ) π \pi π
Proposition 5.3.2. Let ( A , Ψ ) (A,\Psi) ( A , Ψ ) ( B , Φ ) (B,\Phi) ( B , Φ ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) π \pi π G G G H \mathcal{H} H
(i) θ A π g = ( λ g ⊗ I H 0 ) θ A , θ Ψ π g = ( λ g ⊗ I H 0 ) θ Ψ , ∀ g ∈ G \theta_A\pi_g=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A,\theta_\Psi\pi_g=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_\Psi,\ \forall g\in G θ A π g = ( λ g ⊗ I H 0 ) θ A , θ Ψ π g = ( λ g ⊗ I H 0 ) θ Ψ , ∀ g ∈ G
(ii) θ A ∗ θ B , θ Ψ ∗ θ Φ , θ A ∗ θ Φ \theta_A^*\theta_B,\theta_\Psi^*\theta_\Phi,\theta_A^*\theta_\Phi θ A ∗ θ B , θ Ψ ∗ θ Φ , θ A ∗ θ Φ π ( G ) ′ \pi(G)' π ( G ) ′ π ( G ) ′ ′ \pi(G)'' π ( G ) ′′ S A , Ψ ∈ π ( G ) ′ S_{A,\Psi}\in\pi(G)' S A , Ψ ∈ π ( G ) ′
(iii) θ A T θ Φ ∗ , θ A T θ B ∗ , θ Ψ T θ Φ ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ T ∈ π ( G ) ′ \theta_A T\theta_\Phi^*,\theta_A T\theta_B^*,\theta_\Psi T\theta_\Phi^*\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0),\ \forall T\in\pi(G)' θ A T θ Φ ∗ , θ A T θ B ∗ , θ Ψ T θ Φ ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ T ∈ π ( G ) ′ P A , Ψ ∈ R ( G ) ⊗ B ( H 0 ) P_{A,\Psi}\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0) P A , Ψ ∈ R ( G ) ⊗ B ( H 0 )
Proof. Let g , p , q ∈ G g,p,q\in G g , p , q ∈ G h ∈ H 0 h\in\mathcal{H}_0 h ∈ H 0
(i) From the definition of λ g \lambda_g λ g χ q \chi_q χ q λ g χ q = χ g q \lambda_g\chi_q=\chi_{gq} λ g χ q = χ g q L g q h = χ g q ⊗ h = λ g χ q ⊗ h = ( λ g ⊗ I H 0 ) ( χ q ⊗ h ) = ( λ g ⊗ I H 0 ) L q h L_{gq}h=\chi_{gq}\otimes h=\lambda_g\chi_q\otimes h=(\lambda_g\otimes I_{\mathcal{H}_0})(\chi_q\otimes h)=(\lambda_g\otimes I_{\mathcal{H}_0})L_qh L g q h = χ g q ⊗ h = λ g χ q ⊗ h = ( λ g ⊗ I H 0 ) ( χ q ⊗ h ) = ( λ g ⊗ I H 0 ) L q h
θ A π g = ∑ p ∈ G L p A π p π g = ∑ p ∈ G L p A π p − 1 π g = ∑ p ∈ G L p A π p − 1 g = ∑ q ∈ G L g q A π q − 1 = ∑ q ∈ G ( λ g ⊗ I H 0 ) L q A π q − 1 = ( λ g ⊗ I H 0 ) θ A . \begin{aligned}
\theta_A\pi_g
&=\sum_{p\in G}L_pA\pi_p\pi_g
=\sum_{p\in G}L_pA\pi_{p^{-1}}\pi_g
=\sum_{p\in G}L_pA\pi_{p^{-1}g}\\
&=\sum_{q\in G}L_{gq}A\pi_{q^{-1}}
=\sum_{q\in G}(\lambda_g\otimes I_{\mathcal{H}_0})L_qA\pi_{q^{-1}}
=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A.
\end{aligned} θ A π g = p ∈ G ∑ L p A π p π g = p ∈ G ∑ L p A π p − 1 π g = p ∈ G ∑ L p A π p − 1 g = q ∈ G ∑ L g q A π q − 1 = q ∈ G ∑ ( λ g ⊗ I H 0 ) L q A π q − 1 = ( λ g ⊗ I H 0 ) θ A . Similarly θ Ψ π g = ( λ g ⊗ I H 0 ) θ Ψ \theta_\Psi\pi_g=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_\Psi θ Ψ π g = ( λ g ⊗ I H 0 ) θ Ψ
(ii) θ A ∗ θ B π g = θ A ∗ ( λ g ⊗ I H 0 ) θ B = ( ( λ g − 1 ⊗ I H 0 ) θ A ) ∗ θ B = ( θ A π g − 1 ) ∗ θ B = π g θ A ∗ θ B \theta_A^*\theta_B\pi_g=\theta_A^*(\lambda_g\otimes I_{\mathcal{H}_0})\theta_B=((\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_A)^*\theta_B=(\theta_A\pi_{g^{-1}})^*\theta_B=\pi_g\theta_A^*\theta_B θ A ∗ θ B π g = θ A ∗ ( λ g ⊗ I H 0 ) θ B = (( λ g − 1 ⊗ I H 0 ) θ A ) ∗ θ B = ( θ A π g − 1 ) ∗ θ B = π g θ A ∗ θ B θ Ψ ∗ θ Φ , θ A ∗ θ Φ ∈ π ( G ) ′ \theta_\Psi^*\theta_\Phi,\theta_A^*\theta_\Phi\in\pi(G)' θ Ψ ∗ θ Φ , θ A ∗ θ Φ ∈ π ( G ) ′ B = A B=A B = A Φ = Ψ \Phi=\Psi Φ = Ψ S A , Ψ ∈ π ( G ) ′ S_{A,\Psi}\in\pi(G)' S A , Ψ ∈ π ( G ) ′
(iii) Let T ∈ π ( G ) ′ T\in\pi(G)' T ∈ π ( G ) ′
θ A T θ Φ ∗ ( λ g ⊗ I H 0 ) = θ A T ( ( λ g − 1 ⊗ I H 0 ) θ Φ ) ∗ = θ A T π g θ Φ ∗ = θ A π g T θ Φ ∗ = ( λ g ⊗ I H 0 ) θ A T θ Φ ∗ . \begin{aligned}
\theta_A T\theta_\Phi^*(\lambda_g\otimes I_{\mathcal{H}_0})
&=\theta_A T((\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_\Phi)^*
=\theta_A T\pi_g\theta_\Phi^*\\
&=\theta_A\pi_gT\theta_\Phi^*
=(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A T\theta_\Phi^*.
\end{aligned} θ A T θ Φ ∗ ( λ g ⊗ I H 0 ) = θ A T (( λ g − 1 ⊗ I H 0 ) θ Φ ) ∗ = θ A T π g θ Φ ∗ = θ A π g T θ Φ ∗ = ( λ g ⊗ I H 0 ) θ A T θ Φ ∗ . From the construction of L ( G ) \mathcal{L}(G) L ( G ) θ A T θ Φ ∗ ∈ ( L ( G ) ⊗ { I H 0 } ) ′ = L ( G ) ′ ⊗ { I H 0 } ′ = R ( G ) ⊗ B ( H 0 ) \theta_A T\theta_\Phi^*\in(\mathcal{L}(G)\otimes\{I_{\mathcal{H}_0}\})'=\mathcal{L}(G)'\otimes\{I_{\mathcal{H}_0}\}'=\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0) θ A T θ Φ ∗ ∈ ( L ( G ) ⊗ { I H 0 } ) ′ = L ( G ) ′ ⊗ { I H 0 } ′ = R ( G ) ⊗ B ( H 0 ) θ A T θ B ∗ , θ Ψ S θ Φ ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ S ∈ π ( G ) ′ \theta_A T\theta_B^*,\theta_\Psi S\theta_\Phi^*\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0),\ \forall S\in\pi(G)' θ A T θ B ∗ , θ Ψ S θ Φ ∗ ∈ R ( G ) ⊗ B ( H 0 ) , ∀ S ∈ π ( G ) ′ T = S A , Ψ − 1 T=S_{A,\Psi}^{-1} T = S A , Ψ − 1 P A , Ψ ∈ R ( G ) ⊗ B ( H 0 ) P_{A,\Psi}\in\mathcal{R}(G)\otimes\mathcal{B}(\mathcal{H}_0) P A , Ψ ∈ R ( G ) ⊗ B ( H 0 )
▢
Theorem 5.3.3. Let G G G { { A g } g ∈ G , { Ψ g } g ∈ G } \{\{A_g\}_{g\in G},\{\Psi_g\}_{g\in G}\} {{ A g } g ∈ G , { Ψ g } g ∈ G } B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) π \pi π G G G H \mathcal{H} H
A g = A e π g − 1 , Ψ g = Ψ e π g − 1 , ∀ g ∈ G A_g=A_e\pi_{g^{-1}},\qquad\Psi_g=\Psi_e\pi_{g^{-1}},\qquad\forall g\in G A g = A e π g − 1 , Ψ g = Ψ e π g − 1 , ∀ g ∈ G if and only if
A g p A g q ∗ = A p A q ∗ , A g p Ψ g q ∗ = A p Ψ q ∗ , Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . A_{gp}A_{gq}^{*}=A_pA_q^{*},\qquad A_{gp}\Psi_{gq}^{*}=A_p\Psi_q^{*},\qquad\Psi_{gp}\Psi_{gq}^{*}=\Psi_p\Psi_q^{*},\qquad\forall g,p,q\in G. A g p A g q ∗ = A p A q ∗ , A g p Ψ g q ∗ = A p Ψ q ∗ , Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . Proof. ( ⇒ ) (\Rightarrow) ( ⇒ )
A g p Ψ g q ∗ = A e π ( g p ) − 1 ( Ψ e π ( g q ) − 1 ) ∗ = A e π p − 1 π g − 1 π g π q Ψ e ∗ = A p Ψ q ∗ , ∀ g , p , q ∈ G . A_{gp}\Psi_{gq}^{*}=A_e\pi_{(gp)^{-1}}(\Psi_e\pi_{(gq)^{-1}})^{*}=A_e\pi_{p^{-1}}\pi_{g^{-1}}\pi_g\pi_q\Psi_e^{*}=A_p\Psi_q^{*},\qquad\forall g,p,q\in G. A g p Ψ g q ∗ = A e π ( g p ) − 1 ( Ψ e π ( g q ) − 1 ) ∗ = A e π p − 1 π g − 1 π g π q Ψ e ∗ = A p Ψ q ∗ , ∀ g , p , q ∈ G . Similarly we get other two equalities.
( ⇐ ) (\Leftarrow) ( ⇐ ) g ∈ G g\in G g ∈ G
( λ g ⊗ I H 0 ) θ A θ A ∗ = θ A θ A ∗ ( λ g ⊗ I H 0 ) , ( λ g ⊗ I H 0 ) θ A θ Ψ ∗ = θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) , (\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_A^{*}=\theta_A\theta_A^{*}(\lambda_g\otimes I_{\mathcal{H}_0}),\qquad
(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^{*}=\theta_A\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0}), ( λ g ⊗ I H 0 ) θ A θ A ∗ = θ A θ A ∗ ( λ g ⊗ I H 0 ) , ( λ g ⊗ I H 0 ) θ A θ Ψ ∗ = θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) , ( λ g ⊗ I H 0 ) θ Ψ θ Ψ ∗ = θ Ψ θ Ψ ∗ ( λ g ⊗ I H 0 ) . (\lambda_g\otimes I_{\mathcal{H}_0})\theta_\Psi\theta_\Psi^{*}=\theta_\Psi\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0}). ( λ g ⊗ I H 0 ) θ Ψ θ Ψ ∗ = θ Ψ θ Ψ ∗ ( λ g ⊗ I H 0 ) . Noticing λ g \lambda_g λ g ( λ g ⊗ I H 0 ) − 1 = ( λ g ⊗ I H 0 ) ∗ (\lambda_g\otimes I_{\mathcal{H}_0})^{-1}=(\lambda_g\otimes I_{\mathcal{H}_0})^{*} ( λ g ⊗ I H 0 ) − 1 = ( λ g ⊗ I H 0 ) ∗ ( λ g ⊗ I H 0 ) L q = L g q (\lambda_g\otimes I_{\mathcal{H}_0})L_q=L_{gq} ( λ g ⊗ I H 0 ) L q = L g q
( λ g ⊗ I H 0 ) θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) ∗ = ( ∑ p ∈ G ( λ g ⊗ I H 0 ) L p A p ) ( ∑ q ∈ G ( λ g ⊗ I H 0 ) L q Ψ q ) ∗ (\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0})^{*}
=\left(\sum_{p\in G}(\lambda_g\otimes I_{\mathcal{H}_0})L_pA_p\right)
\left(\sum_{q\in G}(\lambda_g\otimes I_{\mathcal{H}_0})L_q\Psi_q\right)^{*} ( λ g ⊗ I H 0 ) θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) ∗ = p ∈ G ∑ ( λ g ⊗ I H 0 ) L p A p q ∈ G ∑ ( λ g ⊗ I H 0 ) L q Ψ q ∗ = ∑ p ∈ G L g p ( ∑ q ∈ G A p Ψ q ∗ L g q ∗ ) = ∑ r ∈ G L r ( ∑ s ∈ G A g − 1 r Ψ g − 1 s ∗ L s ∗ ) =\sum_{p\in G}L_{gp}\left(\sum_{q\in G}A_p\Psi_q^{*}L_{gq}^{*}\right)
=\sum_{r\in G}L_r\left(\sum_{s\in G}A_{g^{-1}r}\Psi_{g^{-1}s}^{*}L_s^{*}\right) = p ∈ G ∑ L g p q ∈ G ∑ A p Ψ q ∗ L g q ∗ = r ∈ G ∑ L r ( s ∈ G ∑ A g − 1 r Ψ g − 1 s ∗ L s ∗ ) = ∑ r ∈ G L r ( ∑ s ∈ G A r Ψ s ∗ L s ∗ ) = θ A θ Ψ ∗ . =\sum_{r\in G}L_r\left(\sum_{s\in G}A_r\Psi_s^{*}L_s^{*}\right)
=\theta_A\theta_\Psi^{*}. = r ∈ G ∑ L r ( s ∈ G ∑ A r Ψ s ∗ L s ∗ ) = θ A θ Ψ ∗ . Define π : G ∋ g ↦ π g : = θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A ∈ B ( H ) \pi:G\ni g\mapsto\pi_g:=\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\in\mathcal{B}(\mathcal{H}) π : G ∋ g ↦ π g := θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A ∈ B ( H )
π g π h = θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A θ Ψ ∗ ( λ h ⊗ I H 0 ) θ A = θ Ψ ∗ θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) ( λ h ⊗ I H 0 ) θ A \pi_g\pi_h=\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^{*}(\lambda_h\otimes I_{\mathcal{H}_0})\theta_A
=\theta_\Psi^{*}\theta_A\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0})(\lambda_h\otimes I_{\mathcal{H}_0})\theta_A π g π h = θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A θ Ψ ∗ ( λ h ⊗ I H 0 ) θ A = θ Ψ ∗ θ A θ Ψ ∗ ( λ g ⊗ I H 0 ) ( λ h ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ g h ⊗ I H 0 ) θ A = π g h , ∀ g , h ∈ G =\theta_\Psi^{*}(\lambda_{gh}\otimes I_{\mathcal{H}_0})\theta_A=\pi_{gh},\qquad\forall g,h\in G = θ Ψ ∗ ( λ g h ⊗ I H 0 ) θ A = π g h , ∀ g , h ∈ G and
π g π g ∗ = θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A θ A ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ \pi_g\pi_g^{*}=\theta_\Psi^{*}(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_A^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_\Psi π g π g ∗ = θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A θ A ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ = θ Ψ ∗ θ A θ A ∗ ( λ g ⊗ I H 0 ) ( λ g − 1 ⊗ I H 0 ) θ Ψ = I H , = \theta_{\Psi}^{*}\theta_{A}\theta_{A}^{*}(\lambda_{g}\otimes I_{\mathcal{H}_0})(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_{\Psi}=I_{\mathcal{H}}, = θ Ψ ∗ θ A θ A ∗ ( λ g ⊗ I H 0 ) ( λ g − 1 ⊗ I H 0 ) θ Ψ = I H , π g ∗ π g = θ A ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A = θ A ∗ ( λ g − 1 ⊗ I H 0 ) ( λ g ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = I H , ∀ g ∈ G . \begin{aligned}
\pi_{g}^{*}\pi_{g}=\theta_{A}^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_{\Psi}\theta_{\Psi}^{*}(\lambda_{g}\otimes I_{\mathcal{H}_0})\theta_{A} \\
&=\theta_{A}^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})(\lambda_{g}\otimes I_{\mathcal{H}_0})\theta_{\Psi}\theta_{\Psi}^{*}\theta_{A}=I_{\mathcal{H}},\qquad\forall g\in G.
\end{aligned} π g ∗ π g = θ A ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ θ Ψ ∗ ( λ g ⊗ I H 0 ) θ A = θ A ∗ ( λ g − 1 ⊗ I H 0 ) ( λ g ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = I H , ∀ g ∈ G . Since G G G π \pi π A g = A e π g − 1 , Ψ g = Ψ e π g − 1 A_g=A_e\pi_{g^{-1}},\Psi_g=\Psi_e\pi_{g^{-1}} A g = A e π g − 1 , Ψ g = Ψ e π g − 1 g ∈ G g\in G g ∈ G
A e π g − 1 = L e ∗ θ A θ Ψ ∗ ( λ g − 1 ⊗ I H 0 ) θ A = L e ∗ ( λ g − 1 ⊗ I H 0 ) θ A θ Ψ ∗ θ A = ( ( λ g ⊗ I H 0 ) L e ) ∗ θ A = L g e ∗ θ A = A g , \begin{aligned}
A_e\pi_{g^{-1}}=L_e^{*}\theta_A\theta_{\Psi}^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_A=L_e^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_A\theta_{\Psi}^{*}\theta_A \\
&=((\lambda_g\otimes I_{\mathcal{H}_0})L_e)^{*}\theta_A=L_{ge}^{*}\theta_A=A_g,
\end{aligned} A e π g − 1 = L e ∗ θ A θ Ψ ∗ ( λ g − 1 ⊗ I H 0 ) θ A = L e ∗ ( λ g − 1 ⊗ I H 0 ) θ A θ Ψ ∗ θ A = (( λ g ⊗ I H 0 ) L e ) ∗ θ A = L g e ∗ θ A = A g , and
Ψ e π g − 1 = L e ∗ θ Ψ θ Ψ ∗ ( λ g − 1 ⊗ I H 0 ) θ A = L e ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = ( ( λ g ⊗ I H 0 ) L e ) ∗ θ Ψ = L g e ∗ θ Ψ = Ψ g . \begin{aligned}
\Psi_e\pi_{g^{-1}}=L_e^{*}\theta_{\Psi}\theta_{\Psi}^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_A=L_e^{*}(\lambda_{g^{-1}}\otimes I_{\mathcal{H}_0})\theta_{\Psi}\theta_{\Psi}^{*}\theta_A \\
&=((\lambda_g\otimes I_{\mathcal{H}_0})L_e)^{*}\theta_{\Psi}=L_{ge}^{*}\theta_{\Psi}=\Psi_g.
\end{aligned} Ψ e π g − 1 = L e ∗ θ Ψ θ Ψ ∗ ( λ g − 1 ⊗ I H 0 ) θ A = L e ∗ ( λ g − 1 ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = (( λ g ⊗ I H 0 ) L e ) ∗ θ Ψ = L g e ∗ θ Ψ = Ψ g . In the direct part of Theorem 5.3.3, we can remove the word ‘Parseval’ since it has not been used in the proof; same is true in the following corollary.
Corollary 5.3.4. Let G G G ( { A g } g ∈ G , { Ψ g } g ∈ G ) \left(\{A_g\}_{g\in G},\{\Psi_g\}_{g\in G}\right) ( { A g } g ∈ G , { Ψ g } g ∈ G ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) π \pi π G G G H \mathcal{H} H
(i) A g = A e S A , Ψ − 1 π g − 1 S A , Ψ , Ψ g = Ψ e π g − 1 A_g=A_eS_{A,\Psi}^{-1}\pi_{g^{-1}}S_{A,\Psi},\Psi_g=\Psi_e\pi_{g^{-1}} A g = A e S A , Ψ − 1 π g − 1 S A , Ψ , Ψ g = Ψ e π g − 1 g ∈ G g\in G g ∈ G
A g p S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A g q ∗ = A p S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A q ∗ , A g p S A , Ψ − 1 Ψ g q ∗ = A p S A , Ψ − 1 Ψ q ∗ , A_{gp}S_{A,\Psi}^{-1}(S_{A,\Psi}^{-1})^{*}A_{gq}^{*}=A_pS_{A,\Psi}^{-1}(S_{A,\Psi}^{-1})^{*}A_q^{*},\qquad A_{gp}S_{A,\Psi}^{-1}\Psi_{gq}^{*}=A_pS_{A,\Psi}^{-1}\Psi_q^{*}, A g p S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A g q ∗ = A p S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A q ∗ , A g p S A , Ψ − 1 Ψ g q ∗ = A p S A , Ψ − 1 Ψ q ∗ , Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . \Psi_{gp}\Psi_{gq}^{*}=\Psi_p\Psi_q^{*},\qquad\forall g,p,q\in G. Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . (ii) A g = A e π g − 1 , Ψ g = Ψ e ( S A , Ψ − 1 ) ∗ π g − 1 S A , Ψ A_g=A_e\pi_{g^{-1}},\Psi_g=\Psi_e(S_{A,\Psi}^{-1})^{*}\pi_{g^{-1}}S_{A,\Psi} A g = A e π g − 1 , Ψ g = Ψ e ( S A , Ψ − 1 ) ∗ π g − 1 S A , Ψ g ∈ G g\in G g ∈ G
A g p A g q ∗ = A p A q ∗ , A g p S A , Ψ − 1 Ψ g q ∗ = A p S A , Ψ − 1 Ψ q ∗ , A_{gp}A_{gq}^{*}=A_pA_q^{*},\qquad A_{gp}S_{A,\Psi}^{-1}\Psi_{gq}^{*}=A_pS_{A,\Psi}^{-1}\Psi_q^{*}, A g p A g q ∗ = A p A q ∗ , A g p S A , Ψ − 1 Ψ g q ∗ = A p S A , Ψ − 1 Ψ q ∗ , Ψ g p ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ g q ∗ = Ψ p ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ q ∗ , ∀ g , p , q ∈ G . \Psi_{gp}(S_{A,\Psi}^{-1})^{*}S_{A,\Psi}^{-1}\Psi_{gq}^{*}=\Psi_p(S_{A,\Psi}^{-1})^{*}S_{A,\Psi}^{-1}\Psi_q^{*},\qquad\forall g,p,q\in G. Ψ g p ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ g q ∗ = Ψ p ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ q ∗ , ∀ g , p , q ∈ G . Proof. (i) We apply Theorem 5.3.3 to the factorable Parseval OVF ( { A g S A , Ψ − 1 } g ∈ G , { Ψ g } g ∈ G ) \left(\{A_gS_{A,\Psi}^{-1}\}_{g\in G},\{\Psi_g\}_{g\in G}\right) ( { A g S A , Ψ − 1 } g ∈ G , { Ψ g } g ∈ G ) π \pi π G G G H \mathcal{H} H A g S A , Ψ − 1 = ( A e S A , Ψ − 1 ) π g − 1 , Ψ g = Ψ e π g − 1 A_gS_{A,\Psi}^{-1}=(A_eS_{A,\Psi}^{-1})\pi_{g^{-1}},\Psi_g=\Psi_e\pi_{g^{-1}} A g S A , Ψ − 1 = ( A e S A , Ψ − 1 ) π g − 1 , Ψ g = Ψ e π g − 1 g ∈ G g\in G g ∈ G
( A g p S A , Ψ − 1 ) ( A g q S A , Ψ − 1 ) ∗ = ( A p S A , Ψ − 1 ) ( A q S A , Ψ − 1 ) ∗ , ( A g p S A , Ψ − 1 ) Ψ g q ∗ = ( A p S A , Ψ − 1 ) Ψ q ∗ , (41) (A_{gp}S_{A,\Psi}^{-1})(A_{gq}S_{A,\Psi}^{-1})^{*}=(A_pS_{A,\Psi}^{-1})(A_qS_{A,\Psi}^{-1})^{*},\qquad(A_{gp}S_{A,\Psi}^{-1})\Psi_{gq}^{*}=(A_pS_{A,\Psi}^{-1})\Psi_q^{*}, \tag*{(41)} ( A g p S A , Ψ − 1 ) ( A g q S A , Ψ − 1 ) ∗ = ( A p S A , Ψ − 1 ) ( A q S A , Ψ − 1 ) ∗ , ( A g p S A , Ψ − 1 ) Ψ g q ∗ = ( A p S A , Ψ − 1 ) Ψ q ∗ , (41) Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . \Psi_{gp}\Psi_{gq}^{*}=\Psi_p\Psi_q^{*},\qquad\forall g,p,q\in G. Ψ g p Ψ g q ∗ = Ψ p Ψ q ∗ , ∀ g , p , q ∈ G . (ii) We apply Theorem 5.3.3 to the factorable Parseval OVF { A g } g ∈ G , { Ψ g ( S A , Ψ − 1 ) ∗ } g ∈ G \{A_g\}_{g\in G},\{\Psi_g(S_{A,\Psi}^{-1})^{*}\}_{g\in G} { A g } g ∈ G , { Ψ g ( S A , Ψ − 1 ) ∗ } g ∈ G π \pi π G G G H \mathcal{H} H A g = A e π g − 1 A_g=A_e\pi_{g^{-1}} A g = A e π g − 1 Ψ g S A , Ψ − 1 = ( Ψ e ( S A , Ψ − 1 ) ∗ ) π g − 1 \Psi_gS_{A,\Psi}^{-1}=(\Psi_e(S_{A,\Psi}^{-1})^{*})\pi_{g^{-1}} Ψ g S A , Ψ − 1 = ( Ψ e ( S A , Ψ − 1 ) ∗ ) π g − 1 g ∈ G g\in G g ∈ G
A g p A g q ∗ = A p A q ∗ , A g p ( Ψ g q ( S A , Ψ − 1 ) ∗ ) ∗ = A p ( Ψ q ( S A , Ψ − 1 ) ∗ ) ∗ , A_{gp}A_{gq}^{*}=A_pA_q^{*},\qquad A_{gp}(\Psi_{gq}(S_{A,\Psi}^{-1})^{*})^{*}=A_p(\Psi_q(S_{A,\Psi}^{-1})^{*})^{*}, A g p A g q ∗ = A p A q ∗ , A g p ( Ψ g q ( S A , Ψ − 1 ) ∗ ) ∗ = A p ( Ψ q ( S A , Ψ − 1 ) ∗ ) ∗ , ( Ψ g p ( S A , Ψ − 1 ) ∗ ) ( Ψ g q ( S A , Ψ − 1 ) ∗ ) ∗ = ( Ψ p ( S A , Ψ − 1 ) ∗ ) ( Ψ q ( S A , Ψ − 1 ) ∗ ) ∗ , ∀ g , p , q ∈ G . (\Psi_{gp}(S_{A,\Psi}^{-1})^{*})(\Psi_{gq}(S_{A,\Psi}^{-1})^{*})^{*}=(\Psi_p(S_{A,\Psi}^{-1})^{*})(\Psi_q(S_{A,\Psi}^{-1})^{*})^{*},\qquad\forall g,p,q\in G. ( Ψ g p ( S A , Ψ − 1 ) ∗ ) ( Ψ g q ( S A , Ψ − 1 ) ∗ ) ∗ = ( Ψ p ( S A , Ψ − 1 ) ∗ ) ( Ψ q ( S A , Ψ − 1 ) ∗ ) ∗ , ∀ g , p , q ∈ G . We next address the situation of factorable weak OVF whenever it is indexed by group-like unitary systems. Group-like unitary systems arose from the study of Weyl-Heisenberg frames. This was first formally defined by Gabardo and Han [112 ] . In the sequel, by T \mathbb{T} T
Definition 5.3.5. (Gabardo and Han [112 ] ) A collection U ⊆ B ( H ) \mathcal{U}\subseteq\mathcal{B}(\mathcal{H}) U ⊆ B ( H ) I H I_{\mathcal{H}} I H unitary system . If the group generated by unitary system U \mathcal{U} U group ( U ) \operatorname{group}(\mathcal{U}) group ( U )
(i) group ( U ) ⊆ T U : = { α U : α ∈ T , U ∈ U } \operatorname{group}(\mathcal{U})\subseteq\mathbb{T}\mathcal{U}:=\{\alpha U:\alpha\in\mathbb{T},U\in\mathcal{U}\} group ( U ) ⊆ T U := { α U : α ∈ T , U ∈ U }
(ii) U \mathcal{U} U T U ≠ T V \mathbb{T}U\ne\mathbb{T}V T U = T V U , V ∈ U U,V\in\mathcal{U} U , V ∈ U U ≠ V U\ne V U = V
then U \mathcal{U} U group-like unitary system .
Let U \mathcal{U} U [113 ] , we define mappings
f : group ( U ) → T and σ : group ( U ) → U . f:\operatorname{group}(\mathcal{U})\to\mathbb{T}\qquad\text{and}\qquad\sigma:\operatorname{group}(\mathcal{U})\to\mathcal{U}. f : group ( U ) → T and σ : group ( U ) → U . in the following way. For each U ∈ group ( U ) U\in\operatorname{group}(\mathcal{U}) U ∈ group ( U ) α ∈ T , V ∈ U \alpha\in\mathbb{T},V\in\mathcal{U} α ∈ T , V ∈ U U = α V U=\alpha V U = α V f ( U ) = α f(U)=\alpha f ( U ) = α σ ( U ) = V \sigma(U)=V σ ( U ) = V f , σ f,\sigma f , σ
U = f ( U ) σ ( U ) , ∀ U ∈ group ( U ) . U=f(U)\sigma(U),\qquad\forall U\in\operatorname{group}(\mathcal{U}). U = f ( U ) σ ( U ) , ∀ U ∈ group ( U ) . These mappings are called as corresponding mappings associated to U \mathcal{U} U
group ( U ) ⊆ T U → f T σ ↓ ↑ U ← T \begin{CD}
\operatorname{group}(\mathcal{U}) \subseteq\mathcal{T}\mathcal{U} @>{f}>> \mathcal{T} \\
@V{\sigma}VV @AA{}A \\
\mathcal{U} @<{}<< \mathcal{T}
\end{CD} group ( U ) ⊆ T U σ ↓ ⏐ U f T ⏐ ↑ T Next result gives certain fundamental properties of corresponding mappings associated with group-like unitary systems.
Proposition 5.3.6. [113 ] For a group-like unitary system U \mathcal{U} U f , σ f,\sigma f , σ
(i) f ( U σ ( V W ) ) f ( V W ) = f ( σ ( U V ) W ) f ( U V ) , ∀ U , V , W ∈ group ( U ) f(U\sigma(VW))f(VW)=f(\sigma(UV)W)f(UV),\ \forall U,V,W\in\operatorname{group}(\mathcal{U}) f ( U σ ( V W )) f ( V W ) = f ( σ ( U V ) W ) f ( U V ) , ∀ U , V , W ∈ group ( U )
(ii) σ ( U σ ( V W ) ) = σ ( σ ( U V ) W ) , ∀ U , V , W ∈ group ( U ) \sigma(U\sigma(VW))=\sigma(\sigma(UV)W),\ \forall U,V,W\in\operatorname{group}(\mathcal{U}) σ ( U σ ( V W )) = σ ( σ ( U V ) W ) , ∀ U , V , W ∈ group ( U )
(iii) σ ( U ) = U \sigma(U)=U σ ( U ) = U f ( U ) = 1 f(U)=1 f ( U ) = 1 U ∈ U U\in\mathcal{U} U ∈ U
(iv) If V , W ∈ group ( U ) V,W\in\operatorname{group}(\mathcal{U}) V , W ∈ group ( U )
U = { σ ( U V ) : U ∈ U } = { σ ( V U − 1 ) : U ∈ U } = { σ ( V U − 1 W ) : U ∈ U } = { σ ( V − 1 U ) : U ∈ U } . \mathcal{U}=\{\sigma(UV):U\in\mathcal{U}\}=\{\sigma(VU^{-1}):U\in\mathcal{U}\}
=\{\sigma(VU^{-1}W):U\in\mathcal{U}\}=\{\sigma(V^{-1}U):U\in\mathcal{U}\}. U = { σ ( U V ) : U ∈ U } = { σ ( V U − 1 ) : U ∈ U } = { σ ( V U − 1 W ) : U ∈ U } = { σ ( V − 1 U ) : U ∈ U } . (v) For fixed V , W ∈ U V,W\in\mathcal{U} V , W ∈ U U \mathcal{U} U
U ↦ σ ( V U ) ( resp. σ ( U V ) , σ ( U V − 1 ) , σ ( V − 1 U ) , σ ( V U − 1 ) , σ ( U − 1 V ) , σ ( V U − 1 W ) ) . U\mapsto\sigma(VU)\quad(\text{resp. }\sigma(UV),\sigma(UV^{-1}),\sigma(V^{-1}U),
\sigma(VU^{-1}),\sigma(U^{-1}V),\sigma(VU^{-1}W)). U ↦ σ ( V U ) ( resp. σ ( U V ) , σ ( U V − 1 ) , σ ( V − 1 U ) , σ ( V U − 1 ) , σ ( U − 1 V ) , σ ( V U − 1 W )) . Since group ( U ) \operatorname{group}(\mathcal{U}) group ( U ) Proposition 5.3.6 , we can replace V V V V − 1 V^{-1} V − 1 V ∈ group ( U ) V\in\operatorname{group}(\mathcal{U}) V ∈ group ( U ) ∑ U ∈ U x U = ∑ U ∈ U x σ ( V U ) \sum_{U\in\mathcal{U}}x_U=\sum_{U\in\mathcal{U}}x_{\sigma(VU)} ∑ U ∈ U x U = ∑ U ∈ U x σ ( V U )
Definition 5.3.7. [113 ] A unitary representation π \pi π U \mathcal{U} U H \mathcal{H} H U \mathcal{U} U H \mathcal{H} H
π ( U ) π ( V ) = f ( U V ) π ( σ ( U V ) ) , π ( U ) − 1 = f ( U − 1 ) π ( σ ( U − 1 ) ) , ∀ U , V ∈ U , \pi(U)\pi(V)=f(UV)\pi(\sigma(UV)),\qquad\pi(U)^{-1}=f(U^{-1})\pi(\sigma(U^{-1})),\ \forall U,V\in\mathcal{U}, π ( U ) π ( V ) = f ( U V ) π ( σ ( U V )) , π ( U ) − 1 = f ( U − 1 ) π ( σ ( U − 1 )) , ∀ U , V ∈ U , where f f f σ \sigma σ U \mathcal{U} U
Since π \pi π U \mathcal{U} U H \mathcal{H} H π ( U ) \pi(\mathcal{U}) π ( U )
Let U \mathcal{U} U { χ U } U ∈ U \{\chi_U\}_{U\in\mathcal{U}} { χ U } U ∈ U ℓ 2 ( U ) \ell^2(\mathcal{U}) ℓ 2 ( U ) λ \lambda λ U \mathcal{U} U λ U χ V = f ( U V ) χ σ ( U V ) , ∀ U , V ∈ U \lambda_U\chi_V=f(UV)\chi_{\sigma(UV)},\ \forall U,V\in\mathcal{U} λ U χ V = f ( U V ) χ σ ( U V ) , ∀ U , V ∈ U λ \lambda λ U \mathcal{U} U U \mathcal{U} U ρ U χ V = f ( V U − 1 ) χ σ ( V U − 1 ) , ∀ U , V ∈ U \rho_U\chi_V=f(VU^{-1})\chi_{\sigma(VU^{-1})},\ \forall U,V\in\mathcal{U} ρ U χ V = f ( V U − 1 ) χ σ ( V U − 1 ) , ∀ U , V ∈ U [113 ] . Like frame generators for groups, we now define the frame generator for group-like unitary systems.
Definition 5.3.8. Let U \mathcal{U} U A A A B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) operator frame generator (resp. a Parseval frame generator) w.r.t. Ψ \Psi Ψ B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( { A U : = A π ( U ) − 1 } U ∈ U , { Ψ U : = Ψ π ( U ) − 1 } U ∈ U ) (\{A_U:=A\pi(U)^{-1}\}_{U\in\mathcal{U}},\{\Psi_U:=\Psi\pi(U)^{-1}\}_{U\in\mathcal{U}}) ({ A U := A π ( U ) − 1 } U ∈ U , { Ψ U := Ψ π ( U ) − 1 } U ∈ U ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ( A , Ψ ) (A,\Psi) ( A , Ψ ) π \pi π
Theorem 5.3.9. Let U \mathcal{U} U I I I U \mathcal{U} U ( { A U } U ∈ U , { Ψ U } U ∈ U ) (\{A_U\}_{U\in\mathcal{U}},\{\Psi_U\}_{U\in\mathcal{U}}) ({ A U } U ∈ U , { Ψ U } U ∈ U ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) θ A ∗ \theta_A^* θ A ∗ π \pi π U \mathcal{U} U H \mathcal{H} H
A U = A I π ( U ) − 1 , Ψ U = Ψ I π ( U ) − 1 , ∀ U ∈ U A_U=A_I\pi(U)^{-1},\qquad\Psi_U=\Psi_I\pi(U)^{-1},\qquad\forall U\in\mathcal{U} A U = A I π ( U ) − 1 , Ψ U = Ψ I π ( U ) − 1 , ∀ U ∈ U if and only if
A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V A W ∗ A_{\sigma(UV)}A_{\sigma(UW)}^*=f(UV)\overline{f(UW)}A_VA_W^* A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) A V A W ∗ A σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V Ψ W ∗ A_{\sigma(UV)}\Psi_{\sigma(UW)}^*=f(UV)\overline{f(UW)}A_V\Psi_W^* A σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) A V Ψ W ∗ Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . \Psi_{\sigma(UV)}\Psi_{\sigma(UW)}^*=f(UV)\overline{f(UW)}\Psi_V\Psi_W^*,\qquad\forall U,V,W\in\mathcal{U}. Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . Proof. ( ⇒ ) (\Rightarrow) ( ⇒ ) U , V , W ∈ U U,V,W\in\mathcal{U} U , V , W ∈ U
A σ ( U V ) A σ ( U W ) ∗ = A I π ( σ ( U V ) ) − 1 ( A I π ( σ ( U W ) ) − 1 ) ∗ = A I ( f ( U V ) ‾ π ( U ) π ( V ) ) − 1 f ( U W ) ‾ π ( U ) π ( W ) A I ∗ = f ( U V ) f ( U W ) ‾ A I π ( V ) − 1 ( A I π ( W ) − 1 ) ∗ = f ( U V ) f ( U W ) ‾ A V A W ∗ . \begin{aligned}
A_{\sigma(UV)}A_{\sigma(UW)}^*
&=A_I\pi(\sigma(UV))^{-1}(A_I\pi(\sigma(UW))^{-1})^*\\
&=A_I\bigl(\overline{f(UV)}\pi(U)\pi(V)\bigr)^{-1}\overline{f(UW)}\pi(U)\pi(W)A_I^*\\
&=f(UV)\overline{f(UW)}A_I\pi(V)^{-1}(A_I\pi(W)^{-1})^*\\
&=f(UV)\overline{f(UW)}A_VA_W^*.
\end{aligned} A σ ( U V ) A σ ( U W ) ∗ = A I π ( σ ( U V ) ) − 1 ( A I π ( σ ( U W ) ) − 1 ) ∗ = A I ( f ( U V ) π ( U ) π ( V ) ) − 1 f ( U W ) π ( U ) π ( W ) A I ∗ = f ( U V ) f ( U W ) A I π ( V ) − 1 ( A I π ( W ) − 1 ) ∗ = f ( U V ) f ( U W ) A V A W ∗ . Others can be shown similarly.
( ⇐ ) (\Leftarrow) ( ⇐ )
tions. Following observation plays an important role in this part. Let h ∈ H h \in\mathcal{H} h ∈ H
L σ ( U V ) h = χ σ ( U V ) ⊗ h = f ( U V ) ‾ λ U χ V ⊗ h = f ( U V ) ‾ ( λ U χ V ⊗ h ) = f ( U V ) ‾ ( λ U ⊗ I H 0 ) ( χ V ⊗ h ) = f ( U V ) ‾ ( λ U ⊗ I H 0 ) L V h . L_{\sigma(UV)}h=\chi_{\sigma(UV)}\otimes h=\overline{f(UV)}\lambda_U\chi_V\otimes h=\overline{f(UV)}(\lambda_U\chi_V\otimes h)
=\overline{f(UV)}(\lambda_U\otimes I_{\mathcal{H}_0})(\chi_V\otimes h)=\overline{f(UV)}(\lambda_U\otimes I_{\mathcal{H}_0})L_Vh. L σ ( U V ) h = χ σ ( U V ) ⊗ h = f ( U V ) λ U χ V ⊗ h = f ( U V ) ( λ U χ V ⊗ h ) = f ( U V ) ( λ U ⊗ I H 0 ) ( χ V ⊗ h ) = f ( U V ) ( λ U ⊗ I H 0 ) L V h . As in the proof of Theorem 5.3.3 , we argue the following, for which now we prove the first. For all U ∈ U U\in\mathcal{U} U ∈ U
( λ U ⊗ I H 0 ) θ A θ A ∗ = θ A θ A ∗ ( λ U ⊗ I H 0 ) , ( λ U ⊗ I H 0 ) θ A θ Ψ ∗ = θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) , (\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_A^*=\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal{H}_0}),\qquad
(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^*=\theta_A\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}), ( λ U ⊗ I H 0 ) θ A θ A ∗ = θ A θ A ∗ ( λ U ⊗ I H 0 ) , ( λ U ⊗ I H 0 ) θ A θ Ψ ∗ = θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) , ( λ U ⊗ I H 0 ) θ Ψ θ Ψ ∗ = θ Ψ θ Ψ ∗ ( λ U ⊗ I H 0 ) . (\lambda_U\otimes I_{\mathcal{H}_0})\theta_\Psi\theta_\Psi^*=\theta_\Psi\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}). ( λ U ⊗ I H 0 ) θ Ψ θ Ψ ∗ = θ Ψ θ Ψ ∗ ( λ U ⊗ I H 0 ) . Consider
( λ U ⊗ I H 0 ) θ A θ A ∗ ( λ U ⊗ I H 0 ) ∗ = ( ∑ V ∈ U ( λ U ⊗ I H 0 ) L V A V ) ( ∑ W ∈ U ( λ U ⊗ I H 0 ) L W A W ) ∗ = ( ∑ V ∈ U f ( U V ) L σ ( U V ) A V ) ( ∑ W ∈ U f ( U W ) L σ ( U W ) A W ) ∗ = ∑ V ∈ U L σ ( U V ) ( ∑ W ∈ U f ( U V ) f ( U W ) ‾ A V A W ∗ L σ ( U W ) ∗ ) = ∑ V ∈ U L σ ( U V ) ( ∑ W ∈ U A σ ( U V ) A σ ( U W ) ∗ L σ ( U W ) ∗ ) = ( ∑ V ∈ U L σ ( U V ) A σ ( U V ) ) ( ∑ W ∈ U L σ ( U W ) A σ ( U W ) ) ∗ = θ A θ A ∗ , \begin{aligned}
(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal{H}_0})^*
&=\left(\sum_{V\in\mathcal{U}}(\lambda_U\otimes I_{\mathcal{H}_0})L_VA_V\right)
\left(\sum_{W\in\mathcal{U}}(\lambda_U\otimes I_{\mathcal{H}_0})L_WA_W\right)^*\\
&=\left(\sum_{V\in\mathcal{U}}f(UV)L_{\sigma(UV)}A_V\right)
\left(\sum_{W\in\mathcal{U}}f(UW)L_{\sigma(UW)}A_W\right)^*\\
&=\sum_{V\in\mathcal{U}}L_{\sigma(UV)}
\left(\sum_{W\in\mathcal{U}}f(UV)\overline{f(UW)}A_VA_W^*L_{\sigma(UW)}^*\right)\\
&=\sum_{V\in\mathcal{U}}L_{\sigma(UV)}
\left(\sum_{W\in\mathcal{U}}A_{\sigma(UV)}A_{\sigma(UW)}^*L_{\sigma(UW)}^*\right)\\
&=\left(\sum_{V\in\mathcal{U}}L_{\sigma(UV)}A_{\sigma(UV)}\right)
\left(\sum_{W\in\mathcal{U}}L_{\sigma(UW)}A_{\sigma(UW)}\right)^*\\
&=\theta_A\theta_A^*,
\end{aligned} ( λ U ⊗ I H 0 ) θ A θ A ∗ ( λ U ⊗ I H 0 ) ∗ = ( V ∈ U ∑ ( λ U ⊗ I H 0 ) L V A V ) ( W ∈ U ∑ ( λ U ⊗ I H 0 ) L W A W ) ∗ = ( V ∈ U ∑ f ( U V ) L σ ( U V ) A V ) ( W ∈ U ∑ f ( U W ) L σ ( U W ) A W ) ∗ = V ∈ U ∑ L σ ( U V ) ( W ∈ U ∑ f ( U V ) f ( U W ) A V A W ∗ L σ ( U W ) ∗ ) = V ∈ U ∑ L σ ( U V ) ( W ∈ U ∑ A σ ( U V ) A σ ( U W ) ∗ L σ ( U W ) ∗ ) = ( V ∈ U ∑ L σ ( U V ) A σ ( U V ) ) ( W ∈ U ∑ L σ ( U W ) A σ ( U W ) ) ∗ = θ A θ A ∗ , where last part of Proposition 5.3.6 is used in the last equality.
Define π : U ∋ U ↦ π ( U ) : = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ∈ B ( H ) \pi:\mathcal{U}\ni U\mapsto\pi(U):=\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\in\mathcal{B}(\mathcal{H}) π : U ∋ U ↦ π ( U ) := θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ∈ B ( H )
π ( U ) π ( V ) = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A = θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ V ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ U λ V ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( U V ) λ σ ( U V ) ⊗ I H 0 ) θ A = f ( U V ) θ Ψ ∗ ( λ σ ( U V ) ⊗ I H 0 ) θ A = f ( U V ) π ( σ ( U V ) ) , ∀ U , V ∈ U \begin{aligned}
\pi(U)\pi(V)
&=\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^*(\lambda_V\otimes I_{\mathcal{H}_0})\theta_A\\
&=\theta_\Psi^*\theta_A\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})(\lambda_V\otimes I_{\mathcal{H}_0})\theta_A\\
&=\theta_\Psi^*(\lambda_U\lambda_V\otimes I_{\mathcal{H}_0})\theta_A\\
&=\theta_\Psi^*(f(UV)\lambda_{\sigma(UV)}\otimes I_{\mathcal{H}_0})\theta_A\\
&=f(UV)\theta_\Psi^*(\lambda_{\sigma(UV)}\otimes I_{\mathcal{H}_0})\theta_A\\
&=f(UV)\pi(\sigma(UV)),\qquad\forall U,V\in\mathcal{U}
\end{aligned} π ( U ) π ( V ) = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A = θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ V ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ U λ V ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( U V ) λ σ ( U V ) ⊗ I H 0 ) θ A = f ( U V ) θ Ψ ∗ ( λ σ ( U V ) ⊗ I H 0 ) θ A = f ( U V ) π ( σ ( U V )) , ∀ U , V ∈ U and
π ( U ) π ( U ) ∗ = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ ( λ U ∗ ⊗ I H 0 ) θ Ψ = θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) ( λ U ∗ ⊗ I H 0 ) θ Ψ = I H , π ( U ) ∗ π ( U ) = θ A ∗ ( λ U ∗ ⊗ I H 0 ) θ Ψ θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A = θ A ∗ ( λ U ∗ ⊗ I H 0 ) ( λ U ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = I H , ∀ U ∈ U . \begin{aligned}
\pi(U)\pi(U)^* &= \theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A\theta_A^*(\lambda_U^*\otimes I_{\mathcal H_0})\theta_\Psi\\
&= \theta_\Psi^*\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal H_0})(\lambda_U^*\otimes I_{\mathcal H_0})\theta_\Psi= I_{\mathcal H},\\
\pi(U)^*\pi(U) &= \theta_A^*(\lambda_U^*\otimes I_{\mathcal H_0})\theta_\Psi\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A \\
&= \theta_A^*(\lambda_U^*\otimes I_{\mathcal H_0})(\lambda_U\otimes I_{\mathcal H_0})\theta_\Psi\theta_\Psi^*\theta_A = I_{\mathcal H}, \qquad\forall U\in\mathcal U.
\end{aligned} π ( U ) π ( U ) ∗ π ( U ) ∗ π ( U ) = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ ( λ U ∗ ⊗ I H 0 ) θ Ψ = θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) ( λ U ∗ ⊗ I H 0 ) θ Ψ = I H , = θ A ∗ ( λ U ∗ ⊗ I H 0 ) θ Ψ θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A = θ A ∗ ( λ U ∗ ⊗ I H 0 ) ( λ U ⊗ I H 0 ) θ Ψ θ Ψ ∗ θ A = I H , ∀ U ∈ U . Further,
π ( U ) f ( U − 1 ) π ( σ ( U − 1 ) ) = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A f ( U − 1 ) θ Ψ ∗ ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( λ U λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( f ( U σ ( U − 1 ) ) λ σ ( U σ ( U − 1 ) ) ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( U σ ( U − 1 ) I ) f ( U − 1 I ) λ σ ( U σ ( U − 1 ) I ) ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( σ ( U U − 1 ) I ) f ( U U − 1 ) λ σ ( σ ( U U − 1 ) I ) ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ I ⊗ I H 0 ) θ A = I H . \begin{aligned}
\pi(U)f(U^{-1})\pi(\sigma(U^{-1})) &= \theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A f(U^{-1})\theta_\Psi^*(\lambda_{\sigma(U^{-1})}\otimes I_{\mathcal H_0})\theta_A \\
&= f(U^{-1})\theta_\Psi^*\theta_A\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})(\lambda_{\sigma(U^{-1})}\otimes I_{\mathcal H_0})\theta_A \\
&= f(U^{-1})\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})(\lambda_{\sigma(U^{-1})}\otimes I_{\mathcal H_0})\theta_A \\
&= f(U^{-1})\theta_\Psi^*(\lambda_U\lambda_{\sigma(U^{-1})}\otimes I_{\mathcal H_0})\theta_A \\
&= f(U^{-1})\theta_\Psi^*(f(U\sigma(U^{-1}))\lambda_{\sigma(U\sigma(U^{-1}))}\otimes I_{\mathcal H_0})\theta_A \\
&= \theta_\Psi^*(f(U\sigma(U^{-1})I)f(U^{-1}I)\lambda_{\sigma(U\sigma(U^{-1})I)}\otimes I_{\mathcal H_0})\theta_A \\
&= \theta_\Psi^*(f(\sigma(UU^{-1})I)f(UU^{-1})\lambda_{\sigma(\sigma(UU^{-1})I)}\otimes I_{\mathcal H_0})\theta_A \\
&= \theta_\Psi^*(\lambda_I\otimes I_{\mathcal H_0})\theta_A = I_{\mathcal H}.
\end{aligned} π ( U ) f ( U − 1 ) π ( σ ( U − 1 )) = θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A f ( U − 1 ) θ Ψ ∗ ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( λ U ⊗ I H 0 ) ( λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( λ U λ σ ( U − 1 ) ⊗ I H 0 ) θ A = f ( U − 1 ) θ Ψ ∗ ( f ( U σ ( U − 1 )) λ σ ( U σ ( U − 1 )) ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( U σ ( U − 1 ) I ) f ( U − 1 I ) λ σ ( U σ ( U − 1 ) I ) ⊗ I H 0 ) θ A = θ Ψ ∗ ( f ( σ ( U U − 1 ) I ) f ( U U − 1 ) λ σ ( σ ( U U − 1 ) I ) ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ I ⊗ I H 0 ) θ A = I H . ⇒ π ( U ) − 1 = f ( U − 1 ) π ( σ ( U − 1 ) ) \Rightarrow\pi(U)^{-1}=f(U^{-1})\pi(\sigma(U^{-1})) ⇒ π ( U ) − 1 = f ( U − 1 ) π ( σ ( U − 1 )) U ∈ U U\in\mathcal U U ∈ U θ A ∗ \theta_A^* θ A ∗ π \pi π π \pi π π ( U ) = π ( V ) \pi(U)=\pi(V) π ( U ) = π ( V )
θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A ⇒ θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ = θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A θ A ∗ ⇒ θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) = θ Ψ ∗ θ A θ A ∗ ( λ V ⊗ I H 0 ) ⇒ λ U ⊗ I H 0 = λ V ⊗ I H 0 . \begin{aligned}
\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A=\theta_\Psi^*(\lambda_V\otimes I_{\mathcal H_0})\theta_A
\Rightarrow\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A\theta_A^*=\theta_\Psi^*(\lambda_V\otimes I_{\mathcal H_0})\theta_A\theta_A^* \\
&\Rightarrow\theta_\Psi^*\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal H_0})=\theta_\Psi^*\theta_A\theta_A^*(\lambda_V\otimes I_{\mathcal H_0})
\Rightarrow\lambda_U\otimes I_{\mathcal H_0}=\lambda_V\otimes I_{\mathcal H_0}.
\end{aligned} θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A = θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A ⇒ θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ = θ Ψ ∗ ( λ V ⊗ I H 0 ) θ A θ A ∗ ⇒ θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) = θ Ψ ∗ θ A θ A ∗ ( λ V ⊗ I H 0 ) ⇒ λ U ⊗ I H 0 = λ V ⊗ I H 0 . We show U U U V V V h ∈ ℓ 2 ( U ) , y ∈ H 0 h\in\ell^2(\mathcal U),y\in\mathcal H_0 h ∈ ℓ 2 ( U ) , y ∈ H 0
( λ U ⊗ I H 0 ) ( h ⊗ y ) = ( λ V ⊗ I H 0 ) ( h ⊗ y ) ⇒ λ U h ⊗ y = λ V h ⊗ y ⇒ ( λ U − λ V ) h ⊗ y = 0 ⇒ 0 = ⟨ ( λ U − λ V ) h ⊗ y , ( λ U − λ V ) h ⊗ y ⟩ = ∥ ( λ U − λ V ) h ∥ 2 ∥ y ∥ 2 (\lambda_U\otimes I_{\mathcal H_0})(h\otimes y)=(\lambda_V\otimes I_{\mathcal H_0})(h\otimes y)\Rightarrow\lambda_Uh\otimes y=\lambda_Vh\otimes y\Rightarrow(\lambda_U-\lambda_V)h\otimes y=0\Rightarrow0=\langle(\lambda_U-\lambda_V)h\otimes y,(\lambda_U-\lambda_V)h\otimes y\rangle=\|(\lambda_U-\lambda_V)h\|^2\|y\|^2 ( λ U ⊗ I H 0 ) ( h ⊗ y ) = ( λ V ⊗ I H 0 ) ( h ⊗ y ) ⇒ λ U h ⊗ y = λ V h ⊗ y ⇒ ( λ U − λ V ) h ⊗ y = 0 ⇒ 0 = ⟨( λ U − λ V ) h ⊗ y , ( λ U − λ V ) h ⊗ y ⟩ = ∥ ( λ U − λ V ) h ∥ 2 ∥ y ∥ 2 y ≠ 0 y\ne0 y = 0 y = 0 y=0 y = 0 h ⊗ y = 0 h\otimes y=0 h ⊗ y = 0 ( λ U − λ V ) ( h ) = 0 (\lambda_U-\lambda_V)(h)=0 ( λ U − λ V ) ( h ) = 0 λ \lambda λ U = V U=V U = V A U = A I π ( U ) − 1 A_U=A_I\pi(U)^{-1} A U = A I π ( U ) − 1 Ψ U = Ψ I π ( U ) − 1 \Psi_U=\Psi_I\pi(U)^{-1} Ψ U = Ψ I π ( U ) − 1 U ∈ U U\in\mathcal U U ∈ U
A I π ( U ) − 1 = L I ∗ θ A ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ ) ∗ = L I ∗ ( θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) ) ∗ = L I ∗ ( θ A ∗ ( λ U ⊗ I H 0 ) ) ∗ = ( θ A ∗ ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ A ∗ f ( U I ) ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ A ∗ L σ ( U I ) ) ∗ = L U ∗ θ A = A U \begin{aligned}
A_I\pi(U)^{-1}
&=L_I^*\theta_A\bigl(\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A\bigr)^*
=L_I^*\bigl(\theta_\Psi^*(\lambda_U\otimes I_{\mathcal H_0})\theta_A\theta_A^*\bigr)^*\\
&=L_I^*\bigl(\theta_\Psi^*\theta_A\theta_A^*(\lambda_U\otimes I_{\mathcal H_0})\bigr)^*
=L_I^*\bigl(\theta_A^*(\lambda_U\otimes I_{\mathcal H_0})\bigr)^*\\
&=\bigl(\theta_A^*(\lambda_U\otimes I_{\mathcal H_0})L_I\bigr)^*
=\bigl(\theta_A^*f(UI)(\lambda_U\otimes I_{\mathcal H_0})L_I\bigr)^* \\
&= (\theta_A^*L_{\sigma(UI)})^* = L_U^*\theta_A = A_U
\end{aligned} A I π ( U ) − 1 = L I ∗ θ A ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ A ∗ ) ∗ = L I ∗ ( θ Ψ ∗ θ A θ A ∗ ( λ U ⊗ I H 0 ) ) ∗ = L I ∗ ( θ A ∗ ( λ U ⊗ I H 0 ) ) ∗ = ( θ A ∗ ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ A ∗ f ( U I ) ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ A ∗ L σ ( U I ) ) ∗ = L U ∗ θ A = A U and
Ψ I π ( U ) − 1 = L I ∗ θ Ψ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ Ψ ) ∗ = L I ∗ ( θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) ) ∗ = ( θ Ψ ∗ ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ Ψ ∗ f ( U I ) ‾ ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ Ψ ∗ L σ ( U I ) ) ∗ = L U ∗ θ Ψ = Ψ U . \begin{aligned}
\Psi_I\pi(U)^{-1}
&= L_I^*\theta_\Psi(\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A)^*
= L_I^*(\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi)^*\\
&= L_I^*(\theta_\Psi^*\theta_A\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}))^*
= L_I^*(\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0}))^*\\
&= (\theta_\Psi^*(\lambda_U\otimes I_{\mathcal{H}_0})L_I)^*
= (\theta_\Psi^*\overline{f(UI)}(\lambda_U\otimes I_{\mathcal{H}_0})L_I)^*\\
&= (\theta_\Psi^*L_{\sigma(UI)})^*
= L_U^*\theta_\Psi
= \Psi_U.
\end{aligned} Ψ I π ( U ) − 1 = L I ∗ θ Ψ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) θ A θ Ψ ) ∗ = L I ∗ ( θ Ψ ∗ θ A θ Ψ ∗ ( λ U ⊗ I H 0 ) ) ∗ = L I ∗ ( θ Ψ ∗ ( λ U ⊗ I H 0 ) ) ∗ = ( θ Ψ ∗ ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ Ψ ∗ f ( U I ) ( λ U ⊗ I H 0 ) L I ) ∗ = ( θ Ψ ∗ L σ ( U I ) ) ∗ = L U ∗ θ Ψ = Ψ U . Note that neither Parsevalness of the frame nor θ A ∗ \theta_A^* θ A ∗ θ A \theta_A θ A θ A ( H ) ‾ = Ker ( θ A ∗ ) ⊥ \overline{\theta_A(\mathcal{H})}=\operatorname{Ker}(\theta_A^*)^\perp θ A ( H ) = Ker ( θ A ∗ ) ⊥ Ker ( θ A ∗ ) = θ A ( H ) ⊥ \operatorname{Ker}(\theta_A^*)=\theta_A(\mathcal{H})^\perp Ker ( θ A ∗ ) = θ A ( H ) ⊥ θ A \theta_A θ A θ A ( H ) = Ker ( θ A ∗ ) ⊥ \theta_A(\mathcal{H})=\operatorname{Ker}(\theta_A^*)^\perp θ A ( H ) = Ker ( θ A ∗ ) ⊥ θ A ∗ \theta_A^* θ A ∗ θ A \theta_A θ A
Corollary 5.3.10. Let U \mathcal{U} U I I I U \mathcal{U} U ( { A U } U ∈ U , { Ψ U } U ∈ U ) (\{A_U\}_{U\in\mathcal{U}},\{\Psi_U\}_{U\in\mathcal{U}}) ({ A U } U ∈ U , { Ψ U } U ∈ U ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) θ A ∗ \theta_A^* θ A ∗ π \pi π U \mathcal{U} U H \mathcal{H} H
(i) A U = A I S A , Ψ − 1 π ( U ) − 1 S A , Ψ , Ψ U = Ψ I π ( U ) − 1 A_U=A_IS_{A,\Psi}^{-1}\pi(U)^{-1}S_{A,\Psi},\Psi_U=\Psi_I\pi(U)^{-1} A U = A I S A , Ψ − 1 π ( U ) − 1 S A , Ψ , Ψ U = Ψ I π ( U ) − 1 U ∈ U U\in\mathcal{U} U ∈ U
A σ ( U V ) S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A W ∗ , A_{\sigma(UV)}S_{A,\Psi}^{-1}(S_{A,\Psi}^{-1})^*A_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}A_VS_{A,\Psi}^{-1}(S_{A,\Psi}^{-1})^*A_W^*, A σ ( U V ) S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A σ ( U W ) ∗ = f ( U V ) f ( U W ) A V S A , Ψ − 1 ( S A , Ψ − 1 ) ∗ A W ∗ , A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V S A , Ψ − 1 Ψ W ∗ , A_{\sigma(UV)}S_{A,\Psi}^{-1}\Psi_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}A_VS_{A,\Psi}^{-1}\Psi_W^*, A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) A V S A , Ψ − 1 Ψ W ∗ , Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . \Psi_{\sigma(UV)}\Psi_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}\Psi_V\Psi_W^*,
\qquad\forall U,V,W\in\mathcal{U}. Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . (ii) A U = A I π ( U ) − 1 , Ψ U = Ψ I ( S A , Ψ − 1 ) ∗ π ( U ) − 1 S A , Ψ A_U=A_I\pi(U)^{-1},\Psi_U=\Psi_I(S_{A,\Psi}^{-1})^*\pi(U)^{-1}S_{A,\Psi} A U = A I π ( U ) − 1 , Ψ U = Ψ I ( S A , Ψ − 1 ) ∗ π ( U ) − 1 S A , Ψ U ∈ U U\in\mathcal{U} U ∈ U
A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V A W ∗ , A_{\sigma(UV)}A_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}A_VA_W^*, A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) A V A W ∗ , A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V S A , Ψ − 1 Ψ W ∗ , A_{\sigma(UV)}S_{A,\Psi}^{-1}\Psi_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}A_VS_{A,\Psi}^{-1}\Psi_W^*, A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) A V S A , Ψ − 1 Ψ W ∗ , Ψ σ ( U V ) ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ Ψ V ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ W ∗ , ∀ U , V , W ∈ U . \Psi_{\sigma(UV)}(S_{A,\Psi}^{-1})^*S_{A,\Psi}^{-1}\Psi_{\sigma(UW)}^*
=f(UV)\overline{f(UW)}\Psi_V(S_{A,\Psi}^{-1})^*S_{A,\Psi}^{-1}\Psi_W^*,
\qquad\forall U,V,W\in\mathcal{U}. Ψ σ ( U V ) ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) Ψ V ( S A , Ψ − 1 ) ∗ S A , Ψ − 1 Ψ W ∗ , ∀ U , V , W ∈ U . Proof. (i) We apply Theorem 5.3.9 to the factorable Parseval OVF ( { A U S A , Ψ − 1 } U ∈ U , { Ψ U } U ∈ U ) (\{A_US_{A,\Psi}^{-1}\}_{U\in\mathcal{U}},\{\Psi_U\}_{U\in\mathcal{U}}) ({ A U S A , Ψ − 1 } U ∈ U , { Ψ U } U ∈ U ) π \pi π U \mathcal{U} U H \mathcal{H} H A U S A , Ψ − 1 = ( A I S A , Ψ − 1 ) π ( U ) − 1 , Ψ U = Ψ I π ( U ) − 1 A_US_{A,\Psi}^{-1}=(A_IS_{A,\Psi}^{-1})\pi(U)^{-1},\Psi_U=\Psi_I\pi(U)^{-1} A U S A , Ψ − 1 = ( A I S A , Ψ − 1 ) π ( U ) − 1 , Ψ U = Ψ I π ( U ) − 1 U ∈ U U\in\mathcal{U} U ∈ U
( A σ ( U V ) S A , Ψ − 1 ) ( A σ ( U W ) S A , Ψ − 1 ) ∗ = f ( U V ) f ( U W ) ‾ ( A V S A , Ψ − 1 ) ( A W S A , Ψ − 1 ) ∗ , (42) (A_{\sigma(UV)}S_{A,\Psi}^{-1})(A_{\sigma(UW)}S_{A,\Psi}^{-1})^*
=f(UV)\overline{f(UW)}(A_VS_{A,\Psi}^{-1})(A_WS_{A,\Psi}^{-1})^*, \tag*{(42)} ( A σ ( U V ) S A , Ψ − 1 ) ( A σ ( U W ) S A , Ψ − 1 ) ∗ = f ( U V ) f ( U W ) ( A V S A , Ψ − 1 ) ( A W S A , Ψ − 1 ) ∗ , (42) ( A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ ) = f ( U V ) f ( U W ) ‾ ( A V S A , Ψ − 1 Ψ W ∗ ) , (A_{\sigma(UV)}S_{A,\Psi}^{-1}\Psi_{\sigma(UW)}^{*})=f(UV)\overline{f(UW)}(A_VS_{A,\Psi}^{-1}\Psi_W^{*}), ( A σ ( U V ) S A , Ψ − 1 Ψ σ ( U W ) ∗ ) = f ( U V ) f ( U W ) ( A V S A , Ψ − 1 Ψ W ∗ ) , Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . \Psi_{\sigma(UV)}\Psi_{\sigma(UW)}^{*}=f(UV)\overline{f(UW)}\Psi_V\Psi_W^{*},\qquad\forall U,V,W\in\mathcal{U}. Ψ σ ( U V ) Ψ σ ( U W ) ∗ = f ( U V ) f ( U W ) Ψ V Ψ W ∗ , ∀ U , V , W ∈ U . (ii) We apply Theorem 5.3.9 to the factorable Parseval OVF ( { A U } U ∈ U , { Ψ U ( S A , Ψ − 1 ) ∗ } U ∈ U ) \left(\{A_U\}_{U\in\mathcal{U}},\{\Psi_U(S_{A,\Psi}^{-1})^{*}\}_{U\in\mathcal{U}}\right) ( { A U } U ∈ U , { Ψ U ( S A , Ψ − 1 ) ∗ } U ∈ U ) π \pi π U \mathcal{U} U H \mathcal{H} H A U = A I π ( U ) − 1 A_U=A_I\pi(U)^{-1} A U = A I π ( U ) − 1 Ψ U ( S A , Ψ − 1 ) ∗ = ( Ψ I ( S A , Ψ − 1 ) ∗ ) π ( U ) − 1 \Psi_U(S_{A,\Psi}^{-1})^{*}=(\Psi_I(S_{A,\Psi}^{-1})^{*})\pi(U)^{-1} Ψ U ( S A , Ψ − 1 ) ∗ = ( Ψ I ( S A , Ψ − 1 ) ∗ ) π ( U ) − 1 U ∈ U U\in\mathcal{U} U ∈ U
A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) ‾ A V A W ∗ , A_{\sigma(UV)}A_{\sigma(UW)}^{*}=f(UV)\overline{f(UW)}A_VA_W^{*}, A σ ( U V ) A σ ( U W ) ∗ = f ( U V ) f ( U W ) A V A W ∗ , A σ ( U V ) ( Ψ σ ( U W ) ( S A , Ψ − 1 ) ∗ ) ∗ = f ( U V ) f ( U W ) ‾ A V ( Ψ W ( S A , Ψ − 1 ) ∗ ) ∗ , A_{\sigma(UV)}\left(\Psi_{\sigma(UW)}(S_{A,\Psi}^{-1})^{*}\right)^{*}
=f(UV)\overline{f(UW)}A_V\left(\Psi_W(S_{A,\Psi}^{-1})^{*}\right)^{*}, A σ ( U V ) ( Ψ σ ( U W ) ( S A , Ψ − 1 ) ∗ ) ∗ = f ( U V ) f ( U W ) A V ( Ψ W ( S A , Ψ − 1 ) ∗ ) ∗ , ( Ψ σ ( U V ) ( S A , Ψ − 1 ) ∗ ) ( Ψ σ ( U W ) ( S A , Ψ − 1 ) ∗ ) ∗ = f ( U V ) f ( U W ) ‾ ( Ψ V ( S A , Ψ − 1 ) ∗ ) ( Ψ W ( S A , Ψ − 1 ) ∗ ) ∗ , \left(\Psi_{\sigma(UV)}(S_{A,\Psi}^{-1})^{*}\right)
\left(\Psi_{\sigma(UW)}(S_{A,\Psi}^{-1})^{*}\right)^{*}
=f(UV)\overline{f(UW)}
\left(\Psi_V(S_{A,\Psi}^{-1})^{*}\right)
\left(\Psi_W(S_{A,\Psi}^{-1})^{*}\right)^{*}, ( Ψ σ ( U V ) ( S A , Ψ − 1 ) ∗ ) ( Ψ σ ( U W ) ( S A , Ψ − 1 ) ∗ ) ∗ = f ( U V ) f ( U W ) ( Ψ V ( S A , Ψ − 1 ) ∗ ) ( Ψ W ( S A , Ψ − 1 ) ∗ ) ∗ , ∀ U , V , W ∈ U . \forall U,V,W\in\mathcal{U}. ∀ U , V , W ∈ U . PERTURBATIONS OF WEAK OPERATOR-VALUED FRAMES
In this section we derive stability results for factorable weak operator-valued frames.
Theorem 5.4.1. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) α , β , γ ≥ 0 \alpha,\beta,\gamma\ge0 α , β , γ ≥ 0 max { α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_\Psi(S_{A,\Psi}^{*})^{-1}\|,\beta\}<1 max { α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ , β } < 1 m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , …
∥ ∑ n = 1 m ( A n ∗ − B n ∗ ) L n ∗ y ∥ ≤ α ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ + β ∥ ∑ n = 1 m B n ∗ L n ∗ y ∥ + γ ( ∑ n = 1 m ∥ L n ∗ y ∥ 2 ) 1 2 , \left\|\sum_{n=1}^{m}(A_n^{*}-B_n^{*})L_n^{*}y\right\|
\leq\alpha\left\|\sum_{n=1}^{m}A_n^{*}L_n^{*}y\right\|
+\beta\left\|\sum_{n=1}^{m}B_n^{*}L_n^{*}y\right\|
+\gamma\left(\sum_{n=1}^{m}\|L_n^{*}y\|^{2}\right)^{\frac12}, n = 1 ∑ m ( A n ∗ − B n ∗ ) L n ∗ y ≤ α n = 1 ∑ m A n ∗ L n ∗ y + β n = 1 ∑ m B n ∗ L n ∗ y + γ ( n = 1 ∑ m ∥ L n ∗ y ∥ 2 ) 2 1 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 \forall y\in\ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 ∀ y ∈ ℓ 2 ( N ) ⊗ H 0
Then ( { B n } n , { Ψ n } n ) \left(\{B_n\}_n,\{\Psi_n\}_n\right) ( { B n } n , { Ψ n } n )
1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) ( 1 + β ) ∥ ( S A , Ψ ∗ ) − 1 ∥ and ∥ θ Ψ ∥ ( ( 1 + α ) ∥ θ A ∥ + γ ) 1 − β . \frac{1-(\alpha+\gamma\|\theta_\Psi(S_{A,\Psi}^{*})^{-1}\|)}
{(1+\beta)\|(S_{A,\Psi}^{*})^{-1}\|}
\qquad\text{and}\qquad
\frac{\|\theta_\Psi\|((1+\alpha)\|\theta_A\|+\gamma)}
{1-\beta}. ( 1 + β ) ∥ ( S A , Ψ ∗ ) − 1 ∥ 1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) and 1 − β ∥ θ Ψ ∥ (( 1 + α ) ∥ θ A ∥ + γ ) . Proof. For m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , … y y y ℓ 2 ( N ) ⊗ H 0 \ell^{2}(\mathbb{N})\otimes\mathcal{H}_0 ℓ 2 ( N ) ⊗ H 0
∥ ∑ n = 1 m B n ∗ L n ∗ y ∥ ≤ ∥ ∑ n = 1 m ( A n ∗ − B n ∗ ) L n ∗ y ∥ + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ ≤ ( 1 + α ) ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ + β ∥ ∑ n = 1 m B n ∗ L n ∗ y ∥ + γ ( ∑ n = 1 m ∥ L n ∗ y ∥ 2 ) 1 2 \begin{aligned}
\left\|\sum_{n=1}^{m}B_n^{*}L_n^{*}y\right\|
\leq
\left\|\sum_{n=1}^{m}(A_n^{*}-B_n^{*})L_n^{*}y\right\|
+
\left\|\sum_{n=1}^{m}A_n^{*}L_n^{*}y\right\| \\
&\leq(1+\alpha)\left\|\sum_{n=1}^{m}A_n^{*}L_n^{*}y\right\|
+\beta\left\|\sum_{n=1}^{m}B_n^{*}L_n^{*}y\right\|
+\gamma\left(\sum_{n=1}^{m}\|L_n^{*}y\|^{2}\right)^{\frac12}
\end{aligned} n = 1 ∑ m B n ∗ L n ∗ y ≤ n = 1 ∑ m ( A n ∗ − B n ∗ ) L n ∗ y + n = 1 ∑ m A n ∗ L n ∗ y ≤ ( 1 + α ) n = 1 ∑ m A n ∗ L n ∗ y + β n = 1 ∑ m B n ∗ L n ∗ y + γ ( n = 1 ∑ m ∥ L n ∗ y ∥ 2 ) 2 1 which implies
∥ ∑ n = 1 m B n ∗ L n ∗ y ∥ ≤ 1 + α 1 − β ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ + γ 1 − β ( ∑ n = 1 m ∥ L n ∗ y ∥ 2 ) 1 2 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . (43) \left\|\sum_{n=1}^{m} B_n^*L_n^*y\right\| \le\frac{1+\alpha}{1-\beta}\left\|\sum_{n=1}^{m} A_n^*L_n^*y\right\|+\frac{\gamma}{1-\beta}\left(\sum_{n=1}^{m}\left\|L_n^*y\right\|^2\right)^{\frac{1}{2}},\qquad\forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0. \tag*{(43)} n = 1 ∑ m B n ∗ L n ∗ y ≤ 1 − β 1 + α n = 1 ∑ m A n ∗ L n ∗ y + 1 − β γ ( n = 1 ∑ m ∥ L n ∗ y ∥ 2 ) 2 1 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . (43) Since
⟨ y , y ⟩ = ⟨ ( I ℓ 2 ( N ) ⊗ I H 0 ) y , y ⟩ = ⟨ ∑ n = 1 ∞ L n L n ∗ y , y ⟩ = ∑ n = 1 ∞ ∥ L n ∗ y ∥ 2 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 , \langle y,y\rangle=\langle(I_{\ell^2(\mathbb{N})}\otimes I_{\mathcal{H}_0})y,y\rangle=\left\langle\sum_{n=1}^{\infty}L_nL_n^*y,y\right\rangle=\sum_{n=1}^{\infty}\left\|L_n^*y\right\|^2,\qquad\forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0, ⟨ y , y ⟩ = ⟨( I ℓ 2 ( N ) ⊗ I H 0 ) y , y ⟩ = ⟨ n = 1 ∑ ∞ L n L n ∗ y , y ⟩ = n = 1 ∑ ∞ ∥ L n ∗ y ∥ 2 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 , Inequality (5.4.2) shows that ∑ n = 1 ∞ B n ∗ L n ∗ y \sum_{n=1}^{\infty}B_n^*L_n^*y ∑ n = 1 ∞ B n ∗ L n ∗ y y ∈ ℓ 2 ( N ) ⊗ H 0 y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 y ∈ ℓ 2 ( N ) ⊗ H 0
∥ ∑ n = 1 ∞ B n ∗ L n ∗ y ∥ ≤ 1 + α 1 − β ∥ ∑ n = 1 ∞ A n ∗ L n ∗ y ∥ + γ 1 − β ( ∑ n = 1 ∞ ∥ L n ∗ y ∥ 2 ) 1 2 = 1 + α 1 − β ∥ θ A ∗ y ∥ + γ 1 − β ∥ y ∥ , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . (44) \begin{aligned}
\left\|\sum_{n=1}^{\infty}B_n^*L_n^*y\right\|\le\frac{1+\alpha}{1-\beta}\left\|\sum_{n=1}^{\infty}A_n^*L_n^*y\right\|+\frac{\gamma}{1-\beta}\left(\sum_{n=1}^{\infty}\left\|L_n^*y\right\|^2\right)^{\frac{1}{2}} \\
&=\frac{1+\alpha}{1-\beta}\left\|\theta_A^*y\right\|+\frac{\gamma}{1-\beta}\|y\|,\qquad\forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0. \tag*{(44)}
\end{aligned} n = 1 ∑ ∞ B n ∗ L n ∗ y ≤ 1 − β 1 + α n = 1 ∑ ∞ A n ∗ L n ∗ y + 1 − β γ ( n = 1 ∑ ∞ ∥ L n ∗ y ∥ 2 ) 2 1 = 1 − β 1 + α ∥ θ A ∗ y ∥ + 1 − β γ ∥ y ∥ , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . (44) and this gives ∑ n = 1 ∞ B n ∗ L n ∗ \sum_{n=1}^{\infty}B_n^*L_n^* ∑ n = 1 ∞ B n ∗ L n ∗ θ B \theta_B θ B ∥ θ B ∗ y ∥ ≤ 1 + α 1 − β ∥ θ A ∗ y ∥ + γ 1 − β ∥ y ∥ \|\theta_B^*y\|\le\frac{1+\alpha}{1-\beta}\|\theta_A^*y\|+\frac{\gamma}{1-\beta}\|y\| ∥ θ B ∗ y ∥ ≤ 1 − β 1 + α ∥ θ A ∗ y ∥ + 1 − β γ ∥ y ∥ ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 \forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0 ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 ∥ θ B ∥ = ∥ θ B ∗ ∥ ≤ 1 + α 1 − β ∥ θ A ∗ ∥ + γ 1 − β = 1 + α 1 − β ∥ θ A ∥ + γ 1 − β \|\theta_B\|=\|\theta_B^*\|\le\frac{1+\alpha}{1-\beta}\|\theta_A^*\|+\frac{\gamma}{1-\beta}=\frac{1+\alpha}{1-\beta}\|\theta_A\|+\frac{\gamma}{1-\beta} ∥ θ B ∥ = ∥ θ B ∗ ∥ ≤ 1 − β 1 + α ∥ θ A ∗ ∥ + 1 − β γ = 1 − β 1 + α ∥ θ A ∥ + 1 − β γ S B , Ψ S_{B,\Psi} S B , Ψ
∥ θ A ∗ y − θ B ∗ y ∥ ≤ α ∥ θ A ∗ y ∥ + β ∥ θ B ∗ y ∥ + γ ∥ y ∥ , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 , \|\theta_A^*y-\theta_B^*y\|\le\alpha\|\theta_A^*y\|+\beta\|\theta_B^*y\|+\gamma\|y\|,\qquad\forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0, ∥ θ A ∗ y − θ B ∗ y ∥ ≤ α ∥ θ A ∗ y ∥ + β ∥ θ B ∗ y ∥ + γ ∥ y ∥ , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 , which implies
∥ θ A ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) − θ B ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) ∥ ≤ α ∥ θ A ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) ∥ + β ∥ θ B ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) ∥ + γ ∥ θ Ψ ( S A , Ψ ) − 1 h ∥ , ∀ h ∈ H . \begin{aligned}
\left\|\theta_A^*(\theta_\Psi(S_{A,\Psi})^{-1}h)-\theta_B^*(\theta_\Psi(S_{A,\Psi})^{-1}h)\right\|\le\alpha\left\|\theta_A^*(\theta_\Psi(S_{A,\Psi})^{-1}h)\right\| \\
&+\beta\left\|\theta_B^*(\theta_\Psi(S_{A,\Psi})^{-1}h)\right\|+\gamma\|\theta_\Psi(S_{A,\Psi})^{-1}h\|,\qquad\forall h\in\mathcal{H}.
\end{aligned} θ A ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) − θ B ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) ≤ α θ A ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) + β θ B ∗ ( θ Ψ ( S A , Ψ ) − 1 h ) + γ ∥ θ Ψ ( S A , Ψ ) − 1 h ∥ , ∀ h ∈ H . But θ A ∗ θ Ψ ( S A , Ψ ) − 1 = I H \theta_A^*\theta_\Psi(S_{A,\Psi})^{-1}=I_{\mathcal{H}} θ A ∗ θ Ψ ( S A , Ψ ) − 1 = I H θ B ∗ θ Ψ ( S A , Ψ ) − 1 = S B , Ψ ∗ ( S A , Ψ ) − 1 \theta_B^*\theta_\Psi(S_{A,\Psi})^{-1}=S_{B,\Psi}^*(S_{A,\Psi})^{-1} θ B ∗ θ Ψ ( S A , Ψ ) − 1 = S B , Ψ ∗ ( S A , Ψ ) − 1
∥ h − S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ ≤ α ∥ h ∥ + β ∥ S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ + γ ∥ θ Ψ ( S A , Ψ ) − 1 h ∥ ≤ ( α + γ ∥ θ Ψ ( S A , Ψ ) − 1 ∥ ) ∥ h ∥ + β ∥ S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ , ∀ h ∈ H . \begin{aligned}
\|h-S_{B,\Psi}^*(S_{A,\Psi})^{-1}h\|\le\alpha\|h\|+\beta\|S_{B,\Psi}^*(S_{A,\Psi})^{-1}h\|+\gamma\|\theta_\Psi(S_{A,\Psi})^{-1}h\| \\
&\le\left(\alpha+\gamma\|\theta_\Psi(S_{A,\Psi})^{-1}\|\right)\|h\|+\beta\|S_{B,\Psi}^*(S_{A,\Psi})^{-1}h\|,\qquad\forall h\in\mathcal{H}.
\end{aligned} ∥ h − S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ ≤ α ∥ h ∥ + β ∥ S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ + γ ∥ θ Ψ ( S A , Ψ ) − 1 h ∥ ≤ ( α + γ ∥ θ Ψ ( S A , Ψ ) − 1 ∥ ) ∥ h ∥ + β ∥ S B , Ψ ∗ ( S A , Ψ ) − 1 h ∥ , ∀ h ∈ H . Since max { α + γ ∥ θ Ψ ( S A , Ψ ) − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_\Psi(S_{A,\Psi})^{-1}\|,\beta\}<1 max { α + γ ∥ θ Ψ ( S A , Ψ ) − 1 ∥ , β } < 1 S B , Ψ ∗ ( S A , Ψ ) − 1 S_{B,\Psi}^*(S_{A,\Psi})^{-1} S B , Ψ ∗ ( S A , Ψ ) − 1 ∥ ( S B , Ψ ∗ ( S A , Ψ ∗ ) − 1 ) − 1 ∥ ≤ 1 + β 1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) \left\|\left(S_{B,\Psi}^{*}(S_{A,\Psi}^{*})^{-1}\right)^{-1}\right\| \leq\frac{1+\beta}{1-(\alpha+\gamma\|\theta_{\Psi}(S_{A,\Psi}^{*})^{-1}\|)} ( S B , Ψ ∗ ( S A , Ψ ∗ ) − 1 ) − 1 ≤ 1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) 1 + β
( S B , Ψ ∗ ( S A , Ψ ∗ ) − 1 ) S A , Ψ ∗ = S B , Ψ ∗ \left(S_{B,\Psi}^{*}(S_{A,\Psi}^{*})^{-1}\right)S_{A,\Psi}^{*}=S_{B,\Psi}^{*} ( S B , Ψ ∗ ( S A , Ψ ∗ ) − 1 ) S A , Ψ ∗ = S B , Ψ ∗ is invertible and
∥ S B , Ψ − 1 ∥ ≤ ∥ ( S A , Ψ ∗ ) − 1 ∥ ∥ S A , Ψ ∗ S B , Ψ − 1 ∥ ≤ ∥ ( S A , Ψ ∗ ) − 1 ∥ ( 1 + β ) 1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) . \|S_{B,\Psi}^{-1}\| \leq\|(S_{A,\Psi}^{*})^{-1}\|\|S_{A,\Psi}^{*}S_{B,\Psi}^{-1}\| \leq\frac{\|(S_{A,\Psi}^{*})^{-1}\|(1+\beta)}{1-(\alpha+\gamma\|\theta_{\Psi}(S_{A,\Psi}^{*})^{-1}\|)}. ∥ S B , Ψ − 1 ∥ ≤ ∥ ( S A , Ψ ∗ ) − 1 ∥∥ S A , Ψ ∗ S B , Ψ − 1 ∥ ≤ 1 − ( α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ) ∥ ( S A , Ψ ∗ ) − 1 ∥ ( 1 + β ) . Therefore ( { B n } n , { Ψ n } n ) \left(\{B_n\}_n,\{\Psi_n\}_n\right) ( { B n } n , { Ψ n } n )
∥ S B , Ψ ∥ ≤ ∥ θ Ψ ∥ ∥ θ B ∥ ≤ ∥ θ Ψ ∥ ( ( 1 + α ) ∥ θ A ∥ + γ ) 1 − β \|S_{B,\Psi}\| \leq\|\theta_{\Psi}\|\|\theta_B\| \leq\frac{\|\theta_{\Psi}\|((1+\alpha)\|\theta_A\|+\gamma)}{1-\beta} ∥ S B , Ψ ∥ ≤ ∥ θ Ψ ∥∥ θ B ∥ ≤ 1 − β ∥ θ Ψ ∥ (( 1 + α ) ∥ θ A ∥ + γ ) and ∥ S B , Ψ − 1 ∥ − 1 \|S_{B,\Psi}^{-1}\|^{-1} ∥ S B , Ψ − 1 ∥ − 1 ∥ S B , Ψ ∥ \|S_{B,\Psi}\| ∥ S B , Ψ ∥ ( { B n } n , { Ψ n } n ) \left(\{B_n\}_n,\{\Psi_n\}_n\right) ( { B n } n , { Ψ n } n )
Corollary 5.4.2. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 )
r : = ∑ n = 1 ∞ ∥ A n − B n ∥ 2 < 1 ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ 2 . r:=\sum_{n=1}^{\infty}\|A_n-B_n\|^2<\frac{1}{\|\theta_{\Psi}(S_{A,\Psi}^{*})^{-1}\|^2}. r := n = 1 ∑ ∞ ∥ A n − B n ∥ 2 < ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ 2 1 . Then ( { B n } n , { Ψ n } n ) \left(\{B_n\}_n,\{\Psi_n\}_n\right) ( { B n } n , { Ψ n } n )
1 − r ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ ∥ ( S A , Ψ ∗ ) − 1 ∥ \frac{1-\sqrt{r}\|\theta_{\Psi}(S_{A,\Psi}^{*})^{-1}\|}{\|(S_{A,\Psi}^{*})^{-1}\|} ∥ ( S A , Ψ ∗ ) − 1 ∥ 1 − r ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ and ∥ θ Ψ ∥ ( ∥ θ A ∥ + r ) \|\theta_{\Psi}\|(\|\theta_A\|+\sqrt{r}) ∥ θ Ψ ∥ ( ∥ θ A ∥ + r )
Proof. We apply Theorem 5.4.1 by taking α = 0 , β = 0 , γ = r \alpha=0,\beta=0,\gamma=\sqrt{r} α = 0 , β = 0 , γ = r max { α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ , β } < 1 \max\{\alpha+\gamma\|\theta_{\Psi}(S_{A,\Psi}^{*})^{-1}\|,\beta\}<1 max { α + γ ∥ θ Ψ ( S A , Ψ ∗ ) − 1 ∥ , β } < 1 m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , …
∥ ∑ n = 1 m ( A n ∗ − B n ∗ ) L n y ∥ ≤ ( ∑ n = 1 m ∥ A n ∗ − B n ∗ ∥ 2 ) 1 2 ( ∑ n = 1 m ∥ L n y ∥ 2 ) 1 2 \left\|\sum_{n=1}^{m}(A_n^{*}-B_n^{*})L_n y\right\|\leq\left(\sum_{n=1}^{m}\|A_n^{*}-B_n^{*}\|^2\right)^{\frac{1}{2}}\left(\sum_{n=1}^{m}\|L_n y\|^2\right)^{\frac{1}{2}} n = 1 ∑ m ( A n ∗ − B n ∗ ) L n y ≤ ( n = 1 ∑ m ∥ A n ∗ − B n ∗ ∥ 2 ) 2 1 ( n = 1 ∑ m ∥ L n y ∥ 2 ) 2 1 ≤ γ ( ∑ n = 1 m ∥ L n y ∥ 2 ) 1 2 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . \leq\gamma\left(\sum_{n=1}^{m}\|L_n y\|^2\right)^{\frac{1}{2}},\qquad\forall y\in\ell^2(\mathbb{N})\otimes\mathcal{H}_0. ≤ γ ( n = 1 ∑ m ∥ L n y ∥ 2 ) 2 1 , ∀ y ∈ ℓ 2 ( N ) ⊗ H 0 . We next derive another stability result with different condition.
Theorem 5.4.3. Let ( { A n } n , { Ψ n } n ) \left(\{A_n\}_n,\{\Psi_n\}_n\right) ( { A n } n , { Ψ n } n ) B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) { B n } n \{B_n\}_n { B n } n B ( H , H 0 ) \mathcal{B}(\mathcal{H},\mathcal{H}_0) B ( H , H 0 ) ∑ n = 1 ∞ ∥ A n − B n ∥ 2 \sum_{n=1}^{\infty}\|A_n-B_n\|^2 ∑ n = 1 ∞ ∥ A n − B n ∥ 2 ∑ n = 1 ∞ ∥ A n − \sum_{n=1}^{\infty}\|A_n- ∑ n = 1 ∞ ∥ A n −
B n B_n B n ∥ B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ < 1 \lVert B_n\rVert\lVert\Psi_n(S^*_{A,\Psi})^{-1}\rVert< 1 ∥ B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ < 1 { B n } n , { Ψ n } n \{B_n\}_n,\{\Psi_n\}_n { B n } n , { Ψ n } n
1 − ∑ n = 1 ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ ∥ ( S A , Ψ ∗ ) − 1 ∥ and ∥ θ Ψ ∥ ( ( ∑ n = 1 ∞ ∥ A n − B n ∥ 2 ) 1 / 2 + ∥ θ A ∥ ) . \frac{1-\sum_{n=1}^{\infty}\lVert A_n-B_n\rVert\lVert\Psi_n(S^*_{A,\Psi})^{-1}\rVert}{\lVert(S^*_{A,\Psi})^{-1}\rVert}
\qquad\text{and}\qquad
\lVert\theta_\Psi\rVert\left(\left(\sum_{n=1}^{\infty}\lVert A_n-B_n\rVert^2\right)^{1/2}+\lVert\theta_A\rVert\right). ∥( S A , Ψ ∗ ) − 1 ∥ 1 − ∑ n = 1 ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ and ∥ θ Ψ ∥ ( n = 1 ∑ ∞ ∥ A n − B n ∥ 2 ) 1/2 + ∥ θ A ∥ . Proof. Let α = ∑ n = 1 ∞ ∥ A n − B n ∥ 2 \alpha=\sum_{n=1}^{\infty}\lVert A_n-B_n\rVert^2 α = ∑ n = 1 ∞ ∥ A n − B n ∥ 2 β = ∑ n = 1 ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ \beta=\sum_{n=1}^{\infty}\lVert A_n-B_n\rVert\lVert\Psi_n(S^*_{A,\Psi})^{-1}\rVert β = ∑ n = 1 ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ m = 1 , 2 , … m=1,2,\ldots m = 1 , 2 , … y y y ℓ 2 ( N ) ⊗ H 0 \ell^2(\mathbb{N})\otimes\mathcal{H}_0 ℓ 2 ( N ) ⊗ H 0
∥ ∑ n = 1 m B n ∗ L n ∗ y ∥ ≤ ∥ ∑ n = 1 m ( A n ∗ − B n ∗ ) L n ∗ y ∥ + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ ≤ ∑ n = 1 m ∥ A n − B n ∥ ∥ L n ∗ y ∥ + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ ≤ ( ∑ n = 1 m ∥ A n − B n ∥ 2 ) 1 / 2 ( ∑ n = 1 m ∥ L n ∗ y ∥ 2 ) 1 / 2 + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ ≤ α 1 / 2 ( ∑ n = 1 m ∥ L n ∗ y ∥ 2 ) 1 / 2 + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ = α 1 / 2 ⟨ ∑ n = 1 m L n L n ∗ y , y ⟩ 1 / 2 + ∥ ∑ n = 1 m A n ∗ L n ∗ y ∥ , \begin{aligned}
\left\lVert\sum_{n=1}^{m}B_n^*L_n^*y\right\rVert
\leq
\left\lVert\sum_{n=1}^{m}(A_n^*-B_n^*)L_n^*y\right\rVert+
\left\lVert\sum_{n=1}^{m}A_n^*L_n^*y\right\rVert\\
&\leq\sum_{n=1}^{m}\lVert A_n-B_n\rVert\lVert L_n^*y\rVert+
\left\lVert\sum_{n=1}^{m}A_n^*L_n^*y\right\rVert\\
&\leq\left(\sum_{n=1}^{m}\lVert A_n-B_n\rVert^2\right)^{1/2}
\left(\sum_{n=1}^{m}\lVert L_n^*y\rVert^2\right)^{1/2}
+\left\lVert\sum_{n=1}^{m}A_n^*L_n^*y\right\rVert\\
&\leq\alpha^{1/2}\left(\sum_{n=1}^{m}\lVert L_n^*y\rVert^2\right)^{1/2}
+\left\lVert\sum_{n=1}^{m}A_n^*L_n^*y\right\rVert\\
&=\alpha^{1/2}\left\langle\sum_{n=1}^{m}L_nL_n^*y,y\right\rangle^{1/2}
+\left\lVert\sum_{n=1}^{m}A_n^*L_n^*y\right\rVert,
\end{aligned} n = 1 ∑ m B n ∗ L n ∗ y ≤ n = 1 ∑ m ( A n ∗ − B n ∗ ) L n ∗ y + n = 1 ∑ m A n ∗ L n ∗ y ≤ n = 1 ∑ m ∥ A n − B n ∥ ∥ L n ∗ y ∥ + n = 1 ∑ m A n ∗ L n ∗ y ≤ ( n = 1 ∑ m ∥ A n − B n ∥ 2 ) 1/2 ( n = 1 ∑ m ∥ L n ∗ y ∥ 2 ) 1/2 + n = 1 ∑ m A n ∗ L n ∗ y ≤ α 1/2 ( n = 1 ∑ m ∥ L n ∗ y ∥ 2 ) 1/2 + n = 1 ∑ m A n ∗ L n ∗ y = α 1/2 ⟨ n = 1 ∑ m L n L n ∗ y , y ⟩ 1/2 + n = 1 ∑ m A n ∗ L n ∗ y , which converges to α ∥ y ∥ + ∥ θ A ∗ y ∥ \sqrt{\alpha}\lVert y\rVert+\lVert\theta_A^*y\rVert α ∥ y ∥ + ∥ θ A ∗ y ∥ θ B \theta_B θ B ∥ θ B ∥ ≤ α + ∥ θ A ∥ \lVert\theta_B\rVert\leq\sqrt{\alpha}+\lVert\theta_A\rVert ∥ θ B ∥ ≤ α + ∥ θ A ∥ S B , Ψ = θ Ψ ∗ θ B = ∑ n = 1 ∞ Ψ n ∗ B n S_{B,\Psi}=\theta_\Psi^*\theta_B=\sum_{n=1}^{\infty}\Psi_n^*B_n S B , Ψ = θ Ψ ∗ θ B = ∑ n = 1 ∞ Ψ n ∗ B n
∥ I H − S B , Ψ ( S A , Ψ ∗ ) − 1 ∥ = ∥ ∑ n = 1 ∞ A n ∗ Ψ n ( S A , Ψ ∗ ) − 1 − ∑ n = 1 ∞ B n ∗ Ψ n ( S A , Ψ ∗ ) − 1 ∥ = ∥ ∑ n = 1 ∞ ( A n ∗ − B n ∗ ) Ψ n ( S A , Ψ ∗ ) − 1 ∥ ≤ ∑ n = 1 ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ = β < 1. \begin{aligned}
\lVert I_{\mathcal H}-S_{B,\Psi}(S^*_{A,\Psi})^{-1}\rVert
&=\left\lVert\sum_{n=1}^{\infty}A_n^*\Psi_n(S^*_{A,\Psi})^{-1}
-\sum_{n=1}^{\infty}B_n^*\Psi_n(S^*_{A,\Psi})^{-1}\right\rVert\\
&=\left\lVert\sum_{n=1}^{\infty}(A_n^*-B_n^*)\Psi_n(S^*_{A,\Psi})^{-1}\right\rVert\\
&\leq\sum_{n=1}^{\infty}\lVert A_n-B_n\rVert\lVert\Psi_n(S^*_{A,\Psi})^{-1}\rVert=\beta<1.
\end{aligned} ∥ I H − S B , Ψ ( S A , Ψ ∗ ) − 1 ∥ = n = 1 ∑ ∞ A n ∗ Ψ n ( S A , Ψ ∗ ) − 1 − n = 1 ∑ ∞ B n ∗ Ψ n ( S A , Ψ ∗ ) − 1 = n = 1 ∑ ∞ ( A n ∗ − B n ∗ ) Ψ n ( S A , Ψ ∗ ) − 1 ≤ n = 1 ∑ ∞ ∥ A n − B n ∥ ∥ Ψ n ( S A , Ψ ∗ ) − 1 ∥ = β < 1. Therefore S B , Ψ ( S A , Ψ ∗ ) − 1 S_{B,\Psi}(S^*_{A,\Psi})^{-1} S B , Ψ ( S A , Ψ ∗ ) − 1 ∥ ( S B , Ψ ( S A , Ψ ∗ ) − 1 ) − 1 ∥ ≤ 1 / ( 1 − β ) \lVert(S_{B,\Psi}(S^*_{A,\Psi})^{-1})^{-1}\rVert\leq1/(1-\beta) ∥( S B , Ψ ( S A , Ψ ∗ ) − 1 ) − 1 ∥ ≤ 1/ ( 1 − β )
CHAPTER 6
CONCLUSION AND FUTURE WORK
In Chapter 2 we initiated the study of frames for metric spaces. Since metric spaces are more general objects and have less structure than Banach spaces, study of frames for metric spaces goes in a different way than that of frames for Hilbert as well as for Banach spaces. Arens-Eells space is used as a tool which allows to use the functional analysis technique to Lipschitz functions. However, this works good only when the codomain of Lipschitz functions is a Banach space. Most of the results in Chapter 2 are concentrated whenever the codomain of Lipschitz function is a Banach space. In future we are interested to work on frames for arbitrary metric spaces.
In Chapter 3 we defined multipliers for metric spaces. We obtained some fundamental properties of multipliers. One of the future work is to explore further on multipliers in metric spaces.
In Chapter 4 we studied a special class of approximate Schauder frames. We characterized a class of approximate Schauder frames and its duals. It is planned to obtain the description of frames and its duals for Banach spaces.
In Chapter 5 we initiated the study of the series ∑ n = 1 ∞ Ψ n ∗ A n \sum_{n=1}^{\infty}\Psi_n^{*}A_n ∑ n = 1 ∞ Ψ n ∗ A n
Dilations of Linear Maps on Vector Spaces
DILATIONS OF FUNCTIONS ON SETS
One of the most useful results in the study of isometries on Hilbert spaces is the Wold decomposition. It describes the structure of an isometry. It uses the notion of a shift.
Definition 6.1.1. (cf. Sz.-Nagy et al. (2010)) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H shift if ⋂ n = 0 ∞ T n ( H ) = { 0 } \bigcap_{n=0}^{\infty}T^n(\mathcal{H})=\{0\} ⋂ n = 0 ∞ T n ( H ) = { 0 }
Theorem 6.1.2. (cf. Sz.-Nagy et al. (2010); Wold (1954)) (Wold decomposition ) Let T T T H \mathcal{H} H H \mathcal{H} H H = H u ⊕ H s \mathcal{H}=\mathcal{H}_u\oplus\mathcal{H}_s H = H u ⊕ H s H u \mathcal{H}_u H u H s \mathcal{H}_s H s T T T H \mathcal{H} H T ∣ H u : H u → H u T|_{\mathcal{H}_u}:\mathcal{H}_u\to\mathcal{H}_u T ∣ H u : H u → H u T ∣ H s : H s → H s T|_{\mathcal{H}_s}:\mathcal{H}_s\to\mathcal{H}_s T ∣ H s : H s → H s
Using functional calculus and Weierstrass polynomial approximation theorem, Halmos in 1950 proved an important result that every contraction on a Hilbert space can be lifted to unitary.
Theorem 6.1.3. (Halmos [125 ] ) (Halmos dilation ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H
U : = ( T I − T T ∗ I − T ∗ T − T ∗ ) U:=\begin{pmatrix} T & \sqrt{I-TT^*} \\ \sqrt{I-T^*T} & -T^* \end{pmatrix} U := ( T I − T ∗ T I − T T ∗ − T ∗ ) is unitary on H ⊕ H \mathcal{H}\oplus\mathcal{H} H ⊕ H
T = P H U ∣ H T=P_{\mathcal{H}}U|_{\mathcal{H}} T = P H U ∣ H where P H : H ⊕ H → H ⊕ H P_{\mathcal{H}}:\mathcal{H}\oplus\mathcal{H}\to\mathcal{H}\oplus\mathcal{H} P H : H ⊕ H → H ⊕ H H \mathcal{H} H
Three years later, Sz. Nagy extended the result of Halmos which reads as follows.
Theorem 6.1.4. (Sz.-Nagy (1953)) (Sz. Nagy dilation ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H K \mathcal{K} K H \mathcal{H} H U : K → K U:\mathcal{K}\to\mathcal{K} U : K → K
T n = P H U n ∣ H , ∀ n = 1 , 2 , … , T^n=P_{\mathcal{H}}U^n|_{\mathcal{H}},\qquad\forall n=1,2,\ldots, T n = P H U n ∣ H , ∀ n = 1 , 2 , … , where P H : K → K P_{\mathcal{H}}:\mathcal{K}\to\mathcal{K} P H : K → K H \mathcal{H} H
Unitary operator U U U K \mathcal{K} K
K = span ‾ { U n h ; n ∈ Z + , h ∈ H } , \mathcal{K}=\overline{\operatorname{span}}\{U^n h_{\,;\,n\in\mathbb{Z}_{+},h\in\mathcal{H}\},} K = span { U n h ; n ∈ Z + , h ∈ H } , then ( K , U ) (\mathcal{K},U) ( K , U ) K \mathcal{K} K ⨁ n = − ∞ ∞ H \bigoplus_{n=-\infty}^{\infty}\mathcal{H} ⨁ n = − ∞ ∞ H
⨁ n = − ∞ ∞ H : = { { h n } n = − ∞ ∞ , h n ∈ H , ∀ n ∈ Z , ∑ n = − ∞ ∞ ∥ h n ∥ 2 < ∞ } \bigoplus_{n=-\infty}^{\infty}\mathcal{H}:=\left\{\{h_n\}_{n=-\infty}^{\infty},h_n\in\mathcal{H},\ \forall n\in\mathbb{Z},\ \sum_{n=-\infty}^{\infty}\lVert h_n\rVert^2<\infty\right\} n = − ∞ ⨁ ∞ H := { { h n } n = − ∞ ∞ , h n ∈ H , ∀ n ∈ Z , n = − ∞ ∑ ∞ ∥ h n ∥ 2 < ∞ } with respect to the inner product
⟨ { h n } n = − ∞ ∞ , { g n } n = − ∞ ∞ ⟩ : = ∑ n = − ∞ ∞ ⟨ h n , g n ⟩ , ∀ { h n } n = − ∞ ∞ , { g n } n = − ∞ ∞ ∈ ⨁ n = − ∞ ∞ H . \left\langle\{h_n\}_{n=-\infty}^{\infty},\{g_n\}_{n=-\infty}^{\infty}\right\rangle:=\sum_{n=-\infty}^{\infty}\langle h_n,g_n\rangle,\quad\forall\{h_n\}_{n=-\infty}^{\infty},\{g_n\}_{n=-\infty}^{\infty}\in\bigoplus_{n=-\infty}^{\infty}\mathcal{H}. ⟨ { h n } n = − ∞ ∞ , { g n } n = − ∞ ∞ ⟩ := n = − ∞ ∑ ∞ ⟨ h n , g n ⟩ , ∀ { h n } n = − ∞ ∞ , { g n } n = − ∞ ∞ ∈ n = − ∞ ⨁ ∞ H . Theorem 6.1.5. (Schaffer (1955) ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H U : = [ u n , m ] − ∞ < n , m < ∞ U:=[u_{n,m}]_{-\infty<n,m<\infty} U := [ u n , m ] − ∞ < n , m < ∞ Schaffer operator defined on ⨁ n = − ∞ ∞ H \bigoplus_{n=-\infty}^{\infty}\mathcal{H} ⨁ n = − ∞ ∞ H infinite matrix defined as follows :
u 0 , 0 : = T , u 0 , 1 : = I − T T ∗ , u − 1 , 0 : = I − T ∗ T , u_{0,0}:=T,\quad u_{0,1}:=\sqrt{I-TT^{*}},\quad u_{-1,0}:=\sqrt{I-T^{*}T}, u 0 , 0 := T , u 0 , 1 := I − T T ∗ , u − 1 , 0 := I − T ∗ T , u − 1 , 1 : = − T ∗ , u n , n + 1 : = I , ∀ n ∈ Z , n ≠ 0 , 1 , u n , m : = 0 , otherwise , u_{-1,1}:=-T^{*},\quad u_{n,n+1}:=I,\ \forall n\in\mathbb{Z},n\ne0,1,\quad u_{n,m}:=0,\quad\text{otherwise}, u − 1 , 1 := − T ∗ , u n , n + 1 := I , ∀ n ∈ Z , n = 0 , 1 , u n , m := 0 , otherwise , i.e.,
U = ( ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ 0 I 0 0 ⋯ ⋯ 0 0 I − T ∗ T − T ∗ 0 ⋯ ⋯ 0 0 T I − T T ∗ 0 ⋯ ⋯ 0 0 0 0 I ⋯ ⋯ 0 0 0 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ) ∞ × ∞ U=\left(
\begin{array}{ccccc}
\vdots& \vdots& \vdots& \vdots& \vdots\\
\cdots& 0 & I & 0 & 0 & \cdots\\
\cdots& 0 & 0 & \sqrt{I-T^{*}T} & -T^{*} & 0 & \cdots\\
\cdots& 0 & 0 & \boxed{T} & \sqrt{I-TT^{*}} & 0 & \cdots\\
\cdots& 0 & 0 & 0 & 0 & I & \cdots\\
\cdots& 0 & 0 & 0 & 0 & 0 & \cdots\\
\vdots& \vdots& \vdots& \vdots& \vdots
\end{array}
\right)_{\infty\times\infty} U = ⋮ ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⋮ 0 0 0 0 0 ⋮ ⋮ I 0 0 0 0 ⋮ ⋮ 0 I − T ∗ T T 0 0 ⋮ ⋮ 0 − T ∗ I − T T ∗ 0 0 ⋮ ⋯ 0 0 I 0 ⋯ ⋯ ⋯ ⋯ ∞ × ∞ where T T T ( 0 , 0 ) (0,0) ( 0 , 0 ) ⨁ n = − ∞ ∞ H \bigoplus_{n=-\infty}^{\infty}\mathcal{H} ⨁ n = − ∞ ∞ H
T n = P H U n ∣ H , ∀ n ∈ N , T^n=P_{\mathcal{H}}U^n|_{\mathcal{H}},\quad\forall n\in\mathbb{N}, T n = P H U n ∣ H , ∀ n ∈ N , where P H : ⨁ n = − ∞ ∞ H → ⨁ n = − ∞ ∞ H P_{\mathcal{H}}:\bigoplus_{n=-\infty}^{\infty}\mathcal{H}\to\bigoplus_{n=-\infty}^{\infty}\mathcal{H} P H : ⨁ n = − ∞ ∞ H → ⨁ n = − ∞ ∞ H H \mathcal{H} H
After a year of work of Sz. Nagy, it was Egervary who observed that Halmos dilation of contraction can be extended finitely so that power of dilation will be dilation of power of contraction.
Theorem 6.1.6. [92 ] (N-dilation ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H N N N
U : = ( T 0 0 ⋯ 0 I − T T ∗ I − T ∗ T 0 0 ⋯ 0 − T ∗ 0 I 0 ⋯ 0 0 0 0 I ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 0 0 0 0 ⋯ I 0 ) ( N + 1 ) × ( N + 1 ) U := \begin{pmatrix}
T & 0 & 0 & \cdots& 0 & \sqrt{I-TT^{*}} \\
\sqrt{I-T^{*}T} & 0 & 0 & \cdots& 0 & -T^{*} \\
0 & I & 0 & \cdots& 0 & 0 \\
0 & 0 & I & \cdots& 0 & 0 \\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & 0 \\
0 & 0 & 0 & \cdots& I & 0
\end{pmatrix}_{(N+1)\times(N+1)} U := T I − T ∗ T 0 0 ⋮ 0 0 0 0 I 0 ⋮ 0 0 0 0 0 I ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 0 ⋮ 0 I I − T T ∗ − T ∗ 0 0 ⋮ 0 0 ( N + 1 ) × ( N + 1 ) is unitary on ⨁ k = 1 N + 1 H \bigoplus_{k=1}^{N+1}\mathcal{H} ⨁ k = 1 N + 1 H
T k = P H U k ∣ H , ∀ k = 1 , … , N , T^{k}=P_{\mathcal{H}}U^{k}|_{\mathcal{H}},\quad\forall k=1,\ldots,N, T k = P H U k ∣ H , ∀ k = 1 , … , N , where P H : ⨁ k = 1 N + 1 H → ⨁ k = 1 N + 1 H P_{\mathcal{H}}:\bigoplus_{k=1}^{N+1}\mathcal{H}\to\bigoplus_{k=1}^{N+1}\mathcal{H} P H : ⨁ k = 1 N + 1 H → ⨁ k = 1 N + 1 H H \mathcal{H} H
A very useful result which can be derived using Theorem 6.1.6 is the von Neumann’s inequality. It was derived by von Neumann (1951) using the theory of analytic functions.
Theorem 6.1.7. (von Neumann inequality ) (Rainone (2007); Shalit (2021); von Neumann (1951)) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H p ∈ C [ z ] p\in\mathbb{C}[z] p ∈ C [ z ]
∥ p ( T ) ∥ ≤ sup ∣ z ∣ = 1 ∣ p ( z ) ∣ . \lVert p(T)\rVert\leq\sup_{|z|=1}|p(z)|. ∥ p ( T )∥ ≤ ∣ z ∣ = 1 sup ∣ p ( z ) ∣. Sz. Nagy’s dilation theorem leads to the study of dilating more than one operator which are commuting. After a decade of work of Sz. Nagy, Ando derived the following result.
Theorem 6.1.8. [15 ] (Ando dilation ) Let H \mathcal{H} H T 1 , T 2 : H → H T_{1},T_{2}:\mathcal{H}\to\mathcal{H} T 1 , T 2 : H → H K \mathcal{K} K
tains H \mathcal{H} H U 1 , U 2 : K → K U_1,U_2:\mathcal{K}\to\mathcal{K} U 1 , U 2 : K → K
T 1 n T 2 m = P H U 1 n U 2 m ∣ H , ∀ n , m = 1 , 2 , … , T_1^nT_2^m=P_{\mathcal{H}}U_1^nU_2^m\big|_{\mathcal{H}},\qquad\forall n,m=1,2,\ldots, T 1 n T 2 m = P H U 1 n U 2 m H , ∀ n , m = 1 , 2 , … , where P H : K → K P_{\mathcal{H}}:\mathcal{K}\to\mathcal{K} P H : K → K H \mathcal{H} H
An easy consequence of Ando dilation is generalization of von Neumann inequality.
Theorem 6.1.9. (Ando-von Neumann inequality ) [cf. 15 , 38 ] Let H \mathcal{H} H T 1 , T 2 : H → H T_1,T_2:\mathcal{H}\to\mathcal{H} T 1 , T 2 : H → H p ∈ C [ z , w ] p\in\mathbb{C}[z,w] p ∈ C [ z , w ]
∥ p ( T 1 , T 2 ) ∥ ≤ sup ∣ z ∣ = ∣ w ∣ = 1 ∣ p ( z , w ) ∣ . \lVert p(T_1,T_2)\rVert\leq\sup_{\lvert z\rvert=\lvert w\rvert=1}\lvert p(z,w)\rvert. ∥ p ( T 1 , T 2 )∥ ≤ ∣ z ∣ = ∣ w ∣ = 1 sup ∣ p ( z , w )∣ . It is known that Ando dilation theorem can not be extended for more than two commuting contractions [cf. 38 , 72 , 87 ] (); Parrott 1970; ); Varopoulos 1974). We next consider inter-twining lifting theorem. This says that any operator which intertwins contractions can be lifted so that the lifted operator intertwins dilation operator.
Theorem 6.1.10. (Sz.-Nagy and Foias (1971) ) (Inter-twining lifting theorem ) Let T 1 : H 1 → H 1 T_1:\mathcal{H}_1\to\mathcal{H}_1 T 1 : H 1 → H 1 T 2 : H 2 → H 2 T_2:\mathcal{H}_2\to\mathcal{H}_2 T 2 : H 2 → H 2 H 1 , H 2 \mathcal{H}_1,\mathcal{H}_2 H 1 , H 2 V 1 : K 1 → K 1 V_1:\mathcal{K}_1\to\mathcal{K}_1 V 1 : K 1 → K 1 V 2 : K 2 → K 2 V_2:\mathcal{K}_2\to\mathcal{K}_2 V 2 : K 2 → K 2 T 1 , T 2 T_1,T_2 T 1 , T 2 S : H 2 → H 1 S:\mathcal{H}_2\to\mathcal{H}_1 S : H 2 → H 1 T 1 S = S T 2 T_1S=ST_2 T 1 S = S T 2 R : K 2 → K 1 R:\mathcal{K}_2\to\mathcal{K}_1 R : K 2 → K 1
V 1 R = R V 2 , P H 1 R H 2 ⊥ = 0 , P H 1 R H 2 = S , ∥ R ∥ = ∥ S ∥ . V_1R=RV_2,\qquad P_{\mathcal{H}_1}R\mathcal{H}_2^\perp=0,\qquad P_{\mathcal{H}_1}R\mathcal{H}_2=S,\qquad\lVert R\rVert=\lVert S\rVert. V 1 R = R V 2 , P H 1 R H 2 ⊥ = 0 , P H 1 R H 2 = S , ∥ R ∥ = ∥ S ∥ . Conversely if R : K 2 → K 1 R:\mathcal{K}_2\to\mathcal{K}_1 R : K 2 → K 1 V 1 R = R V 2 V_1R=RV_2 V 1 R = R V 2 P H 1 R H 2 ⊥ = 0 P_{\mathcal{H}_1}R\mathcal{H}_2^\perp=0 P H 1 R H 2 ⊥ = 0 S : = P H 1 R H 2 S:=P_{\mathcal{H}_1}R\mathcal{H}_2 S := P H 1 R H 2 T 1 S = S T 2 T_1S=ST_2 T 1 S = S T 2
Next theorem gives a characterization which gives a condition that a given operator in a larger space becomes a dilation of compression of it to a smaller space.
Theorem 6.1.11. (Sarason (1965) ) (Sarason’s lemma ) Let H \mathcal{H} H K \mathcal{K} K V : K → K V:\mathcal{K}\to\mathcal{K} V : K → K T : = P H V ∣ H T:=P_{\mathcal{H}}V|_{\mathcal{H}} T := P H V ∣ H T n = P H V n ∣ H T^n=P_{\mathcal{H}}V^n|_{\mathcal{H}} T n = P H V n ∣ H n ∈ N n\in\mathbb{N} n ∈ N M ⊆ N ⊆ K \mathcal{M}\subseteq\mathcal{N}\subseteq\mathcal{K} M ⊆ N ⊆ K V V V
H = N ⊖ M , \mathcal{H}=\mathcal{N}\ominus\mathcal{M}, H = N ⊖ M , where N ⊖ M \mathcal{N} \ominus\mathcal{M} N ⊖ M M \mathcal{M} M N \mathcal{N} N
Following Theorems 6.1.3 , 6.1.4 , 6.1.6 , 6.1.8 , 6.1.10 and 6.1.11 , extension of contractions on Hilbert spaces became an active area of research, known as dilation theory ([3 , 14 , 17 , 154 ] (); Paulsen 2002; ); Pisier 2001; ); Shalit 2021; ); Sz.-Nagy et al. 2010)). This study of contractions motivated the study of contractions and other classes of operators not only on Hilbert spaces, but also on Banach spaces ([5 , 96 ] (); Stroescu 1973)).
Recently, Bhat, De, and Rakshit abstracted the key ingredients in Halmos and Sz.-Nagy dilation theorem and set up a set theoretic version of dilation theory. Following is the fundamental observation which lead Bhat, De, and Rakshit to set up a set theoretic notion of dilation theory.
There is an embedding i i i
There is a nice map in the larger space.
There is an idempotent from the larger space onto the given space.
These observations can be picturized using the following commutative diagram.
K → U n K i ↑ ↓ P K H → T n H \begin{CD}
\mathcal{K} @>{U^n}>> \mathcal{K}\\
@A{i}AA @VV{P_{\mathcal{K}}}V\\
\mathcal{H} @>{T^n}>> \mathcal{H}
\end{CD} K i ⏐ ↑ H U n T n K ↓ ⏐ P K H Definition 6.1.12. (Bhat et al. [36 ] ) Let A \mathcal{A} A f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A injective power dilation of f f f ( B , i , g , p ) (\mathcal{B},i,g,p) ( B , i , g , p ) B \mathcal{B} B i : A → B i:\mathcal{A}\to\mathcal{B} i : A → B v : B → B v:\mathcal{B}\to\mathcal{B} v : B → B p : B → B p:\mathcal{B}\to\mathcal{B} p : B → B p ( B ) = i ( A ) p(\mathcal{B})=i(\mathcal{A}) p ( B ) = i ( A )
i ( f n ( a ) ) = p ( g n ( i ( a ) ) ) , ∀ a ∈ A , ∀ n ∈ Z + . (45) i(f^n(a))=p(g^n(i(a))),\qquad\forall a\in\mathcal{A},\forall n\in\mathbb{Z}_{+}. \tag*{(45)} i ( f n ( a )) = p ( g n ( i ( a ))) , ∀ a ∈ A , ∀ n ∈ Z + . (45) A dilation ( B , i , g , p ) (\mathcal{B},i,g,p) ( B , i , g , p ) f f f minimal if
B = ⋃ n = 0 ∞ g n ( i ( A ) ) . \mathcal{B}=\bigcup_{n=0}^{\infty}g^n(i(\mathcal{A})). B = n = 0 ⋃ ∞ g n ( i ( A )) . Equation 6.1.1 says that the following diagram commutes for all $n.
B → g n B → p B i ↑ i ↑ A → f n A \begin{CD}
\mathcal{B} @>{g^{n}}>> \mathcal{B} @>{p}>> \mathcal{B} \\
@A{i}AA & & @AiAA \\
\mathcal{A} @>{f^{n}}>> \mathcal{A}
\end{CD} B i ⏐ ↑ i ⏐ ↑ A g n f n B A p B Bhat, De, and Rakshit succeeded in obtaining fundamental theorems of dilations. We now recall these results.
Definition 6.1.13. (Bhat et al. [36 ] ) (Set shifts ) Let A \mathcal{A} A f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A
⋂ n = 0 ∞ f n ( A ) = ∅ . \bigcap_{n=0}^{\infty}f^{n}(\mathcal{A})=\varnothing. n = 0 ⋂ ∞ f n ( A ) = ∅ . Theorem 6.1.14. (Bhat et al. [36 ] ) (Wold decomposition for sets ) Let f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A A \mathcal{A} A A = A b ⊔ A s \mathcal{A}=\mathcal{A}_{b}\sqcup\mathcal{A}_{s} A = A b ⊔ A s A b \mathcal{A}_{b} A b A s \mathcal{A}_{s} A s f f f f ∣ A b f|_{\mathcal{A}_{b}} f ∣ A b f ∣ A s f|_{\mathcal{A}_{s}} f ∣ A s
Theorem 6.1.15. (Bhat et al. [36 ] ) (Halmos dilation for sets ) Let f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A B : = A × { 0 , 1 } \mathcal{B}:=\mathcal{A}\times\{0,1\} B := A × { 0 , 1 }
i : A ∋ a ↦ ( a , 0 ) ∈ B i:\mathcal{A}\ni a\mapsto(a,0)\in\mathcal{B} i : A ∋ a ↦ ( a , 0 ) ∈ B g : B ∋ ( a , m ) ↦ ( a , 1 − m ) ∈ B g:\mathcal{B}\ni(a,m)\mapsto(a,1-m)\in\mathcal{B} g : B ∋ ( a , m ) ↦ ( a , 1 − m ) ∈ B and p : B → B p:\mathcal{B}\to\mathcal{B} p : B → B
p ( a , m ) : = { ( a , 0 ) if m = 0 , ( f ( a ) , 0 ) if m = 1. p(a,m):=
\begin{cases}
(a,0) & \text{if }m=0,\\
(f(a),0) & \text{if }m=1.
\end{cases} p ( a , m ) := { ( a , 0 ) ( f ( a ) , 0 ) if m = 0 , if m = 1. Then i i i g g g p p p
i ( f ( a ) ) = p ( g ( i ( a ) ) ) , ∀ a ∈ A . i(f(a))=p(g(i(a))),\qquad\forall a\in\mathcal{A}. i ( f ( a )) = p ( g ( i ( a ))) , ∀ a ∈ A . Theorem 6.1.16. (Bhat et al. [36 ] ) (Sz. Nagy dilation for sets ) Every map f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A
In Bhat et al. [36 ] , a particular type of minimal injective dilation, called as standard dilation was defined. This dilation is defined as follows. Let f : A → A f:\mathcal{A}\to\mathcal{A} f : A → A
B : = A × Z + , \mathcal{B}:=\mathcal{A}\times\mathbb{Z}_{+}, B := A × Z + , i ( a ) : = ( a , 0 ) , ∀ a ∈ A , i(a):=(a,0),\qquad\forall a\in\mathcal{A}, i ( a ) := ( a , 0 ) , ∀ a ∈ A , g ( a , m ) : = ( a , m + 1 ) , ∀ ( a , m ) ∈ B , g(a,m):=(a,m+1),\qquad\forall(a,m)\in\mathcal{B}, g ( a , m ) := ( a , m + 1 ) , ∀ ( a , m ) ∈ B , p ( a , m ) ≔ ( f m ( a ) , 0 ) , ∀ ( a , m ) ∈ B . p(a,m) \coloneqq(f^{m}(a),0),\quad\forall(a,m) \in\mathcal{B}. p ( a , m ) : = ( f m ( a ) , 0 ) , ∀ ( a , m ) ∈ B . Then ( B , i , g , p ) (\mathcal{B},i,g,p) ( B , i , g , p ) f f f
Theorem 6.1.17. (Bhat et al. [36 ] ) (Inter-twining lifting theorem for sets) Let f 1 : A 1 → A 1 f_1:\mathcal{A}_1 \to\mathcal{A}_1 f 1 : A 1 → A 1 f 2 : A 2 → A 2 f_2:\mathcal{A}_2 \to\mathcal{A}_2 f 2 : A 2 → A 2 ( B 1 , i 1 , g 1 , p 1 ) (\mathcal{B}_1,i_1,g_1,p_1) ( B 1 , i 1 , g 1 , p 1 ) ( B 2 , i 2 , g 2 , p 2 ) (\mathcal{B}_2,i_2,g_2,p_2) ( B 2 , i 2 , g 2 , p 2 ) s : A 2 → A 1 s:\mathcal{A}_2 \to\mathcal{A}_1 s : A 2 → A 1 s f 2 = f 1 s sf_2=f_1s s f 2 = f 1 s r : B 2 → B 1 r:\mathcal{B}_2 \to\mathcal{B}_1 r : B 2 → B 1
r g 2 = g 1 r , r p 2 = p 1 r , r i 2 = i 1 s . rg_2=g_1r,\qquad rp_2=p_1r,\qquad ri_2=i_1s. r g 2 = g 1 r , r p 2 = p 1 r , r i 2 = i 1 s . Conversely if r : B 2 → B 1 r:\mathcal{B}_2 \to\mathcal{B}_1 r : B 2 → B 1 r g 2 = g 1 r rg_2=g_1r r g 2 = g 1 r r p 2 = p 1 r rp_2=p_1r r p 2 = p 1 r s : A 2 → A 1 s:\mathcal{A}_2 \to\mathcal{A}_1 s : A 2 → A 1 r i 2 = i 1 s ri_2=i_1s r i 2 = i 1 s s f 2 = f 1 s sf_2=f_1s s f 2 = f 1 s
Theorem 6.1.18. (Bhat et al. [36 ] ) (Ando dilation for sets) Let J J J { f j } j ∈ J \{f_j\}_{j\in J} { f j } j ∈ J A \mathcal{A} A ( B , i , { g j } j ∈ J , p ) (\mathcal{B},i,\{g_j\}_{j\in J},p) ( B , i , { g j } j ∈ J , p ) B \mathcal{B} B i : A → B i:\mathcal{A} \to\mathcal{B} i : A → B { g j } j ∈ J \{g_j\}_{j\in J} { g j } j ∈ J B \mathcal{B} B p : B → B p:\mathcal{B} \to\mathcal{B} p : B → B
i ( f j 1 f j 2 ⋯ f j k ( a ) ) = p ( g j 1 g j 2 ⋯ g j k ( i ( a ) ) ) , ∀ j 1 , … , j k ∈ J , ∀ a ∈ A . i(f_{j_1}f_{j_2}\cdots f_{j_k}(a))=p(g_{j_1}g_{j_2}\cdots g_{j_k}(i(a))),\quad\forall j_1,\ldots,j_k\in J,\forall a\in\mathcal{A}. i ( f j 1 f j 2 ⋯ f j k ( a )) = p ( g j 1 g j 2 ⋯ g j k ( i ( a ))) , ∀ j 1 , … , j k ∈ J , ∀ a ∈ A . Theorem 6.1.19. (Bhat et al. [36 ] ) (Sarason’s lemma for sets) Let g : B → B g:\mathcal{B} \to\mathcal{B} g : B → B A ⊆ B \mathcal{A}\subseteq\mathcal{B} A ⊆ B f : A → A f:\mathcal{A} \to\mathcal{A} f : A → A f ( a ) = g ( a ) f(a)=g(a) f ( a ) = g ( a ) a ∈ A a\in\mathcal{A} a ∈ A g ( a ) ∈ A g(a)\in\mathcal{A} g ( a ) ∈ A A = A 2 ∖ A 1 \mathcal{A}=\mathcal{A}_2\setminus\mathcal{A}_1 A = A 2 ∖ A 1 A 1 \mathcal{A}_1 A 1 A 2 \mathcal{A}_2 A 2 g g g p : B → B p:\mathcal{B} \to\mathcal{B} p : B → B p 2 = p p^2=p p 2 = p p ( B ) = A p(\mathcal{B})=\mathcal{A} p ( B ) = A
p g n ( a ) = f n ( a ) , ∀ n ∈ N , ∀ a ∈ A . pg^n(a)=f^n(a),\quad\forall n\in\mathbb{N},\forall a\in\mathcal{A}. p g n ( a ) = f n ( a ) , ∀ n ∈ N , ∀ a ∈ A . WOLD DECOMPOSITION, HALMOS DILATION AND N-DILATION FOR VECTOR SPACES
In this appendix we consider vector spaces (need not be finite dimensional) over arbitrary fields. We note that the Definition 6.1.1 of shift of an operator on a Hilbert space does not use the Hilbert space structure. Thus it can be formulated for vector spaces without modifications.
Definition 6.2.1. Let V \mathcal{V} V T : V → V T:\mathcal{V} \to\mathcal{V} T : V → V T T T shift if ⋂ n = 0 ∞ T n ( V ) = { 0 } \bigcap_{n=0}^{\infty}T^n(\mathcal{V})=\{0\} ⋂ n = 0 ∞ T n ( V ) = { 0 }
Theorem 6.2.2. (Wold decomposition for vector spaces) Let T T T V \mathcal{V} V V \mathcal{V} V V = V b ⊕ V s \mathcal{V} = \mathcal{V}_b \oplus\mathcal{V}_s V = V b ⊕ V s V b \mathcal{V}_b V b T T T V \mathcal{V} V T ∣ V b : V b → V b T|_{\mathcal{V}_b} : \mathcal{V}_b \to\mathcal{V}_b T ∣ V b : V b → V b T ∣ V s : V s → V T|_{\mathcal{V}_s} : \mathcal{V}_s \to\mathcal{V} T ∣ V s : V s → V
Proof. Define V b : = ⋂ n = 0 ∞ T n ( V ) \mathcal{V}_b := \bigcap_{n=0}^{\infty} T^n(\mathcal{V}) V b := ⋂ n = 0 ∞ T n ( V ) V s \mathcal{V}_s V s V b \mathcal{V}_b V b V \mathcal{V} V V = V b ⊕ V s \mathcal{V} = \mathcal{V}_b \oplus\mathcal{V}_s V = V b ⊕ V s T ( V b ) = T ( ⋂ n = 0 ∞ T n ( V ) ) ⊆ ⋂ n = 0 ∞ T n ( V ) = V b T(\mathcal{V}_b) = T\left(\bigcap_{n=0}^{\infty} T^n(\mathcal{V})\right) \subseteq\bigcap_{n=0}^{\infty} T^n(\mathcal{V}) = \mathcal{V}_b T ( V b ) = T ( ⋂ n = 0 ∞ T n ( V ) ) ⊆ ⋂ n = 0 ∞ T n ( V ) = V b V b \mathcal{V}_b V b T T T V \mathcal{V} V T ∣ V b T|_{\mathcal{V}_b} T ∣ V b T T T T ∣ V b T|_{\mathcal{V}_b} T ∣ V b y ∈ V b y \in\mathcal{V}_b y ∈ V b { x n } n = 1 ∞ \{x_n\}_{n=1}^{\infty} { x n } n = 1 ∞ V \mathcal{V} V y = T x 1 = T 2 x 2 = T 3 x 3 = ⋯ y = Tx_1 = T^2x_2 = T^3x_3 = \cdots y = T x 1 = T 2 x 2 = T 3 x 3 = ⋯ T T T x 1 = T x 2 = T 2 x 2 = ⋯ x_1 = Tx_2 = T^2x_2 = \cdots x 1 = T x 2 = T 2 x 2 = ⋯ y = T x 1 y = Tx_1 y = T x 1 x 1 ∈ V b x_1 \in\mathcal{V}_b x 1 ∈ V b T ∣ V b T|_{\mathcal{V}_b} T ∣ V b T ∣ V s T|_{\mathcal{V}_s} T ∣ V s y ∈ ⋂ n = 0 ∞ ( T ∣ V s ) n ( V s ) ⊆ ( ⋂ n = 0 ∞ T n ( V ) ) ∩ V s = V b ∩ V s y \in\bigcap_{n=0}^{\infty}(T|_{\mathcal{V}_s})^n(\mathcal{V}_s) \subseteq\left(\bigcap_{n=0}^{\infty}T^n(\mathcal{V})\right) \cap\mathcal{V}_s = \mathcal{V}_b \cap\mathcal{V}_s y ∈ ⋂ n = 0 ∞ ( T ∣ V s ) n ( V s ) ⊆ ( ⋂ n = 0 ∞ T n ( V ) ) ∩ V s = V b ∩ V s y = 0 y = 0 y = 0
Since vector space complements are not unique, we do not have uniqueness in Wold decomposition for vector spaces. We next derive Halmos dilation for vector spaces.
Theorem 6.2.3. (Halmos dilation for vector spaces) Let V \mathcal{V} V T : V → V T : \mathcal{V} \to\mathcal{V} T : V → V
U : = ( T I I 0 ) U := \begin{pmatrix} T & I \\ I & 0 \end{pmatrix} U := ( T I I 0 ) is invertible on V ⊕ V \mathcal{V} \oplus\mathcal{V} V ⊕ V
T = P V U ∣ V , T = P_{\mathcal{V}}U|_{\mathcal{V}}, T = P V U ∣ V , where P V : V ⊕ V → V ⊕ V P_{\mathcal{V}} : \mathcal{V} \oplus\mathcal{V} \to\mathcal{V} \oplus\mathcal{V} P V : V ⊕ V → V ⊕ V V \mathcal{V} V
Proof. It suffices to produce inverse map for U U U
V : = ( 0 I I − T ) V := \begin{pmatrix} 0 & I \\ I & -T \end{pmatrix} V := ( 0 I I − T ) is the inverse of U U U
In the sequel, any invertible operator of the form
( T B C D ) , \begin{pmatrix} T & B \\ C & D \end{pmatrix}, ( T C B D ) , where B , C , D : V → V B,C,D:\mathcal{V}\to\mathcal{V} B , C , D : V → V Halmos dilation of T T T
Now we observe that Halmos dilation for vector spaces is not unique. Using the theory of block matrices (Lu and Shiou (2002)) we can produce a variety of Halmos dilations for a given operator. Following are some classes of Halmos dilations.
(i) If T : V → V T:\mathcal{V}\to\mathcal{V} T : V → V B , C , D : V → V B,C,D:\mathcal{V}\to\mathcal{V} B , C , D : V → V D − C T − 1 B D-CT^{-1}B D − C T − 1 B
U : = ( T B C D ) U:=\begin{pmatrix} T & B \\ C & D \end{pmatrix} U := ( T C B D ) is a Halmos dilation of T T T V ⊕ V \mathcal{V}\oplus\mathcal{V} V ⊕ V
( T − 1 + T − 1 B ( D − C T − 1 B ) − 1 − T − 1 B ( D − C T − 1 B ) − 1 − ( D − C T − 1 B ) − 1 C T − 1 ( D − C T − 1 B ) − 1 ) . \begin{pmatrix}
T^{-1}+T^{-1}B(D-CT^{-1}B)^{-1} & -T^{-1}B(D-CT^{-1}B)^{-1}\\
-(D-CT^{-1}B)^{-1}CT^{-1} & (D-CT^{-1}B)^{-1}
\end{pmatrix}. ( T − 1 + T − 1 B ( D − C T − 1 B ) − 1 − ( D − C T − 1 B ) − 1 C T − 1 − T − 1 B ( D − C T − 1 B ) − 1 ( D − C T − 1 B ) − 1 ) . (ii) D : V → V D:\mathcal{V}\to\mathcal{V} D : V → V B , C : V → V B,C:\mathcal{V}\to\mathcal{V} B , C : V → V T − B D − 1 C T-BD^{-1}C T − B D − 1 C
( T B C D ) \begin{pmatrix} T & B \\ C & D \end{pmatrix} ( T C B D ) is a Halmos dilation of T T T V ⊕ V \mathcal{V}\oplus\mathcal{V} V ⊕ V
( ( T − B D − 1 C ) − 1 − ( T − B D − 1 C ) − 1 B D − 1 − D − 1 C ( T − B D − 1 C ) − 1 D − 1 + D − 1 C ( T − B D − 1 C ) − 1 B D − 1 ) . \begin{pmatrix}
(T-BD^{-1}C)^{-1} & -(T-BD^{-1}C)^{-1}BD^{-1}\\
-D^{-1}C(T-BD^{-1}C)^{-1} & D^{-1}+D^{-1}C(T-BD^{-1}C)^{-1}BD^{-1}
\end{pmatrix}. ( ( T − B D − 1 C ) − 1 − D − 1 C ( T − B D − 1 C ) − 1 − ( T − B D − 1 C ) − 1 B D − 1 D − 1 + D − 1 C ( T − B D − 1 C ) − 1 B D − 1 ) . (iii) B : V → V B:\mathcal{V}\to\mathcal{V} B : V → V C , D : V → V C,D:\mathcal{V}\to\mathcal{V} C , D : V → V C − D B − 1 T C-DB^{-1}T C − D B − 1 T
( T B C D ) \begin{pmatrix} T & B \\ C & D \end{pmatrix} ( T C B D ) is a Halmos dilation of T T T V ⊕ V \mathcal{V}\oplus\mathcal{V} V ⊕ V
( − ( C − D B − 1 T ) − 1 D B − 1 ( C − D B − 1 T ) − 1 B − 1 + B − 1 T ( C − D B − 1 T ) − 1 D B − 1 − B − 1 T ( C − D B − 1 T ) − 1 ) . \begin{pmatrix}
-(C-DB^{-1}T)^{-1}DB^{-1} & (C-DB^{-1}T)^{-1}\\
B^{-1}+B^{-1}T(C-DB^{-1}T)^{-1}DB^{-1} & -B^{-1}T(C-DB^{-1}T)^{-1}
\end{pmatrix}. ( − ( C − D B − 1 T ) − 1 D B − 1 B − 1 + B − 1 T ( C − D B − 1 T ) − 1 D B − 1 ( C − D B − 1 T ) − 1 − B − 1 T ( C − D B − 1 T ) − 1 ) . (iv) C : V → V C:\mathcal{V}\to\mathcal{V} C : V → V B , D : V → V B,D:\mathcal{V}\to\mathcal{V} B , D : V → V B − T C − 1 D B-TC^{-1}D B − T C − 1 D
( T B C D ) \begin{pmatrix} T & B \\ C & D \end{pmatrix} ( T C B D ) is a Halmos dilation of T T T V ⊕ V \mathcal{V}\oplus\mathcal{V} V ⊕ V
( − C − 1 D ( B − T C − 1 D ) − 1 C − 1 + C − 1 D ( B − T C − 1 D ) − 1 T C − 1 ( B − T C − 1 D ) − 1 − ( B − T C − 1 D ) − 1 T C − 1 ) \begin{pmatrix}
-C^{-1}D(B-TC^{-1}D)^{-1} & C^{-1}+C^{-1}D(B-TC^{-1}D)^{-1}TC^{-1} \\
(B-TC^{-1}D)^{-1} & -(B-TC^{-1}D)^{-1}TC^{-1}
\end{pmatrix} ( − C − 1 D ( B − T C − 1 D ) − 1 ( B − T C − 1 D ) − 1 C − 1 + C − 1 D ( B − T C − 1 D ) − 1 T C − 1 − ( B − T C − 1 D ) − 1 T C − 1 ) Recently, Bhat and Mukherjee [37 ] proved that there is certain kind of uniqueness of Halmos dilation for strict contractions in Hilbert spaces, as shown below.
Theorem 6.2.4. (Bhat and Mukherjee [37 ] ) Let H \mathcal{H} H T : H → H T:\mathcal{H}\to\mathcal{H} T : H → H ∥ T ∥ < 1 \lVert T\rVert<1 ∥ T ∥ < 1 T T T H ⊕ H \mathcal{H}\oplus\mathcal{H} H ⊕ H
( T − I − T T ∗ W I − T ∗ T T ∗ W ) , \begin{pmatrix}
T & -\sqrt{I-TT^{*}}W \\
\sqrt{I-T^{*}T} & T^{*}W
\end{pmatrix}, ( T I − T ∗ T − I − T T ∗ W T ∗ W ) , for some unitary operator W : H → H W:\mathcal{H}\to\mathcal{H} W : H → H
We next derive a negative result to Theorem 6.2.4 for Halmos dilation in vector spaces.
Theorem 6.2.5. Let V \mathcal{V} V T : V → V T:\mathcal{V}\to\mathcal{V} T : V → V T T T
Proof. Note that
( T T − I T + I T ) \begin{pmatrix}
T & T-I \\
T+I & T
\end{pmatrix} ( T T + I T − I T ) is an invertible operator and hence is a Halmos dilation of T T T
( T T − I T + I T ) and ( T I I 0 ) \begin{pmatrix}
T & T-I \\
T+I & T
\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix}
T & I \\
I & 0
\end{pmatrix} ( T T + I T − I T ) and ( T I I 0 ) are not similar. Since V \mathcal{V} V
:::
Theorem 6.2.3 can be generalized which gives vector space version of Theorem 6.1.6.
Theorem 6.2.6. (N-dilation for vector spaces ) Let V \mathcal{V} V T : V → V T:\mathcal{V}\to\mathcal{V} T : V → V
be a linear map. Let N N N
U : = ( T 0 0 ⋯ 0 I I 0 0 ⋯ 0 0 0 I 0 ⋯ 0 0 0 0 I ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 0 0 0 0 ⋯ I 0 ) ( N + 1 ) × ( N + 1 ) U := \begin{pmatrix}
T & 0 & 0 & \cdots& 0 & I \\
I & 0 & 0 & \cdots& 0 & 0 \\
0 & I & 0 & \cdots& 0 & 0 \\
0 & 0 & I & \cdots& 0 & 0 \\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & 0 \\
0 & 0 & 0 & \cdots& I & 0
\end{pmatrix}_{(N+1)\times(N+1)} U := T I 0 0 ⋮ 0 0 0 0 I 0 ⋮ 0 0 0 0 0 I ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 0 ⋮ 0 I I 0 0 0 ⋮ 0 0 ( N + 1 ) × ( N + 1 ) is invertible on ⨁ k = 1 N + 1 Y \bigoplus_{k=1}^{N+1}\mathcal{Y} ⨁ k = 1 N + 1 Y
T k = P Y U k ∣ Y , ∀ k = 1 , … , N , (46) T^k=P_{\mathcal{Y}}U^k|_{\mathcal{Y}},\qquad\forall k=1,\ldots,N, \tag*{(46)} T k = P Y U k ∣ Y , ∀ k = 1 , … , N , (46) where P Y : ⨁ k = 1 N + 1 Y → ⨁ k = 1 N + 1 Y P_{\mathcal{Y}}:\bigoplus_{k=1}^{N+1}\mathcal{Y}\to\bigoplus_{k=1}^{N+1}\mathcal{Y} P Y : ⨁ k = 1 N + 1 Y → ⨁ k = 1 N + 1 Y Y \mathcal{Y} Y
Proof. A direct calculation of power of U U U U U U
V : = ( 0 I 0 0 ⋯ 0 0 0 0 I 0 ⋯ 0 0 0 0 0 I ⋯ 0 0 0 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 0 ⋯ 0 I I − T 0 0 ⋯ 0 0 ) ( N + 1 ) × ( N + 1 ) . V := \begin{pmatrix}
0 & I & 0 & 0 & \cdots& 0 & 0 \\
0 & 0 & I & 0 & \cdots& 0 & 0 \\
0 & 0 & 0 & I & \cdots& 0 & 0 \\
0 & 0 & 0 & 0 & \cdots& 0 & 0 \\
\vdots& \vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & 0 & \cdots& 0 & I \\
I & -T & 0 & 0 & \cdots& 0 & 0
\end{pmatrix}_{(N+1)\times(N+1)}. V := 0 0 0 0 ⋮ 0 I I 0 0 0 ⋮ 0 − T 0 I 0 0 ⋮ 0 0 0 0 I 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 0 ⋮ 0 0 0 0 0 0 ⋮ I 0 ( N + 1 ) × ( N + 1 ) . Then U V = V U = I UV=VU=I U V = V U = I V V V U U U
Note that the Equation (6.2.1) holds only upto N N N N + 1 N+1 N + 1 ⨁ n = − ∞ ∞ Y \bigoplus_{n=-\infty}^{\infty}\mathcal{Y} ⨁ n = − ∞ ∞ Y
⨁ n = − ∞ ∞ Y : = { { x n } n = − ∞ ∞ , x n ∈ Y , ∀ n ∈ Z , x n ≠ 0 only for finitely many n ’s } \bigoplus_{n=-\infty}^{\infty}\mathcal{Y}:=\left\{\{x_n\}_{n=-\infty}^{\infty},x_n\in\mathcal{Y},\ \forall n\in\mathbb{Z},x_n\ne0\text{ only for finitely many }n\text{'s}\right\} n = − ∞ ⨁ ∞ Y := { { x n } n = − ∞ ∞ , x n ∈ Y , ∀ n ∈ Z , x n = 0 only for finitely many n ’s } with respect to natural operations.
Theorem 6.2.7. (Sz. Nagy dilation for vector spaces ) Let Y \mathcal{Y} Y T : Y → Y T:\mathcal{Y}\to\mathcal{Y} T : Y → Y U : = [ u n , m ] − ∞ ≤ n , m ≤ ∞ U:=[u_{n,m}]_{-\infty\le n,m\le\infty} U := [ u n , m ] − ∞ ≤ n , m ≤ ∞
⨁ n = − ∞ ∞ Y \bigoplus_{n=-\infty}^{\infty}\mathcal{Y} ⨁ n = − ∞ ∞ Y
u 0 , 0 : = T , u n , n + 1 : = I , ∀ n ∈ Z , u n , m : = 0 otherwise, u_{0,0}:=T,\qquad u_{n,n+1}:=I,\qquad\forall n\in\mathbb{Z},\qquad u_{n,m}:=0\quad\text{otherwise,} u 0 , 0 := T , u n , n + 1 := I , ∀ n ∈ Z , u n , m := 0 otherwise, i.e.,
U = ( ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ 0 I 0 0 0 ⋯ ⋯ 0 0 I 0 0 ⋯ ⋯ 0 0 T I 0 ⋯ ⋯ 0 0 0 0 I ⋯ ⋯ 0 0 0 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ) ∞ × ∞ U=
\left(
\begin{array}{ccccc}
\vdots& \vdots& \vdots& \vdots& \vdots\\
\cdots& 0 & I & 0 & 0 & 0 & \cdots\\
\cdots& 0 & 0 & I & 0 & 0 & \cdots\\
\cdots& 0 & 0 & T & I & 0 & \cdots\\
\cdots& 0 & 0 & 0 & 0 & I & \cdots\\
\cdots& 0 & 0 & 0 & 0 & 0 & \cdots\\
\vdots& \vdots& \vdots& \vdots& \vdots
\end{array}
\right)_{\infty\times\infty} U = ⋮ ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⋮ 0 0 0 0 0 ⋮ ⋮ I 0 0 0 0 ⋮ ⋮ 0 I T 0 0 ⋮ ⋮ 0 0 I 0 0 ⋮ 0 0 0 I 0 ⋯ ⋯ ⋯ ⋯ ⋯ ∞ × ∞ where T T T ( 0 , 0 ) (0,0) ( 0 , 0 ) ⨁ n = − ∞ ∞ Y \bigoplus_{n=-\infty}^{\infty}\mathcal{Y} ⨁ n = − ∞ ∞ Y
T n = P Y U Y n , ∀ n ∈ N , (47) T^{n}=P_{\mathcal{Y}}U_{\mathcal{Y}}^{n},\qquad\forall n\in\mathbb{N}, \tag*{(47)} T n = P Y U Y n , ∀ n ∈ N , (47) where P Y : ⨁ n = − ∞ ∞ Y → ⨁ n = − ∞ ∞ Y P_{\mathcal{Y}}:\bigoplus_{n=-\infty}^{\infty}\mathcal{Y}\to\bigoplus_{n=-\infty}^{\infty}\mathcal{Y} P Y : ⨁ n = − ∞ ∞ Y → ⨁ n = − ∞ ∞ Y Y \mathcal{Y} Y
Proof. We get Equation (6.2.2) by calculation of powers of operator U U U V : = [ v n , m ] − ∞ < n , m < ∞ V:=[v_{n,m}]_{-\infty<n,m<\infty} V := [ v n , m ] − ∞ < n , m < ∞
v 0 , 0 : = 0 , v 1 , − 1 : = − T , v n , n − 1 : = I , ∀ n ∈ Z , v n , m : = 0 otherwise, v_{0,0}:=0,\qquad v_{1,-1}:=-T,\qquad v_{n,n-1}:=I,\qquad\forall n\in\mathbb{Z},\qquad v_{n,m}:=0\quad\text{otherwise,} v 0 , 0 := 0 , v 1 , − 1 := − T , v n , n − 1 := I , ∀ n ∈ Z , v n , m := 0 otherwise, i.e.,
V = ( ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ I 0 0 0 0 ⋯ ⋯ 0 I 0 0 0 ⋯ ⋯ 0 − T I 0 0 ⋯ ⋯ 0 0 0 I 0 ⋯ ⋯ 0 0 0 0 I ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ) ∞ × ∞ V=
\left(
\begin{array}{ccccc}
\vdots& \vdots& \vdots& \vdots& \vdots\\
\cdots& I & 0 & 0 & 0 & 0 & \cdots\\
\cdots& 0 & I & 0 & 0 & 0 & \cdots\\
\cdots& 0 & -T & I & 0 & 0 & \cdots\\
\cdots& 0 & 0 & 0 & I & 0 & \cdots\\
\cdots& 0 & 0 & 0 & 0 & I & \cdots\\
\vdots& \vdots& \vdots& \vdots& \vdots
\end{array}
\right)_{\infty\times\infty} V = ⋮ ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⋮ I 0 0 0 0 ⋮ ⋮ 0 I − T 0 0 ⋮ ⋮ 0 0 I 0 0 ⋮ ⋮ 0 0 0 I 0 ⋮ 0 0 0 0 I ⋯ ⋯ ⋯ ⋯ ⋯ ∞ × ∞ where 0 0 0 ( 0 , 0 ) (0,0) ( 0 , 0 ) U V = V U = I UV=VU=I U V = V U = I U U U
MINIMAL DILATION, INTERTWINING LIFTING THEOREM AND VARIANT OF ANDO DILATION FOR VECTOR SPACES
An important observation associated with Theorems 6.2.5 , 6.2.6 and 6.2.7 is that the dilation is not optimal, i.e., even if the given operator is invertible, then also U U U T T T [36 ] . Set theoretic definition of dilation, given in Definition 6.1.12 motivated Bhat, De, and Rakshit, to introduce the dilation of linear maps on vector spaces.
Definition 6.3.1. [36 ] Let V \mathcal{V} V T : V → V T:\mathcal{V}\to\mathcal{V} T : V → V linear injective dilation of T T T ( W , I , U , P ) (\mathcal{W},I,U,P) ( W , I , U , P ) W \mathcal{W} W I : V → W I:\mathcal{V}\to\mathcal{W} I : V → W U : W → W U:\mathcal{W}\to\mathcal{W} U : W → W P : W → W P:\mathcal{W}\to\mathcal{W} P : W → W P ( W ) = I ( W ) P(\mathcal{W})=I(\mathcal{W}) P ( W ) = I ( W )
(Dilation equation)
I T n x = P U n I x , ∀ n ∈ Z + , ∀ x ∈ V . IT^n x = PU^n Ix,\qquad\forall n\in\mathbb{Z}_{+},\ \forall x\in\mathcal{V}. I T n x = P U n I x , ∀ n ∈ Z + , ∀ x ∈ V . A dilation ( W , I , U , P ) (\mathcal{W},I,U,P) ( W , I , U , P ) T T T minimal if
W = span { U n I x : n ∈ Z + , x ∈ V } . \mathcal{W}=\operatorname{span}\{U^nIx:n\in\mathbb{Z}_{+},x\in\mathcal{V}\}. W = span { U n I x : n ∈ Z + , x ∈ V } . An easier way to remember the dilation equation is the following commutative diagram.
W → U n W → P W I ↑ ↑ I V → T n V \begin{CD}
\mathcal{W} @>{U^n}>> \mathcal{W} @>{P}>> \mathcal{W}\\
@A{I}AA && @AA{I}A\\
\mathcal{V} @>{T^n}>> \mathcal{V}
\end{CD} W I ⏐ ↑ ⏐ ↑ I V U n T n W V P W Following result is the vector space version of Theorem 6.1.16 .
Theorem 6.3.2. [36 ] (Minimal Sz. Nagy dilation for sets ) Every linear map T : V → V T:\mathcal{V}\to\mathcal{V} T : V → V
Proof. We reproduce the proof given by Bhat et al. [36 ] for the sake of future use. Define
W : = { ( x n ) n = 0 ∞ : x n ∈ V , ∀ n ∈ Z + , x n ≠ 0 only for finitely many n ’s } . \mathcal{W}:=\left\{(x_n)_{n=0}^{\infty}:x_n\in\mathcal{V},\ \forall n\in\mathbb{Z}_{+},x_n\ne0\text{ only for finitely many }n\text{'s}\right\}. W := { ( x n ) n = 0 ∞ : x n ∈ V , ∀ n ∈ Z + , x n = 0 only for finitely many n ’s } . Clearly W \mathcal{W} W
I : V ∋ x ↦ ( x , 0 , … ) ∈ W , I:\mathcal{V}\ni x\mapsto(x,0,\ldots)\in\mathcal{W}, I : V ∋ x ↦ ( x , 0 , … ) ∈ W , U : W ∋ ( x n ) n = 0 ∞ ↦ ( 0 , x 0 , … ) ∈ W , U:\mathcal{W}\ni(x_n)_{n=0}^{\infty}\mapsto(0,x_0,\ldots)\in\mathcal{W}, U : W ∋ ( x n ) n = 0 ∞ ↦ ( 0 , x 0 , … ) ∈ W , P : W ∋ ( x n ) n = 0 ∞ ↦ ∑ n = 0 ∞ I T n x n ∈ W . P:\mathcal{W}\ni(x_n)_{n=0}^{\infty}\mapsto\sum_{n=0}^{\infty}IT^n x_n\in\mathcal{W}. P : W ∋ ( x n ) n = 0 ∞ ↦ n = 0 ∑ ∞ I T n x n ∈ W . Then ( W , I , U , P ) (\mathcal{W},I,U,P) ( W , I , U , P ) T T T
We call the dilation ( W , I , U , P ) (\mathcal{W},I,U,P) ( W , I , U , P ) standard dilation of T T T
Theorem 6.3.3. (Inter-twining lifting theorem for vector spaces) Let V 1 \mathcal{V}_1 V 1 V 2 \mathcal{V}_2 V 2 T 1 : V 1 → V 1 T_1:\mathcal{V}_1\to\mathcal{V}_1 T 1 : V 1 → V 1 T 2 : V 2 → V 2 T_2:\mathcal{V}_2\to\mathcal{V}_2 T 2 : V 2 → V 2 ( W 1 , I 1 , U 1 , P 1 ) (\mathcal{W}_1,I_1,U_1,P_1) ( W 1 , I 1 , U 1 , P 1 ) ( W 2 , I 2 , U 2 , P 2 ) (\mathcal{W}_2,I_2,U_2,P_2) ( W 2 , I 2 , U 2 , P 2 ) T 1 T_1 T 1 T 2 T_2 T 2 S : V 2 → V 1 S:\mathcal{V}_2\to\mathcal{V}_1 S : V 2 → V 1 T 1 S = S T 2 T_1S=ST_2 T 1 S = S T 2 R : W 2 → W 1 R:\mathcal{W}_2\to\mathcal{W}_1 R : W 2 → W 1
U 1 R = R U 2 , R P 2 = P 1 R , R I 2 = I 1 S . (48) U_1R=RU_2,\qquad RP_2=P_1R,\qquad RI_2=I_1S. \tag*{(48)} U 1 R = R U 2 , R P 2 = P 1 R , R I 2 = I 1 S . (48) Conversely if R : W 2 → W 1 R:\mathcal{W}_2\to\mathcal{W}_1 R : W 2 → W 1 U 1 R = R U 2 , R P 2 = P 1 R U_1R=RU_2,RP_2=P_1R U 1 R = R U 2 , R P 2 = P 1 R S : V 2 → V 1 S:\mathcal{V}_2\to\mathcal{V}_1 S : V 2 → V 1
R I 2 = I 1 S , T 1 S = S T 2 . (49) RI_2=I_1S,\qquad T_1S=ST_2. \tag*{(49)} R I 2 = I 1 S , T 1 S = S T 2 . (49) Proof. Define R : W 2 ∋ ( x n ) n = 0 ∞ ↦ ( S x n ) n = 0 ∞ ∈ W 1 R:\mathcal{W}_2\ni(x_n)_{n=0}^{\infty}\mapsto(Sx_n)_{n=0}^{\infty}\in\mathcal{W}_1 R : W 2 ∋ ( x n ) n = 0 ∞ ↦ ( S x n ) n = 0 ∞ ∈ W 1 ( x n ) n = 0 ∞ ∈ W 2 (x_n)_{n=0}^{\infty}\in\mathcal{W}_2 ( x n ) n = 0 ∞ ∈ W 2
U 1 R ( x n ) n = 0 ∞ = U 1 ( S x n ) n = 0 ∞ = ( 0 , S x 0 , S x 1 , … ) , U_1R(x_n)_{n=0}^{\infty}=U_1(Sx_n)_{n=0}^{\infty}=(0,Sx_0,Sx_1,\ldots), U 1 R ( x n ) n = 0 ∞ = U 1 ( S x n ) n = 0 ∞ = ( 0 , S x 0 , S x 1 , … ) , R U 2 ( x n ) n = 0 ∞ = R ( 0 , x 0 , x 1 , … ) = ( 0 , S x 0 , S x 1 , … ) , RU_2(x_n)_{n=0}^{\infty}=R(0,x_0,x_1,\ldots)=(0,Sx_0,Sx_1,\ldots), R U 2 ( x n ) n = 0 ∞ = R ( 0 , x 0 , x 1 , … ) = ( 0 , S x 0 , S x 1 , … ) , R P 2 ( x n ) n = 0 ∞ = R ( ∑ n = 0 ∞ I 2 T 2 n x n ) = ∑ n = 0 ∞ R I 2 T 2 n x n = ∑ n = 0 ∞ R ( T 2 n x n , 0 , 0 , … ) = ∑ n = 0 ∞ ( S T 2 n x n , 0 , 0 , … ) , \begin{aligned}
RP_2(x_n)_{n=0}^{\infty}
&=R\left(\sum_{n=0}^{\infty}I_2T_2^n x_n\right)
=\sum_{n=0}^{\infty}RI_2T_2^n x_n\\
&=\sum_{n=0}^{\infty}R(T_2^n x_n,0,0,\ldots)
=\sum_{n=0}^{\infty}(ST_2^n x_n,0,0,\ldots),
\end{aligned} R P 2 ( x n ) n = 0 ∞ = R ( n = 0 ∑ ∞ I 2 T 2 n x n ) = n = 0 ∑ ∞ R I 2 T 2 n x n = n = 0 ∑ ∞ R ( T 2 n x n , 0 , 0 , … ) = n = 0 ∑ ∞ ( S T 2 n x n , 0 , 0 , … ) , P 1 R ( x n ) n = 0 ∞ = P 1 ( S x n ) n = 0 ∞ = ∑ n = 0 ∞ I 1 T 1 n S x n = ∑ n = 0 ∞ I 1 S T 2 n x n = ∑ n = 0 ∞ ( S T 2 n x n , 0 , 0 , … ) . \begin{aligned}
P_1R(x_n)_{n=0}^{\infty}
&=P_1(Sx_n)_{n=0}^{\infty}
=\sum_{n=0}^{\infty}I_1T_1^nSx_n\\
&=\sum_{n=0}^{\infty}I_1ST_2^n x_n
=\sum_{n=0}^{\infty}(ST_2^n x_n,0,0,\ldots).
\end{aligned} P 1 R ( x n ) n = 0 ∞ = P 1 ( S x n ) n = 0 ∞ = n = 0 ∑ ∞ I 1 T 1 n S x n = n = 0 ∑ ∞ I 1 S T 2 n x n = n = 0 ∑ ∞ ( S T 2 n x n , 0 , 0 , … ) . R I 2 x = R ( x , 0 , 0 , … ) = ( S x , 0 , 0 , … ) , I 1 S x = ( S x , 0 , 0 , … ) . R I_2 x = R(x,0,0,\ldots) = (Sx,0,0,\ldots), \qquad I_1 Sx = (Sx,0,0,\ldots). R I 2 x = R ( x , 0 , 0 , … ) = ( S x , 0 , 0 , … ) , I 1 S x = ( S x , 0 , 0 , … ) . We now consider the converse part. For this, first we have to define linear map S S S y ∈ V 2 y \in\mathcal{V}_2 y ∈ V 2 R P 2 ( y , 0 , … ) = P 1 R ( y , 0 , … ) ∈ I 1 ( V 1 ) R P_2(y,0,\ldots) = P_1 R(y,0,\ldots) \in I_1(\mathcal{V}_1) R P 2 ( y , 0 , … ) = P 1 R ( y , 0 , … ) ∈ I 1 ( V 1 ) I 1 I_1 I 1 x ∈ V 2 x \in\mathcal{V}_2 x ∈ V 2 R P 2 ( y , 0 , … ) = P 1 R ( y , 0 , … ) = I 1 ( x ) R P_2(y,0,\ldots) = P_1 R(y,0,\ldots) = I_1(x) R P 2 ( y , 0 , … ) = P 1 R ( y , 0 , … ) = I 1 ( x ) S y : = x Sy := x S y := x S S S y ∈ V 2 y \in\mathcal{V}_2 y ∈ V 2 x ∈ V 2 x \in\mathcal{V}_2 x ∈ V 2 S y = x Sy = x S y = x I 1 S y = R P 2 ( y , 0 , … ) = R I 2 y I_1Sy = R P_2(y,0,\ldots) = R I_2y I 1 S y = R P 2 ( y , 0 , … ) = R I 2 y
R P 2 U 2 ( x , 0 , … ) = R P 2 ( 0 , x , 0 , … ) = R I 2 T 2 x (50) R P_2 U_2(x,0,\ldots) = R P_2(0,x,0,\ldots) = R I_2 T_2x \tag*{(50)} R P 2 U 2 ( x , 0 , … ) = R P 2 ( 0 , x , 0 , … ) = R I 2 T 2 x (50) and
P 1 U 1 R ( x , 0 , … ) = P 1 R U 2 ( x , 0 , … ) = P 1 R ( 0 , x , 0 , … ) (51) P_1 U_1 R(x,0,\ldots) = P_1 R U_2(x,0,\ldots) = P_1 R(0,x,0,\ldots) \tag*{(51)} P 1 U 1 R ( x , 0 , … ) = P 1 R U 2 ( x , 0 , … ) = P 1 R ( 0 , x , 0 , … ) (51) = R P 2 ( 0 , x , 0 , … ) = R I 2 T 2 x , ∀ x ∈ V 2 . (52) = R P_2(0,x,0,\ldots) = R I_2 T_2x,\qquad\forall x \in\mathcal{V}_2. \tag*{(52)} = R P 2 ( 0 , x , 0 , … ) = R I 2 T 2 x , ∀ x ∈ V 2 . (52) Given conditions produce
R P 2 U 2 = P 1 R U 2 = P 1 U 1 R . (53) R P_2 U_2 = P_1 R U_2 = P_1 U_1 R. \tag*{(53)} R P 2 U 2 = P 1 R U 2 = P 1 U 1 R . (53) Equation (6.3.6) says that (6.3.3) and (6.3.4) are equal which completes the proof. □
Following is a variant of Ando dilation for vector spaces.
Theorem 6.3.4. (Ando like dilation for vector spaces ) Let V \mathcal{V} V T , S : V → V T,S:\mathcal{V}\to\mathcal{V} T , S : V → V ( W , I , U 1 , P ) (\mathcal{W},I,U_1,P) ( W , I , U 1 , P ) ( W , I , U 2 , P ) (\mathcal{W},I,U_2,P) ( W , I , U 2 , P ) T , S T,S T , S
( 0 c U ) = ( 0 r V ) \begin{pmatrix} 0_c & U \end{pmatrix}
=
\begin{pmatrix} 0_r \\ V \end{pmatrix} ( 0 c U ) = ( 0 r V ) and
I T n S m x = P U n V m I x , ∀ n , m ∈ Z + , ∀ x ∈ V , I T^n S^m x = P U^n V^m I x,\qquad\forall n,m \in\mathbb{Z}_+,\ \forall x \in\mathcal{V}, I T n S m x = P U n V m I x , ∀ n , m ∈ Z + , ∀ x ∈ V , where 0 c 0_c 0 c 0 r 0_r 0 r
Proof. We extend the construction in the proof of Theorem 6.3.2. Define
W : = { ( x 0 , 0 x 0 , 1 x 0 , 2 ⋯ x 1 , 0 x 1 , 1 x 1 , 2 ⋯ x 2 , 0 x 2 , 1 x 2 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) ∞ × ∞ : x n , m ∈ Y , ∀ n , m ∈ Z + , x n , m ≠ 0 only for finitely many ( n , m ) } . \mathscr{W}:=\left\{\left(\begin{array}{ccccc}
x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
x_{2,0} & x_{2,1} & x_{2,2} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)_{\infty\times\infty}:x_{n,m}\in\mathcal{Y},\ \forall n,m\in\mathbb{Z}_{+},x_{n,m}\ne0\ \text{only for finitely many }(n,m)\right\}. W := ⎩ ⎨ ⎧ x 0 , 0 x 1 , 0 x 2 , 0 ⋮ x 0 , 1 x 1 , 1 x 2 , 1 ⋮ x 0 , 2 x 1 , 2 x 2 , 2 ⋮ ⋯ ⋯ ⋯ ⋱ ∞ × ∞ : x n , m ∈ Y , ∀ n , m ∈ Z + , x n , m = 0 only for finitely many ( n , m ) ⎭ ⎬ ⎫ . Then W \mathscr{W} W
I : W ∋ x ↦ ( x 0 0 ⋯ 0 0 0 ⋯ 0 0 0 ⋯ ⋮ ⋮ ⋮ ⋱ ) ∈ W I:\mathscr{W}\ni x\mapsto\left(\begin{array}{ccccc}
x & 0 & 0 & \cdots\\
0 & 0 & 0 & \cdots\\
0 & 0 & 0 & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\in\mathscr{W} I : W ∋ x ↦ x 0 0 ⋮ 0 0 0 ⋮ 0 0 0 ⋮ ⋯ ⋯ ⋯ ⋱ ∈ W U : W ∋ ( x 0 , 0 x 0 , 1 x 0 , 2 ⋯ x 1 , 0 x 1 , 1 x 1 , 2 ⋯ x 2 , 0 x 2 , 1 x 2 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) ↦ ( 0 0 0 ⋯ x 0 , 0 x 0 , 1 x 0 , 2 ⋯ x 1 , 0 x 1 , 1 x 1 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) ∈ W U:\mathscr{W}\ni\left(\begin{array}{ccccc}
x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
x_{2,0} & x_{2,1} & x_{2,2} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\mapsto
\left(\begin{array}{ccccc}
0 & 0 & 0 & \cdots\\
x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\in\mathscr{W} U : W ∋ x 0 , 0 x 1 , 0 x 2 , 0 ⋮ x 0 , 1 x 1 , 1 x 2 , 1 ⋮ x 0 , 2 x 1 , 2 x 2 , 2 ⋮ ⋯ ⋯ ⋯ ⋱ ↦ 0 x 0 , 0 x 1 , 0 ⋮ 0 x 0 , 1 x 1 , 1 ⋮ 0 x 0 , 2 x 1 , 2 ⋮ ⋯ ⋯ ⋯ ⋱ ∈ W V : W ∋ ( x 0 , 0 x 0 , 1 x 0 , 2 ⋯ x 1 , 0 x 1 , 1 x 1 , 2 ⋯ x 2 , 0 x 2 , 1 x 2 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) ↦ ( 0 x 0 , 0 x 0 , 1 ⋯ 0 x 1 , 0 x 1 , 1 ⋯ 0 x 2 , 0 x 2 , 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) ∈ W V:\mathscr{W}\ni\left(\begin{array}{ccccc}
x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
x_{2,0} & x_{2,1} & x_{2,2} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\mapsto
\left(\begin{array}{ccccc}
0 & x_{0,0} & x_{0,1} & \cdots\\
0 & x_{1,0} & x_{1,1} & \cdots\\
0 & x_{2,0} & x_{2,1} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\in\mathscr{W} V : W ∋ x 0 , 0 x 1 , 0 x 2 , 0 ⋮ x 0 , 1 x 1 , 1 x 2 , 1 ⋮ x 0 , 2 x 1 , 2 x 2 , 2 ⋮ ⋯ ⋯ ⋯ ⋱ ↦ 0 0 0 ⋮ x 0 , 0 x 1 , 0 x 2 , 0 ⋮ x 0 , 1 x 1 , 1 x 2 , 1 ⋮ ⋯ ⋯ ⋯ ⋱ ∈ W P : W ∋ ( x 0 , 0 x 0 , 1 x 0 , 2 ⋯ x 1 , 0 x 1 , 1 x 1 , 2 ⋯ x 2 , 0 x 2 , 1 x 2 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) ↦ ∑ m = 0 ∞ ∑ n = 0 ∞ I T n S m x n , m ∈ W . P:\mathscr{W}\ni\left(\begin{array}{ccccc}
x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
x_{2,0} & x_{2,1} & x_{2,2} & \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{array}\right)\mapsto\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}I T^{n}S^{m}x_{n,m}\in\mathscr{W}. P : W ∋ x 0 , 0 x 1 , 0 x 2 , 0 ⋮ x 0 , 1 x 1 , 1 x 2 , 1 ⋮ x 0 , 2 x 1 , 2 x 2 , 2 ⋮ ⋯ ⋯ ⋯ ⋱ ↦ m = 0 ∑ ∞ n = 0 ∑ ∞ I T n S m x n , m ∈ W . We then have
( O c U ) = ( 0 0 0 0 ⋯ 0 x 0 , 0 x 0 , 1 x 0 , 2 ⋯ 0 x 1 , 0 x 1 , 1 x 1 , 2 ⋯ 0 x 2 , 0 x 2 , 1 x 2 , 2 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ) = ( O r V ) . \left(O_{c}\ \ U\right)=
\left(\begin{array}{ccccc}
0 & 0 & 0 & 0 & \cdots\\
0 & x_{0,0} & x_{0,1} & x_{0,2} & \cdots\\
0 & x_{1,0} & x_{1,1} & x_{1,2} & \cdots\\
0 & x_{2,0} & x_{2,1} & x_{2,2} & \cdots\\
0 & \vdots& \vdots& \vdots& \ddots
\end{array}\right)
=\left(\begin{array}{c}
O_{r}\\
V
\end{array}\right). ( O c U ) = 0 0 0 0 0 0 x 0 , 0 x 1 , 0 x 2 , 0 ⋮ 0 x 0 , 1 x 1 , 1 x 2 , 1 ⋮ 0 x 0 , 2 x 1 , 2 x 2 , 2 ⋮ ⋯ ⋯ ⋯ ⋯ ⋱ = ( O r V ) . Now P U n I x = I T n x PU^{n}Ix = IT^{n}x P U n I x = I T n x P V n I x = I S m x PV^{n}Ix = IS^{m}x P V n I x = I S m x ∀ x ∈ V \forall x \in\mathcal{V} ∀ x ∈ V ∀ n , m ∈ Z + \forall n,m \in\mathbb{Z}_{+} ∀ n , m ∈ Z + ( W , I , U 1 , P ) (\mathcal{W},I,U_{1},P) ( W , I , U 1 , P ) ( W , I , U 2 , P ) (\mathcal{W},I,U_{2},P) ( W , I , U 2 , P ) T , S T,S T , S I T n S m x = P U n V m I x IT^{n}S^{m}x = PU^{n}V^{m}Ix I T n S m x = P U n V m I x ∀ n , m ∈ Z + \forall n,m \in\mathbb{Z}_{+} ∀ n , m ∈ Z + ∀ x ∈ V \forall x \in\mathcal{V} ∀ x ∈ V
Conclusion and future work : In this appendix we derived some basic results on dilation of linear maps. Since vector spaces are more general than Hilbert spaces and tools of Hilbert spaces will not work in vector space, we are interested to explore algebraic aspects of dilation theory.
C*-ALGEBRAS
Israel M. Gelfand defined abstractly the notion of a complete algebra [116 ] . These are Banach spaces in which we can multiply the elements and the multiplication enjoys continuity.
Definition 6.4.1. (cf. Zhu (1993)) A Banach space A \mathcal{A} A C \mathbb{C} C
(i) ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ \lVert xy\rVert\le\lVert x\rVert\lVert y\rVert ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ ∀ x , y ∈ A \forall x,y \in\mathcal{A} ∀ x , y ∈ A
(ii) ∥ e ∥ = 1 \lVert e\rVert= 1 ∥ e ∥ = 1 e e e A \mathcal{A} A
Example 6.4.2. (cf. Zhu (1993))
(i) If K K K C ( K ) \mathcal{C}(K) C ( K ) K K K
(ii) If X \mathcal{X} X B ( X ) \mathcal{B}(\mathcal{X}) B ( X ) X \mathcal{X} X
Proposition 6.4.3. [cf. 13 ] Every unital Banach algebra A \mathcal{A} A B ( A ) \mathcal{B}(\mathcal{A}) B ( A )
One of the most important notion associated with the study of Banach algebras is the notion of spectrum.
Definition 6.4.4. (cf. Zhu (1993)) Let A \mathcal{A} A e e e x x x A \mathcal{A} A λ \lambda λ λ e − x \lambda e-x λ e − x
Theorem 6.4.5. (cf. Zhu (1993)) Spectrum of every element of a unital Banach algebra is a nonempty compact subset of C \mathbb{C} C
Following is the first fundamental theorem in the study of Banach algebras which characterizes Banach algebras using the information of spectrum.
Theorem 6.4.6. (cf. Zhu (1993)) (Gelfand-Mazur theorem ) If every nonzero element of a Banach algebra is invertible, then it is isometrically isomorphic to C \mathbb{C} C
A subclass of Banach algebras known as C ∗ C^* C ∗ C ∗ C^* C ∗ [117 ] .
Definition 6.4.7. (cf. Zhu (1993)) A unital Banach algebra A \mathcal{A} A C ∗ C^* C ∗ ∗ : A ∋ x ↦ x ∗ ∈ A *:\mathcal{A}\ni x\mapsto x^*\in\mathcal{A} ∗ : A ∋ x ↦ x ∗ ∈ A
(i) ( x ∗ ) ∗ = x , ∀ x ∈ A (x^*)^*=x,\ \forall x\in\mathcal{A} ( x ∗ ) ∗ = x , ∀ x ∈ A
(ii) ( x + y ) ∗ = x ∗ + y ∗ , ∀ x , y ∈ A (x+y)^*=x^*+y^*,\ \forall x,y\in\mathcal{A} ( x + y ) ∗ = x ∗ + y ∗ , ∀ x , y ∈ A
(iii) ( α x ) ∗ = α ‾ x ∗ , ∀ α ∈ K , ∀ x ∈ A (\alpha x)^*=\overline{\alpha}x^*,\ \forall\alpha\in\mathbb{K},\forall x\in\mathcal{A} ( α x ) ∗ = α x ∗ , ∀ α ∈ K , ∀ x ∈ A
(iv) ( x y ) ∗ = y ∗ x ∗ , ∀ x , y ∈ A (xy)^*=y^*x^*,\ \forall x,y\in\mathcal{A} ( x y ) ∗ = y ∗ x ∗ , ∀ x , y ∈ A
(v) ∥ x ∗ x ∥ = ∥ x ∥ 2 , ∀ x ∈ A \lVert x^*x\rVert=\lVert x\rVert^2,\ \forall x\in\mathcal{A} ∥ x ∗ x ∥ = ∥ x ∥ 2 , ∀ x ∈ A
A map ∗ : A ∋ x ↦ x ∗ ∈ A *:\mathcal{A}\ni x\mapsto x^*\in\mathcal{A} ∗ : A ∋ x ↦ x ∗ ∈ A involution .
Segal called the term C ∗ C^* C ∗ C ∗ C^* C ∗
Example 6.4.8. (cf. Zhu (1993))
(i) If K K K C ( K ) \mathcal{C}(K) C ( K ) C ∗ C^* C ∗ f ∗ ( x ) : = f ( x ) ‾ , ∀ x ∈ K f^*(x):=\overline{f(x)},\forall x\in K f ∗ ( x ) := f ( x ) , ∀ x ∈ K
(ii) If H \mathcal{H} H B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) C ∗ C^* C ∗
(iii) If H \mathcal{H} H K ( H ) \mathcal{K}(\mathcal{H}) K ( H ) C ∗ C^* C ∗ B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) H \mathcal{H} H
Following two results characterize unital C ∗ C^* C ∗
Theorem 6.4.9. (cf. Zhu (1993)) (Gelfand-Naimark theorem ) If A \mathcal{A} A C ∗ C^* C ∗ A \mathcal{A} A ∗ * ∗ C ( K ) \mathcal{C}(K) C ( K ) K K K
Theorem 6.4.10. (cf. Zhu (1993)) (Gelfand-Naimark-Segal theorem ) Let A \mathcal{A} A C ∗ C^* C ∗ H \mathcal{H} H A \mathcal{A} A ∗ * ∗ C ∗ C^* C ∗ B ( H ) \mathcal{B}(\mathcal{H}) B ( H )
Example 6.4.11. ([cf. 143 ] ) Consider the unital C*-algebra C [ 0 , 1 ] \mathcal{C}[0,1] C [ 0 , 1 ]
π : C [ 0 , 1 ] ∋ f ↦ π ( f ) ∈ B ( L 2 [ 0 , 1 ] ) ; \pi:\mathcal{C}[0,1]\ni f\mapsto\pi(f)\in\mathcal{B}(\mathcal{L}^{2}[0,1]); π : C [ 0 , 1 ] ∋ f ↦ π ( f ) ∈ B ( L 2 [ 0 , 1 ]) ; π ( f ) : L 2 [ 0 , 1 ] ∋ g ↦ ( π ( f ) ) ( g ) : = f g ∈ L 2 [ 0 , 1 ] \pi(f):\mathcal{L}^{2}[0,1]\ni g\mapsto(\pi(f))(g):=fg\in\mathcal{L}^{2}[0,1] π ( f ) : L 2 [ 0 , 1 ] ∋ g ↦ ( π ( f )) ( g ) := f g ∈ L 2 [ 0 , 1 ] is an isometric -isomorphism to a C -subalgebra of B ( L 2 [ 0 , 1 ] ) \mathcal{B}(\mathcal{L}^{2}[0,1]) B ( L 2 [ 0 , 1 ])
COMMUTATORS CLOSE TO THE IDENTITY IN B(H)
Let n ∈ N n\in\mathbb{N} n ∈ N M n ( K ) M_n(\mathbb{K}) M n ( K ) n n n n n n K \mathbb{K} K D , X ∈ M n ( K ) D,X\in M_n(\mathbb{K}) D , X ∈ M n ( K ) D X − X D = 1 M n ( K ) DX-XD=1_{M_n(\mathbb{K})} D X − X D = 1 M n ( K ) [126 ] ). This argument will not work for bounded linear operators on infinite dimensional Hilbert space since the map trace is not defined on the algebra B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) H \mathcal{H} H B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) D X − X D DX-XD D X − X D commutator of D D D X X X [ D , X ] [D,X] [ D , X ] T ∈ B ( H ) T\in\mathcal{B}(\mathcal{H}) T ∈ B ( H ) T = [ D , X ] T=[D,X] T = [ D , X ] D , X ∈ B ( H ) D,X\in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
Using the property of spectrum of bounded linear operator, Winter in 1947 proved that the following result.
Theorem 6.5.1. ((Wintner (1947))) Let H \mathcal{H} H D , X ∈ B ( H ) D,X\in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
[ D , X ] = 1 B ( H ) . (54) [D,X]=1_{\mathcal{B}(\mathcal{H})}. \tag*{(54)} [ D , X ] = 1 B ( H ) . (54) After two years, Wielandt (1949) gave a simple proof for the failure of Equation 6.5.1. We note that the boundedness of operators is crucial in Theorem 6.5.1. Following example shows that Theorem 6.5.1 fails for unbounded operators
Example 6.5.2. ([126 ] ) Let H : = L 2 ( R ) \mathcal{H}:=\mathcal{L}^{2}(\mathbb{R}) H := L 2 ( R )
( D f ) ( x ) : = d d x f , ( X f ) ( x ) : = x f ( x ) (Df)(x):=\frac{d}{dx}f,\qquad(Xf)(x):=xf(x) ( D f ) ( x ) := d x d f , ( X f ) ( x ) := x f ( x ) Then [ D , X ] = 1 B ( H ) [D,X]=1_{\mathcal{B}(\mathcal{H})} [ D , X ] = 1 B ( H )
Theorem 6.5.1 leads to the question that which operators on infinite dimensional Hilbert spaces can be written as commutators of operators? First partial answer was given by Halmos.
Theorem 6.5.3. (cf. Putnam (1967)) Let H \mathcal{H} H C ∈ B ( H ) C \in\mathcal{B}(\mathcal{H}) C ∈ B ( H ) I B ( H ) + C I_{\mathcal{B}(\mathcal{H})}+C I B ( H ) + C
Brown and Pearcy [43 ] characterized the set of bounded operators which can be written as commutators.
Theorem 6.5.4. [43 ] Let H \mathcal{H} H B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) λ I B ( H ) + C \lambda I_{\mathcal{B}(\mathcal{H})}+C λ I B ( H ) + C λ \lambda λ C C C
Following the paper of Brown and Pearcy [43 ] there is a series of papers devoted to the study of commutators on sequence spaces, L p \mathcal{L}^{p} L p C ∗ C^{*} C ∗ [84 , 85 , 86 , 90 , 91 , 142 , 145 , 151 ] (); Marcoux 2006; ); Marcoux 2010; ); Schneeberger 1971; ); Stasinski 2016; ); Yood 2008).
It was Popa (1982) who started a quantitative study of commutators close to the identity operator. He gave the following quantitative bound given by Popa for the product of norm of operators whenever the commutator is close to the identity.
Theorem 6.5.5. (Popa (1982)) Let H \mathcal{H} H D , X ∈ B ( H ) D,X \in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
∥ [ D , X ] − 1 B ( H ) ∥ ≤ ε \lVert[D,X]-1_{\mathcal{B}(\mathcal{H})}\rVert\leq\varepsilon ∥[ D , X ] − 1 B ( H ) ∥ ≤ ε for some ε > 0 \varepsilon> 0 ε > 0
∥ D ∥ ∥ X ∥ ≥ 1 2 log 1 ε . \lVert D\rVert\lVert X\rVert\geq\frac{1}{2}\log\frac{1}{\varepsilon}. ∥ D ∥ ∥ X ∥ ≥ 2 1 log ε 1 . Now the problem in Theorem 6.5.5, is the existence of D , X ∈ B ( H ) D,X \in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H ) [ D , X ] [D,X] [ D , X ] r r r s s s r = O ( s ) r=O(s) r = O ( s ) γ \gamma γ ∣ r ∣ ≤ γ s \lvert r\rvert\leq\gamma s ∣ r ∣ ≤ γ s
Theorem 6.5.6. (Popa (1982); Tao (2019)) Let H \mathcal{H} H 0 < ε ≤ 1 0 < \varepsilon\leq1 0 < ε ≤ 1 D , X ∈ B ( H ) D,X \in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
∥ [ D , X ] − 1 B ( H ) ∥ ≤ ε \lVert[D,X]-1_{\mathcal{B}(\mathcal{H})}\rVert\leq\varepsilon ∥[ D , X ] − 1 B ( H ) ∥ ≤ ε and
∥ D ∥ ∥ X ∥ = O ( ε − 2 ) . \lVert D\rVert\lVert X\rVert= O(\varepsilon^{-2}). ∥ D ∥ ∥ X ∥ = O ( ε − 2 ) . Terence Tao improved Theorem 6.5.6 and obtained the following theorem.
Theorem 6.5.7. (Tao (2019)) Let H \mathcal{H} H 0 < ε ≤ 1 / 2 0 < \varepsilon\le1/2 0 < ε ≤ 1/2 D , X ∈ B ( H ) D,X \in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
∥ [ D , X ] − 1 B ( H ) ∥ ≤ ε \left\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\right\| \le\varepsilon [ D , X ] − 1 B ( H ) ≤ ε such that
∥ D ∥ ∥ X ∥ = O ( log 5 1 ε ) . \|D\|\|X\| = O\left(\log^5 \frac{1}{\varepsilon}\right). ∥ D ∥∥ X ∥ = O ( log 5 ε 1 ) . In (Popa (1982)) there is another result about commutators. Let K ( H ) \mathcal{K}(\mathcal{H}) K ( H ) B ( H ) \mathcal{B}(\mathcal{H}) B ( H )
C + K ( H ) : = { λ .1 B ( H ) + T : λ ∈ C , T ∈ K ( H ) } . \mathbb{C}+\mathcal{K}(\mathcal{H}) := \{\lambda.1_{\mathcal{B}(\mathcal{H})}+T:\lambda\in\mathbb{C},T \in\mathcal{K}(\mathcal{H})\}. C + K ( H ) := { λ . 1 B ( H ) + T : λ ∈ C , T ∈ K ( H )} . Theorem 6.5.8. (Popa (1982)) If K ∈ B ( H ) K \in\mathcal{B}(\mathcal{H}) K ∈ B ( H )
∥ A ∥ = O ( 1 ) , ∥ A ∥ = O ( dist ( A , C + K ( H ) ) 2 3 ) , \|A\| = O(1), \qquad\|A\| = O\left(\operatorname{dist}(A,\mathbb{C}+\mathcal{K}(\mathcal{H}))^{\frac{2}{3}}\right), ∥ A ∥ = O ( 1 ) , ∥ A ∥ = O ( dist ( A , C + K ( H ) ) 3 2 ) , then there exist D , X ∈ B ( H ) D,X \in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H )
∥ D ∥ ∥ X ∥ = O ( 1 ) such that A = [ D , X ] . \|D\|\|X\| = O(1) \qquad\text{such that} \qquad A=[D,X]. ∥ D ∥∥ X ∥ = O ( 1 ) such that A = [ D , X ] . COMMUTATORS CLOSE TO THE IDENTITY IN UNITAL C*-ALGEBRAS
We recall fundamentals of matrices over unital C*-algebras as given in (Murphy (1990)).
Let A \mathcal{A} A n ∈ N n \in\mathbb{N} n ∈ N M n ( A ) M_n(\mathcal{A}) M n ( A ) n n n n n n A \mathcal{A} A A : = [ a i , j ] 1 ≤ i , j ≤ n ∈ M n ( A ) A := [a_{i,j}]_{1\le i,j\le n} \in M_n(\mathcal{A}) A := [ a i , j ] 1 ≤ i , j ≤ n ∈ M n ( A ) A ∗ : = [ a j , i ∗ ] 1 ≤ i , j ≤ n A^* := [a^*_{j,i}]_{1\le i,j\le n} A ∗ := [ a j , i ∗ ] 1 ≤ i , j ≤ n M n ( A ) M_n(\mathcal{A}) M n ( A ) ( H , π ) (\mathcal{H},\pi) ( H , π ) H \mathcal{H} H π : M n ( A ) → M n ( B ( H ) ) \pi:M_n(\mathcal{A})\to M_n(\mathcal{B}(\mathcal{H})) π : M n ( A ) → M n ( B ( H )) M n ( A ) M_n(\mathcal{A}) M n ( A )
∥ A ∥ : = ∥ π ( A ) ∥ , ∀ A ∈ M n ( A ) . \|A\| := \|\pi(A)\|,\qquad\forall A \in M_n(\mathcal{A}). ∥ A ∥ := ∥ π ( A ) ∥ , ∀ A ∈ M n ( A ) . This norm makes M n ( A ) M_n(\mathcal{A}) M n ( A )
In the sequel, A \mathcal{A} A
Lemma 6.6.1. (Commutator calculation ) Let u , v , b 1 , … , b n ∈ A u,v,b_1,\ldots,b_n \in\mathcal{A} u , v , b 1 , … , b n ∈ A δ > 0 \delta> 0 δ > 0
X : = ( 0 0 0 ⋯ 0 δ b 1 1 A 0 0 ⋯ 0 δ b 2 0 1 A 0 ⋯ 0 δ b 3 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 δ b n − 1 0 0 0 ⋯ 1 A δ b n ) ∈ M n ( A ) X := \begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & \delta b_1 \\
1_{\mathcal{A}} & 0 & 0 & \cdots& 0 & \delta b_2 \\
0 & 1_{\mathcal{A}} & 0 & \cdots& 0 & \delta b_3 \\
\vdots& \vdots& \vdots& \cdots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & \delta b_{n-1} \\
0 & 0 & 0 & \cdots& 1_{\mathcal{A}} & \delta b_n
\end{pmatrix} \in M_n(\mathcal{A}) X := 0 1 A 0 ⋮ 0 0 0 0 1 A ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 0 ⋮ 0 1 A δ b 1 δ b 2 δ b 3 ⋮ δ b n − 1 δ b n ∈ M n ( A ) and
D : = ( v δ 1 A 0 ⋯ 0 δ b 1 u u δ v δ 21 A ⋯ 0 δ b 2 u 0 u δ v δ ⋯ 0 δ b 3 u ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ v δ ( n − 1 ) 1 A + δ b n − 1 u 0 0 0 ⋯ u δ v δ + δ b n u ) ∈ M n ( A ) . D := \begin{pmatrix}
\frac{v}{\delta} & 1_{\mathcal{A}} & 0 & \cdots& 0 & \delta b_1u \\
\frac{u}{\delta} & \frac{v}{\delta} & 2 1_{\mathcal{A}} & \cdots& 0 & \delta b_2u \\
0 & \frac{u}{\delta} & \frac{v}{\delta} & \cdots& 0 & \delta b_3u \\
\vdots& \vdots& \vdots& \cdots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& \frac{v}{\delta} & (n-1)1_{\mathcal{A}}+\delta b_{n-1}u \\
0 & 0 & 0 & \cdots& \frac{u}{\delta} & \frac{v}{\delta}+\delta b_nu
\end{pmatrix} \in M_n(\mathcal{A}). D := δ v δ u 0 ⋮ 0 0 1 A δ v δ u ⋮ 0 0 0 2 1 A δ v ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 0 ⋮ δ v δ u δ b 1 u δ b 2 u δ b 3 u ⋮ ( n − 1 ) 1 A + δ b n − 1 u δ v + δ b n u ∈ M n ( A ) . Then
[ D , X ] = 1 M n ( A ) + ( 0 0 0 ⋯ 0 [ v , b 1 ] + 0 + δ b 2 + δ b 1 [ u , b n ] 0 0 0 ⋯ 0 [ v , b 2 ] + [ u , b 1 ] + 2 δ b 3 + δ b 2 [ u , b n ] 0 0 0 ⋯ 0 [ v , b 3 ] + [ u , b 2 ] + 3 δ b 4 + δ b 3 [ u , b n ] ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 0 [ v , b n − 1 ] + [ u , b n − 2 ] + ( n − 1 ) δ b n + δ b n − 1 [ u , b n ] 0 0 0 ⋯ 0 [ v , b n ] + [ u , b n − 1 ] + 0 + δ b n [ u , b n ] − n 1 A ) . [D,X] = 1_{M_n(\mathcal{A})}+
\begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & [v,b_1]+0+\delta b_2+\delta b_1[u,b_n] \\
0 & 0 & 0 & \cdots& 0 & [v,b_2]+[u,b_1]+2\delta b_3+\delta b_2[u,b_n] \\
0 & 0 & 0 & \cdots& 0 & [v,b_3]+[u,b_2]+3\delta b_4+\delta b_3[u,b_n] \\
\vdots& \vdots& \vdots& \cdots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & [v,b_{n-1}]+[u,b_{n-2}]+(n-1)\delta b_n+\delta b_{n-1}[u,b_n] \\
0 & 0 & 0 & \cdots& 0 & [v,b_n]+[u,b_{n-1}]+0+\delta b_n[u,b_n]-n1_{\mathcal{A}}
\end{pmatrix}. [ D , X ] = 1 M n ( A ) + 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 0 ⋮ 0 0 [ v , b 1 ] + 0 + δ b 2 + δ b 1 [ u , b n ] [ v , b 2 ] + [ u , b 1 ] + 2 δ b 3 + δ b 2 [ u , b n ] [ v , b 3 ] + [ u , b 2 ] + 3 δ b 4 + δ b 3 [ u , b n ] ⋮ [ v , b n − 1 ] + [ u , b n − 2 ] + ( n − 1 ) δ b n + δ b n − 1 [ u , b n ] [ v , b n ] + [ u , b n − 1 ] + 0 + δ b n [ u , b n ] − n 1 A . Corollary 6.6.2. Let u , v , b 1 , … , b n ∈ A u,v,b_1,\ldots,b_n \in\mathcal{A} u , v , b 1 , … , b n ∈ A δ > 0 \delta> 0 δ > 0
[ v , b i ] + [ u , b i − 1 ] + i δ b i + 1 + δ b i [ u , b n ] = 0 , ∀ i = 2 , … , n − 1 (55) [v,b_i]+[u,b_{i-1}]+i\delta b_{i+1}+\delta b_i[u,b_n]=0,\qquad\forall i=2,\ldots,n-1 \tag*{(55)} [ v , b i ] + [ u , b i − 1 ] + i δ b i + 1 + δ b i [ u , b n ] = 0 , ∀ i = 2 , … , n − 1 (55) and
[ v , b n ] + [ u , b n − 1 ] + δ b n [ u , b n ] = n ⋅ 1 M n ( A ) . (56) [v,b_n]+[u,b_{n-1}]+\delta b_n[u,b_n]=n\cdot1_{M_n(\mathcal{A})}. \tag*{(56)} [ v , b n ] + [ u , b n − 1 ] + δ b n [ u , b n ] = n ⋅ 1 M n ( A ) . (56) Then for any μ > 0 \mu> 0 μ > 0 D μ , X μ ∈ M n ( A ) D_{\mu},X_{\mu} \in M_n(\mathcal{A}) D μ , X μ ∈ M n ( A )
∥ D μ ∥ ≤ ∥ u ∥ μ 2 δ + ∥ v ∥ μ δ + ( n − 1 ) + δ ∑ i = 1 n μ n − i − 1 ∥ b i ∥ ∥ u ∥ . \lVert D_{\mu}\rVert\le\frac{\lVert u\rVert}{\mu^{2}\delta}+\frac{\lVert v\rVert}{\mu\delta}+(n-1)+\delta\sum_{i=1}^{n}\mu^{n-i-1}\lVert b_i\rVert\lVert u\rVert. ∥ D μ ∥ ≤ μ 2 δ ∥ u ∥ + μ δ ∥ v ∥ + ( n − 1 ) + δ i = 1 ∑ n μ n − i − 1 ∥ b i ∥ ∥ u ∥ . ∥ X μ ∥ ≤ 1 + δ ∑ i = 1 n μ n − i + 1 ∥ b i ∥ and \lVert X_{\mu}\rVert\le1+\delta\sum_{i=1}^{n}\mu^{n-i+1}\lVert b_i\rVert\qquad\text{and} ∥ X μ ∥ ≤ 1 + δ i = 1 ∑ n μ n − i + 1 ∥ b i ∥ and ∥ [ D μ , X μ ] − 1 M n ( A ) ∥ ≤ μ n − 1 ∥ [ v , b 1 ] ∥ + δ b 2 + δ b 1 [ u , b n ] ∥ . \lVert[D_{\mu},X_{\mu}]-1_{M_n(\mathcal{A})}\rVert\le\mu^{n-1}\lVert[v,b_1]\rVert+\delta b_2+\delta b_1[u,b_n]\rVert. ∥[ D μ , X μ ] − 1 M n ( A ) ∥ ≤ μ n − 1 ∥[ v , b 1 ]∥ + δ b 2 + δ b 1 [ u , b n ]∥ . Proof. Let D D D X X X
S μ : = ( μ n − 1 0 0 ⋯ 0 0 0 μ n − 2 0 ⋯ 0 0 0 0 μ n − 3 ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ μ 0 0 0 0 ⋯ 0 1 ) ∈ M n ( K ) , D μ : = 1 μ S μ D S μ − 1 , X μ : = μ S μ X S μ − 1 . S_{\mu}:=
\begin{pmatrix}
\mu^{n-1} & 0 & 0 & \cdots& 0 & 0\\
0 & \mu^{n-2} & 0 & \cdots& 0 & 0\\
0 & 0 & \mu^{n-3} & \cdots& 0 & 0\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& \mu& 0\\
0 & 0 & 0 & \cdots& 0 & 1
\end{pmatrix}\in M_n(\mathbb{K}),
\qquad
D_{\mu}:=\frac{1}{\mu}S_{\mu}DS_{\mu}^{-1},
\qquad
X_{\mu}:=\mu S_{\mu}XS_{\mu}^{-1}. S μ := μ n − 1 0 0 ⋮ 0 0 0 μ n − 2 0 ⋮ 0 0 0 0 μ n − 3 ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ μ 0 0 0 0 ⋮ 0 1 ∈ M n ( K ) , D μ := μ 1 S μ D S μ − 1 , X μ := μ S μ X S μ − 1 . Then
∥ D μ ∥ = ∥ ( v μ δ 1 A 0 ⋯ 0 μ n − 2 δ b 1 u u μ 2 δ v μ δ 2 ⋅ 1 A ⋯ 0 μ n − 3 δ b 2 u 0 u μ 2 δ v μ δ ⋯ 0 μ n − 4 δ b 3 u ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ v μ δ ( n − 1 ) 1 A + δ b n − 1 u 0 0 0 ⋯ u μ 2 δ v μ δ + μ − 1 δ b n u ) ∥ ≤ ∥ u μ 2 δ ∥ + ∥ v μ δ ∥ + ∥ ( n − 1 ) 1 A ∥ + ∥ ( 0 0 0 ⋯ 0 μ n − 2 δ b 1 u 0 0 0 ⋯ 0 μ n − 3 δ b 2 u 0 0 0 ⋯ 0 μ n − 4 δ b 3 u ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 δ b n − 1 u 0 0 0 ⋯ 0 μ − 1 δ b n u ) ∥ ≤ ∥ u ∥ μ 2 δ + ∥ v ∥ μ δ + ( n − 1 ) + δ ∑ i = 1 n μ n − i − 1 ∥ b i ∥ ∥ u ∥ . \begin{aligned}
\lVert D_{\mu}\rVert
=
\left\lVert
\begin{pmatrix}
\frac{v}{\mu\delta} & 1_{\mathcal A} & 0 & \cdots& 0 & \mu^{n-2}\delta b_1u\\
\frac{u}{\mu^{2}\delta} & \frac{v}{\mu\delta} & 2\cdot1_{\mathcal A} & \cdots& 0 & \mu^{n-3}\delta b_2u\\
0 & \frac{u}{\mu^{2}\delta} & \frac{v}{\mu\delta} & \cdots& 0 & \mu^{n-4}\delta b_3u\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& \frac{v}{\mu\delta} & (n-1)1_{\mathcal A}+\delta b_{n-1}u\\
0 & 0 & 0 & \cdots& \frac{u}{\mu^{2}\delta} & \frac{v}{\mu\delta}+\mu^{-1}\delta b_nu
\end{pmatrix}
\right\rVert\\
&\le
\left\lVert\frac{u}{\mu^{2}\delta}\right\rVert+
\left\lVert\frac{v}{\mu\delta}\right\rVert+
\lVert(n-1)1_{\mathcal A}\rVert+
\left\lVert
\begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & \mu^{n-2}\delta b_1u\\
0 & 0 & 0 & \cdots& 0 & \mu^{n-3}\delta b_2u\\
0 & 0 & 0 & \cdots& 0 & \mu^{n-4}\delta b_3u\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & \delta b_{n-1}u\\
0 & 0 & 0 & \cdots& 0 & \mu^{-1}\delta b_nu
\end{pmatrix}
\right\rVert\\
&\le\frac{\lVert u\rVert}{\mu^{2}\delta}+\frac{\lVert v\rVert}{\mu\delta}+(n-1)+\delta\sum_{i=1}^{n}\mu^{n-i-1}\lVert b_i\rVert\lVert u\rVert.
\end{aligned} ∥ D μ ∥ = μ δ v μ 2 δ u 0 ⋮ 0 0 1 A μ δ v μ 2 δ u ⋮ 0 0 0 2 ⋅ 1 A μ δ v ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ μ δ v μ 2 δ u μ n − 2 δ b 1 u μ n − 3 δ b 2 u μ n − 4 δ b 3 u ⋮ ( n − 1 ) 1 A + δ b n − 1 u μ δ v + μ − 1 δ b n u ≤ μ 2 δ u + μ δ v + ∥( n − 1 ) 1 A ∥ + 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ 0 0 μ n − 2 δ b 1 u μ n − 3 δ b 2 u μ n − 4 δ b 3 u ⋮ δ b n − 1 u μ − 1 δ b n u ≤ μ 2 δ ∥ u ∥ + μ δ ∥ v ∥ + ( n − 1 ) + δ i = 1 ∑ n μ n − i − 1 ∥ b i ∥ ∥ u ∥ . and
∥ X μ ∥ = ∥ ( 0 0 0 ⋯ 0 μ n δ b 1 1 A 0 0 ⋯ 0 μ n − 1 δ b 2 0 1 A 0 ⋯ 0 μ n − 2 δ b 3 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 μ 2 δ b n − 1 0 0 0 ⋯ 1 A μ δ b n ) ∥ ≤ ∥ 1 A ∥ + ∥ ( 0 0 0 ⋯ 0 μ n δ b 1 0 0 0 ⋯ 0 μ n − 1 δ b 2 0 0 0 ⋯ 0 μ n − 2 δ b 3 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 μ 2 δ b n − 1 0 0 0 ⋯ 0 μ δ b n ) ∥ ≤ 1 + δ ∑ i = 1 n μ n − i + 1 ∥ b i ∥ . \begin{aligned}
\lVert X_{\mu}\rVert
&=
\left\|
\begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & \mu^{n}\delta b_{1}\\
1_{\mathcal A} & 0 & 0 & \cdots& 0 & \mu^{n-1}\delta b_{2}\\
0 & 1_{\mathcal A} & 0 & \cdots& 0 & \mu^{n-2}\delta b_{3}\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & \mu^{2}\delta b_{n-1}\\
0 & 0 & 0 & \cdots& 1_{\mathcal A} & \mu\delta b_{n}
\end{pmatrix}
\right\|\\
&\leq\lVert1_{\mathcal A}\rVert+
\left\|
\begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & \mu^{n}\delta b_{1}\\
0 & 0 & 0 & \cdots& 0 & \mu^{n-1}\delta b_{2}\\
0 & 0 & 0 & \cdots& 0 & \mu^{n-2}\delta b_{3}\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & \mu^{2}\delta b_{n-1}\\
0 & 0 & 0 & \cdots& 0 & \mu\delta b_{n}
\end{pmatrix}
\right\|\\
&\leq1+\delta\sum_{i=1}^{n}\mu^{n-i+1}\lVert b_i\rVert.
\end{aligned} ∥ X μ ∥ = 0 1 A 0 ⋮ 0 0 0 0 1 A ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ 0 1 A μ n δ b 1 μ n − 1 δ b 2 μ n − 2 δ b 3 ⋮ μ 2 δ b n − 1 μ δ b n ≤ ∥ 1 A ∥ + 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ 0 0 μ n δ b 1 μ n − 1 δ b 2 μ n − 2 δ b 3 ⋮ μ 2 δ b n − 1 μ δ b n ≤ 1 + δ i = 1 ∑ n μ n − i + 1 ∥ b i ∥ . Now using (6.6.1) and (6.6.2) we get
∥ [ D μ , X μ ] − 1 M n ( A ) ∥ = ∥ ( 0 0 0 ⋯ 0 μ n − 1 ( [ v , b 1 ] + δ b 2 + δ b 1 [ v , b n ] ) 0 0 0 ⋯ 0 0 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 0 0 0 0 ⋯ 0 0 ) ∥ ≤ μ n − 1 ∥ [ v , b 1 ] + δ b 2 + δ b 1 [ v , b n ] ∥ . \left\|[D_{\mu},X_{\mu}]-1_{M_n(\mathcal A)}\right\|
=
\left\|
\begin{pmatrix}
0 & 0 & 0 & \cdots& 0 & \mu^{n-1}([v,b_1]+\delta b_2+\delta b_1[v,b_n])\\
0 & 0 & 0 & \cdots& 0 & 0\\
0 & 0 & 0 & \cdots& 0 & 0\\
\vdots& \vdots& \vdots& \ddots& \vdots& \vdots\\
0 & 0 & 0 & \cdots& 0 & 0\\
0 & 0 & 0 & \cdots& 0 & 0
\end{pmatrix}
\right\|
\leq\mu^{n-1}\left\|[v,b_1]+\delta b_2+\delta b_1[v,b_n]\right\|. [ D μ , X μ ] − 1 M n ( A ) = 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 0 0 0 ⋮ 0 0 ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ 0 0 0 ⋮ 0 0 μ n − 1 ([ v , b 1 ] + δ b 2 + δ b 1 [ v , b n ]) 0 0 ⋮ 0 0 ≤ μ n − 1 ∥ [ v , b 1 ] + δ b 2 + δ b 1 [ v , b n ] ∥ . Let A \mathcal A A u , v ∈ A u,v\in\mathcal A u , v ∈ A
u ∗ u = v ∗ v = u u ∗ + v v ∗ = 1 A and u ∗ v = v ∗ u = 0. (57) u^*u=v^*v=uu^*+vv^*=1_{\mathcal A}
\quad\text{and}\quad
u^*v=v^*u=0. \tag*{(57)} u ∗ u = v ∗ v = u u ∗ + v v ∗ = 1 A and u ∗ v = v ∗ u = 0. (57) Examples of such unital C*-algebras are B ( H ) \mathcal B(\mathcal H) B ( H ) H \mathcal H H O 2 \mathcal O_2 O 2 [73 ] . Note that whenever a unital C*-algebra admits a trace map there are no isometries satisfying Equation (57) . In particular, any finite dimensional unital C*-algebra does not have such elements. It is also clear that no commutative unital C*-algebra can have isometries satisfying Equation (57) .
It is shown in Tao (2019) that whenever H \mathcal{H} H B ( H ) \mathcal{B}(\mathcal{H}) B ( H ) M 2 ( B ( H ) ) M_{2}(\mathcal{B}(\mathcal{H})) M 2 ( B ( H )) (57) . To do so we first need a result from the theory of C*-algebras.
Theorem 6.6.3 . (cf. Pedersen (2018); Takesaki (2002))
(i) Every -homomorphism between C -algebras is norm decreasing.
(ii) If a -homomorphism between C -algebras is injective, then it is isometric.
Theorem 6.6.4 . Let A \mathcal{A} A u , v ∈ A u,v \in\mathcal{A} u , v ∈ A (57) holds, then the map
ϕ : A ∋ x ↦ ( u ∗ x u u ∗ x v v ∗ x u v ∗ x v ) ∈ M 2 ( A ) (58) \phi: \mathcal{A} \ni x \mapsto\begin{pmatrix} u^{*}xu & u^{*}xv \\ v^{*}xu & v^{*}xv \end{pmatrix} \in M_{2}(\mathcal{A}) \tag*{(58)} ϕ : A ∋ x ↦ ( u ∗ xu v ∗ xu u ∗ xv v ∗ xv ) ∈ M 2 ( A ) (58) is a C*-algebra isomorphism with the inverse map
ψ : M 2 ( A ) ∋ ( a b c d ) ↦ u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ∈ A . (59) \psi: M_{2}(\mathcal{A}) \ni\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto uau^{*}+ubv^{*}+vcu^{*}+vdv^{*} \in\mathcal{A}. \tag*{(59)} ψ : M 2 ( A ) ∋ ( a c b d ) ↦ u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ∈ A . (59) Proof. Using Equation (57) , a direct computation gives
ϕ ψ ( a b c d ) = ϕ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) = ( u ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) u u ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) v v ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) u v ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) v ) = ( 1 A a 1 A + 1 A b 0 + 0 c 1 A + 0 d 0 1 A a 0 + 1 A b 1 A + 0 c 0 + 0 d 1 A 0 a 1 A + 0 b 0 + 1 A c 1 A + 1 A d 0 0 a 0 + 0 b 1 A + 1 A c 0 + 1 A d 1 A ) = ( a b c d ) , ∀ ( a b c d ) ∈ M 2 ( A ) \begin{aligned}
\phi\psi\begin{pmatrix} a & b \\ c & d \end{pmatrix}
= \phi(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*}) \\
&= \begin{pmatrix}
u^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})u &
u^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})v \\
v^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})u &
v^{*}(uau^{*}+ubv^{*}+vcu^{*}+vdv^{*})v
\end{pmatrix} \\
&= \begin{pmatrix}
1_{\mathcal{A}}a1_{\mathcal{A}}+1_{\mathcal{A}}b0+0c1_{\mathcal{A}}+0d0 &
1_{\mathcal{A}}a0+1_{\mathcal{A}}b1_{\mathcal{A}}+0c0+0d1_{\mathcal{A}} \\
0a1_{\mathcal{A}}+0b0+1_{\mathcal{A}}c1_{\mathcal{A}}+1_{\mathcal{A}}d0 &
0a0+0b1_{\mathcal{A}}+1_{\mathcal{A}}c0+1_{\mathcal{A}}d1_{\mathcal{A}}
\end{pmatrix} \\
&= \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad\forall\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_{2}(\mathcal{A})
\end{aligned} ϕ ψ ( a c b d ) = ϕ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) = ( u ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) u v ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) u u ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) v v ∗ ( u a u ∗ + u b v ∗ + v c u ∗ + v d v ∗ ) v ) = ( 1 A a 1 A + 1 A b 0 + 0 c 1 A + 0 d 0 0 a 1 A + 0 b 0 + 1 A c 1 A + 1 A d 0 1 A a 0 + 1 A b 1 A + 0 c 0 + 0 d 1 A 0 a 0 + 0 b 1 A + 1 A c 0 + 1 A d 1 A ) = ( a c b d ) , ∀ ( a c b d ) ∈ M 2 ( A ) and
ψ ϕ x = ψ ( u ∗ x u u ∗ x v v ∗ x u v ∗ x v ) = u ( u ∗ x u ) u ∗ + u ( u ∗ x v ) v ∗ + v ( v ∗ x u ) u ∗ + v ( v ∗ x v ) v ∗ = u u ∗ x ( u u ∗ + v v ∗ ) + v v ∗ x ( u u ∗ + v v ∗ ) = u u ∗ x 1 A + v v ∗ x 1 A = ( u u ∗ + v v ∗ ) x = x , ∀ x ∈ A . \begin{aligned}
\psi\phi x
= \psi\begin{pmatrix} u^{*}xu & u^{*}xv \\ v^{*}xu & v^{*}xv \end{pmatrix} \\
&= u(u^{*}xu)u^{*}+u(u^{*}xv)v^{*}+v(v^{*}xu)u^{*}+v(v^{*}xv)v^{*} \\
&= uu^{*}x(uu^{*}+vv^{*})+vv^{*}x(uu^{*}+vv^{*}) \\
&= uu^{*}x1_{\mathcal{A}}+vv^{*}x1_{\mathcal{A}}=(uu^{*}+vv^{*})x=x,\qquad\forall x\in\mathcal{A}.
\end{aligned} ψ ϕ x = ψ ( u ∗ xu v ∗ xu u ∗ xv v ∗ xv ) = u ( u ∗ xu ) u ∗ + u ( u ∗ xv ) v ∗ + v ( v ∗ xu ) u ∗ + v ( v ∗ xv ) v ∗ = u u ∗ x ( u u ∗ + v v ∗ ) + v v ∗ x ( u u ∗ + v v ∗ ) = u u ∗ x 1 A + v v ∗ x 1 A = ( u u ∗ + v v ∗ ) x = x , ∀ x ∈ A . Further,
( ϕ ( x ) ) ∗ = ( u ∗ x u u ∗ x v v ∗ x u v ∗ x v ) ∗ = ( u ∗ x ∗ u ( v ∗ x u ) ∗ ( u ∗ x v ) ∗ v ∗ x ∗ v ) = ( u ∗ x ∗ u u ∗ x ∗ v v ∗ x ∗ u v ∗ x ∗ v ) = ϕ ( x ∗ ) , ∀ x ∈ A . (\phi(x))^{*}=
\begin{pmatrix}
u^{*}xu & u^{*}xv\\
v^{*}xu & v^{*}xv
\end{pmatrix}^{*}
=
\begin{pmatrix}
u^{*}x^{*}u & (v^{*}xu)^{*}\\
(u^{*}xv)^{*} & v^{*}x^{*}v
\end{pmatrix}
=
\begin{pmatrix}
u^{*}x^{*}u & u^{*}x^{*}v\\
v^{*}x^{*}u & v^{*}x^{*}v
\end{pmatrix}
=\phi(x^{*}),\qquad\forall x\in\mathcal{A}. ( ϕ ( x ) ) ∗ = ( u ∗ xu v ∗ xu u ∗ xv v ∗ xv ) ∗ = ( u ∗ x ∗ u ( u ∗ xv ) ∗ ( v ∗ xu ) ∗ v ∗ x ∗ v ) = ( u ∗ x ∗ u v ∗ x ∗ u u ∗ x ∗ v v ∗ x ∗ v ) = ϕ ( x ∗ ) , ∀ x ∈ A . Hence ϕ \phi ϕ -isomorphism. Using Theorem 6.6.3 , to show ϕ \phi ϕ -algebra isomorphism (i.e., isometric isomorphism), it suffices to show that ϕ \phi ϕ x ∈ A x\in\mathcal{A} x ∈ A ϕ x = 0 \phi x=0 ϕ x = 0
u ∗ x u = u ∗ x v = 0 , v ∗ x v = v ∗ x u = 0. u^{*}xu=u^{*}xv=0,\qquad v^{*}xv=v^{*}xu=0. u ∗ xu = u ∗ xv = 0 , v ∗ xv = v ∗ xu = 0. Using the first equation we get u u ∗ x u u ∗ = u u ∗ x v v ∗ = 0 uu^{*}xuu^{*}=uu^{*}xvv^{*}=0 u u ∗ xu u ∗ = u u ∗ xv v ∗ = 0 u u ∗ x = u u ∗ x ( u u ∗ + v v ∗ ) = 0 uu^{*}x=uu^{*}x(uu^{*}+vv^{*})=0 u u ∗ x = u u ∗ x ( u u ∗ + v v ∗ ) = 0 v v ∗ x = 0 vv^{*}x=0 v v ∗ x = 0 x = ( u u ∗ + v v ∗ ) x = 0 x=(uu^{*}+vv^{*})x=0 x = ( u u ∗ + v v ∗ ) x = 0 ϕ \phi ϕ
Along with the lines of Theorem 6.6.4 we can easily derive the following result.
Theorem 6.6.5. Let A \mathcal{A} A -algebra and n ∈ N n\in\mathbb{N} n ∈ N u , v ∈ A u,v\in\mathcal{A} u , v ∈ A
ϕ : M n ( A ) ∋ X ↦ ( u ∗ X u u ∗ X v v ∗ X u v ∗ X v ) ∈ M 2 n ( A ) \phi:M_{n}(\mathcal{A})\ni X\mapsto
\begin{pmatrix}
u^{*}Xu & u^{*}Xv\\
v^{*}Xu & v^{*}Xv
\end{pmatrix}\in M_{2n}(\mathcal{A}) ϕ : M n ( A ) ∋ X ↦ ( u ∗ X u v ∗ X u u ∗ X v v ∗ X v ) ∈ M 2 n ( A ) is a C -algebra isomorphism with the inverse map*
ψ : M 2 n ( A ) ∋ ( A B C D ) ↦ u A u ∗ + u B v ∗ + v C u ∗ + v D v ∗ ∈ M n ( A ) , \psi:M_{2n}(\mathcal{A})\ni
\begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
\mapsto uAu^{*}+uBv^{*}+vCu^{*}+vDv^{*}\in M_{n}(\mathcal{A}), ψ : M 2 n ( A ) ∋ ( A C B D ) ↦ u A u ∗ + u B v ∗ + v C u ∗ + v D v ∗ ∈ M n ( A ) , where if X : = [ x i , j ] i , j X:=[x_{i,j}]_{i,j} X := [ x i , j ] i , j a , b ∈ A a,b\in\mathcal{A} a , b ∈ A a X b aXb a X b [ a x i , j b ] i , j [ax_{i,j}b]_{i,j} [ a x i , j b ] i , j -algebras A , M 2 ( A ) , M 4 ( A ) , … , M 2 n ( A ) , … \mathcal{A},M_{2}(\mathcal{A}),M_{4}(\mathcal{A}),\ldots,M_{2^{n}}(\mathcal{A}),\ldots A , M 2 ( A ) , M 4 ( A ) , … , M 2 n ( A ) , … -isometrically isomorphic.
In the rest of this chapter, we assume that unital C*-algebra A \mathcal{A} A u , v u,v u , v x ∈ A n x\in\mathcal{A}^{n} x ∈ A n x i x_i x i i t h i^{\mathrm{th}} i th
Proposition 6.6.6. Let n ≥ 2 n \ge2 n ≥ 2 T : A n → A n − 1 T : \mathcal{A}^{n} \to\mathcal{A}^{n-1} T : A n → A n − 1
T ( b i ) i = 1 n : = ( [ v , b i ] + [ u , b i − 1 ] ) i = 2 n , ∀ ( b i ) i = 1 n ∈ A n . T(b_i)_{i=1}^{n} := \left([v,b_i] + [u,b_{i-1}]\right)_{i=2}^{n}, \qquad\forall(b_i)_{i=1}^{n} \in\mathcal{A}^{n}. T ( b i ) i = 1 n := ( [ v , b i ] + [ u , b i − 1 ] ) i = 2 n , ∀ ( b i ) i = 1 n ∈ A n . Then there exists a bounded linear right-inverse R : A n − 1 → A n R : \mathcal{A}^{n-1} \to\mathcal{A}^{n} R : A n − 1 → A n T T T
∥ R b ∥ = sup 1 ≤ i ≤ n ∥ ( R b ) i ∥ ≤ 8 2 n 2 sup 2 ≤ i ≤ n ∥ b i ∥ \|Rb\| = \sup_{1 \le i \le n} \|(Rb)_i\| \le8\sqrt{2}n^2 \sup_{2 \le i \le n} \|b_i\| ∥ R b ∥ = 1 ≤ i ≤ n sup ∥ ( R b ) i ∥ ≤ 8 2 n 2 2 ≤ i ≤ n sup ∥ b i ∥ ≤ 8 2 n 2 sup 1 ≤ i ≤ n ∥ b i ∥ = 8 2 n 2 ∥ b ∥ , ∀ b ∈ A n . \le8\sqrt{2}n^2 \sup_{1 \le i \le n} \|b_i\| = 8\sqrt{2}n^2\|b\|, \qquad\forall b \in\mathcal{A}^{n}. ≤ 8 2 n 2 1 ≤ i ≤ n sup ∥ b i ∥ = 8 2 n 2 ∥ b ∥ , ∀ b ∈ A n . Proof. Define
L : A n − 1 ∋ ( x i ) i = 2 n ↦ ( − 1 2 x i v ∗ − 1 2 x i + 1 u ∗ ) i = 1 n ∈ A n , where x 1 : = 0 , x n + 1 : = 0 L : \mathcal{A}^{n-1} \ni(x_i)_{i=2}^{n} \mapsto\left(-\frac{1}{2}x_i v^*-\frac{1}{2}x_{i+1}u^*\right)_{i=1}^{n} \in\mathcal{A}^{n}, \qquad\text{where } x_1 := 0, x_{n+1} := 0 L : A n − 1 ∋ ( x i ) i = 2 n ↦ ( − 2 1 x i v ∗ − 2 1 x i + 1 u ∗ ) i = 1 n ∈ A n , where x 1 := 0 , x n + 1 := 0 and
E : A n − 1 ∋ ( x i ) i = 2 n ↦ ( 1 2 ( v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ) ) i = 2 n ∈ A n − 1 . E : \mathcal{A}^{n-1} \ni(x_i)_{i=2}^{n} \mapsto\left(\frac{1}{2}(v x_i v^* + v x_{i+1}u^* + u x_{i-1}v^* + u x_i u^*)\right)_{i=2}^{n} \in\mathcal{A}^{n-1}. E : A n − 1 ∋ ( x i ) i = 2 n ↦ ( 2 1 ( v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ) ) i = 2 n ∈ A n − 1 . Then
T L ( x i ) i = 2 n = T ( − 1 2 x i v ∗ − 1 2 x i + 1 u ∗ ) i = 1 n = − 1 2 ( T ( x i v ∗ ) i = 2 n + T ( x i + 1 u ∗ ) i = 2 n ) = − 1 2 ( ( [ v , x i v ∗ ] + [ u , x i − 1 v ∗ ] ) i = 2 n + ( [ v , x i + 1 u ∗ ] + [ u , x i u ∗ ] ) i = 2 n ) = − 1 2 ( v x i v ∗ − x i v ∗ v + u x i − 1 v ∗ − x i − 1 v ∗ u + v x i + 1 u ∗ − x i + 1 u ∗ v + u x i u ∗ − x i u ∗ u ) i = 2 n = − 1 2 ( v x i v ∗ − x i + u x i − 1 v ∗ − x i − 1 v ∗ u + v x i + 1 u ∗ + u x i u ∗ − x i ) i = 2 n = ( x i ) i = 2 n − 1 2 ( v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ) i = 2 n = ( 1 − E ) ( x i ) i = 2 n , ∀ ( x i ) i = 2 n ∈ A n − 1 , where 1 ( x i ) i = 2 n : = ( x i ) i = 2 n . \begin{aligned}
TL(x_i)_{i=2}^{n}
&= T\left(-\frac{1}{2}x_i v^*-\frac{1}{2}x_{i+1}u^*\right)_{i=1}^{n} \\
&= -\frac{1}{2}\left(T(x_i v^*)_{i=2}^{n}+T(x_{i+1}u^*)_{i=2}^{n}\right) \\
&= -\frac{1}{2}\left(\left([v,x_i v^*]+[u,x_{i-1}v^*]\right)_{i=2}^{n}+\left([v,x_{i+1}u^*]+[u,x_i u^*]\right)_{i=2}^{n}\right) \\
&= -\frac{1}{2}\left(vx_i v^*-x_i v^*v+ux_{i-1}v^*-x_{i-1}v^*u+vx_{i+1}u^*-x_{i+1}u^*v+ux_i u^*-x_i u^*u\right)_{i=2}^{n} \\
&= -\frac{1}{2}\left(vx_i v^*-x_i+ux_{i-1}v^*-x_{i-1}v^*u+vx_{i+1}u^*+ux_i u^*-x_i\right)_{i=2}^{n} \\
&= (x_i)_{i=2}^{n}-\frac{1}{2}\left(vx_i v^*+vx_{i+1}u^*+ux_{i-1}v^*+ux_i u^*\right)_{i=2}^{n} \\
&= (1-E)(x_i)_{i=2}^{n}, \qquad\forall(x_i)_{i=2}^{n}\in\mathcal{A}^{n-1}, \qquad\text{where } 1(x_i)_{i=2}^{n}:=(x_i)_{i=2}^{n}.
\end{aligned} T L ( x i ) i = 2 n = T ( − 2 1 x i v ∗ − 2 1 x i + 1 u ∗ ) i = 1 n = − 2 1 ( T ( x i v ∗ ) i = 2 n + T ( x i + 1 u ∗ ) i = 2 n ) = − 2 1 ( ( [ v , x i v ∗ ] + [ u , x i − 1 v ∗ ] ) i = 2 n + ( [ v , x i + 1 u ∗ ] + [ u , x i u ∗ ] ) i = 2 n ) = − 2 1 ( v x i v ∗ − x i v ∗ v + u x i − 1 v ∗ − x i − 1 v ∗ u + v x i + 1 u ∗ − x i + 1 u ∗ v + u x i u ∗ − x i u ∗ u ) i = 2 n = − 2 1 ( v x i v ∗ − x i + u x i − 1 v ∗ − x i − 1 v ∗ u + v x i + 1 u ∗ + u x i u ∗ − x i ) i = 2 n = ( x i ) i = 2 n − 2 1 ( v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ) i = 2 n = ( 1 − E ) ( x i ) i = 2 n , ∀ ( x i ) i = 2 n ∈ A n − 1 , where 1 ( x i ) i = 2 n := ( x i ) i = 2 n . We next try to show that the operator 1 − E 1-E 1 − E A n \mathcal{A}^{n} A n A n − 1 \mathcal{A}^{n-1} A n − 1
∥ ( x i ) i = 2 n ∥ ′ ′ : = sup 2 ≤ i ≤ n ( 2 − i 2 n 2 ) − 1 2 ∥ x i ∥ . \left\|(x_i)_{i=2}^{n}\right\|^{\prime\prime}:=\sup_{2\le i\le n}\left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|x_i\|. ∥ ( x i ) i = 2 n ∥ ′′ := 2 ≤ i ≤ n sup ( 2 − n 2 i 2 ) − 2 1 ∥ x i ∥. ::: Let x = ( x i ) i = 2 n ∈ A n − 1 x=(x_i)_{i=2}^n \in\mathcal{A}^{n-1} x = ( x i ) i = 2 n ∈ A n − 1 ∥ ( x i ) i = 2 n ∥ ′ ′ ≤ 1 \|(x_i)_{i=2}^n\|'' \le1 ∥ ( x i ) i = 2 n ∥ ′′ ≤ 1
( 2 − i 2 n 2 ) − 1 2 ∥ x i ∥ ≤ sup 2 ≤ i ≤ n ( 2 − i 2 n 2 ) − 1 2 ∥ x i ∥ ≤ 1 , ∀ 2 ≤ i ≤ n . \left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|x_i\| \le\sup_{2\le i\le n}\left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|x_i\| \le1,\qquad\forall2\le i\le n. ( 2 − n 2 i 2 ) − 2 1 ∥ x i ∥ ≤ 2 ≤ i ≤ n sup ( 2 − n 2 i 2 ) − 2 1 ∥ x i ∥ ≤ 1 , ∀2 ≤ i ≤ n . Hence ∥ x i ∥ ≤ ( 2 − i 2 n 2 ) 1 2 \|x_i\|\le\left(2-\frac{i^2}{n^2}\right)^{\frac{1}{2}} ∥ x i ∥ ≤ ( 2 − n 2 i 2 ) 2 1 2 ≤ i ≤ n 2\le i\le n 2 ≤ i ≤ n
∥ ( E x ) i ∥ = 1 2 ∥ v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ∥ = 1 2 ∥ ( x i x i + 1 x i − 1 x i ) ∥ ≤ 1 2 ∥ ( ∥ x i ∥ ∥ x i + 1 ∥ ∥ x i − 1 ∥ ∥ x i ∥ ) ∥ ≤ 1 2 ( ∥ x i ∥ 2 + ∥ x i + 1 ∥ 2 + ∥ x i − 1 ∥ 2 + ∥ x i ∥ 2 ) 1 2 ≤ 1 2 ( ( 2 − i 2 n 2 ) + ( 2 − ( i + 1 ) 2 n 2 ) + ( 2 − ( i − 1 ) 2 n 2 ) + ( 2 − i 2 n 2 ) ) 1 2 = ( 2 − i 2 n 2 − 1 2 n 2 ) 1 2 ≤ ( 1 − 1 8 n 2 ) 1 2 ( 2 − i 2 n 2 ) 1 2 ≤ ( 1 − 1 8 n 2 ) ( 2 − i 2 n 2 ) 1 2 , ∀ 2 ≤ i ≤ n . \begin{aligned}
\|(Ex)_i\|&=\frac{1}{2}\|v x_i v^*+v x_{i+1}u^*+u x_{i-1}v^*+u x_i u^*\|\\
&=\frac{1}{2}\left\|\begin{pmatrix}x_i&x_{i+1}\\x_{i-1}&x_i\end{pmatrix}\right\|
\le\frac{1}{2}\left\|\begin{pmatrix}\|x_i\|&\|x_{i+1}\|\\\|x_{i-1}\|&\|x_i\|\end{pmatrix}\right\|\\
&\le\frac{1}{2}\left(\|x_i\|^2+\|x_{i+1}\|^2+\|x_{i-1}\|^2+\|x_i\|^2\right)^{\frac{1}{2}}\\
&\le\frac{1}{2}\left(\left(2-\frac{i^2}{n^2}\right)+\left(2-\frac{(i+1)^2}{n^2}\right)+\left(2-\frac{(i-1)^2}{n^2}\right)+\left(2-\frac{i^2}{n^2}\right)\right)^{\frac{1}{2}}\\
&=\left(2-\frac{i^2}{n^2}-\frac{1}{2n^2}\right)^{\frac{1}{2}}\le\left(1-\frac{1}{8n^2}\right)^{\frac{1}{2}}\left(2-\frac{i^2}{n^2}\right)^{\frac{1}{2}}\\
&\le\left(1-\frac{1}{8n^2}\right)\left(2-\frac{i^2}{n^2}\right)^{\frac{1}{2}},\qquad\forall2\le i\le n.
\end{aligned} ∥ ( E x ) i ∥ = 2 1 ∥ v x i v ∗ + v x i + 1 u ∗ + u x i − 1 v ∗ + u x i u ∗ ∥ = 2 1 ( x i x i − 1 x i + 1 x i ) ≤ 2 1 ( ∥ x i ∥ ∥ x i − 1 ∥ ∥ x i + 1 ∥ ∥ x i ∥ ) ≤ 2 1 ( ∥ x i ∥ 2 + ∥ x i + 1 ∥ 2 + ∥ x i − 1 ∥ 2 + ∥ x i ∥ 2 ) 2 1 ≤ 2 1 ( ( 2 − n 2 i 2 ) + ( 2 − n 2 ( i + 1 ) 2 ) + ( 2 − n 2 ( i − 1 ) 2 ) + ( 2 − n 2 i 2 ) ) 2 1 = ( 2 − n 2 i 2 − 2 n 2 1 ) 2 1 ≤ ( 1 − 8 n 2 1 ) 2 1 ( 2 − n 2 i 2 ) 2 1 ≤ ( 1 − 8 n 2 1 ) ( 2 − n 2 i 2 ) 2 1 , ∀2 ≤ i ≤ n . Hence
∥ E x ′ ∥ ′ ′ = sup 2 ≤ i ≤ n ( 2 − i 2 n 2 ) − 1 2 ∥ ( E x ) i ∥ ≤ ( 1 − 1 8 n 2 ) ∥ x ∥ ′ , ∀ x ∈ A n − 1 . \|Ex'\|''=\sup_{2\le i\le n}\left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|(Ex)_i\|\le\left(1-\frac{1}{8n^2}\right)\|x\|',\qquad\forall x\in\mathcal{A}^{n-1}. ∥ E x ′ ∥ ′′ = 2 ≤ i ≤ n sup ( 2 − n 2 i 2 ) − 2 1 ∥ ( E x ) i ∥ ≤ ( 1 − 8 n 2 1 ) ∥ x ∥ ′ , ∀ x ∈ A n − 1 . Since 1 − 1 8 n 2 < 1 1-\frac{1}{8n^2}<1 1 − 8 n 2 1 < 1 1 − E 1-E 1 − E ∥ ( 1 − E ) − 1 x ′ ∥ ≤ 8 n 2 ∥ x ∥ ′ \|(1-E)^{-1}x'\|\le8n^2\|x\|' ∥ ( 1 − E ) − 1 x ′ ∥ ≤ 8 n 2 ∥ x ∥ ′
1 2 ∥ ( ( 1 − E ) − 1 x ) i ∥ ≤ sup 2 ≤ i ≤ n ( 2 − i 2 n 2 ) − 1 2 ∥ ( ( 1 − E ) − 1 x ) i ∥ = ∥ ( 1 − E ) − 1 x ′ ∥ ≤ 8 n 2 ∥ x ∥ ′ = 8 n 2 sup 2 ≤ i ≤ n ( 2 − i 2 n 2 ) − 1 2 ∥ x i ∥ ≤ 8 n 2 sup 2 ≤ i ≤ n ∥ x i ∥ = 8 n 2 ∥ x ∥ , ∀ x ∈ A n − 1 . \begin{aligned}
\frac{1}{\sqrt{2}}\|((1-E)^{-1}x)_i\|&\le\sup_{2\le i\le n}\left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|((1-E)^{-1}x)_i\|\\
&=\|(1-E)^{-1}x'\|\le8n^2\|x\|'\\
&=8n^2\sup_{2\le i\le n}\left(2-\frac{i^2}{n^2}\right)^{-\frac{1}{2}}\|x_i\|\\
&\le8n^2\sup_{2\le i\le n}\|x_i\|=8n^2\|x\|,\qquad\forall x\in\mathcal{A}^{n-1}.
\end{aligned} 2 1 ∥ (( 1 − E ) − 1 x ) i ∥ ≤ 2 ≤ i ≤ n sup ( 2 − n 2 i 2 ) − 2 1 ∥ (( 1 − E ) − 1 x ) i ∥ = ∥ ( 1 − E ) − 1 x ′ ∥ ≤ 8 n 2 ∥ x ∥ ′ = 8 n 2 2 ≤ i ≤ n sup ( 2 − n 2 i 2 ) − 2 1 ∥ x i ∥ ≤ 8 n 2 2 ≤ i ≤ n sup ∥ x i ∥ = 8 n 2 ∥ x ∥ , ∀ x ∈ A n − 1 . Define R : = L ( 1 − E ) − 1 R := L(1-E)^{-1} R := L ( 1 − E ) − 1 T R = T L ( 1 − E ) − 1 = ( 1 − E ) ( 1 − E ) − 1 = 1 TR = TL(1-E)^{-1} = (1-E)(1-E)^{-1} = 1 T R = T L ( 1 − E ) − 1 = ( 1 − E ) ( 1 − E ) − 1 = 1
∥ R b ∥ = sup 1 ≤ i ≤ n ∥ ( R b ) i ∥ = ∥ L ( 1 − E ) − 1 b ∥ ≤ ∥ L ∥ ∥ ( 1 − E ) − 1 b ∥ ≤ ∥ ( 1 − E ) − 1 b ∥ = sup 2 ≤ i ≤ n ∥ ( ( 1 − E ) − 1 b ) i ∥ ≤ 8 2 n 2 ∥ b ∥ = 8 2 n 2 sup 2 ≤ i ≤ n ∥ b i ∥ , ∀ b ∈ A n − 1 . \begin{aligned}
\lVert Rb\rVert&= \sup_{1\leq i\leq n}\lVert(Rb)_i\rVert= \lVert L(1-E)^{-1}b\rVert\leq\lVert L\rVert\lVert(1-E)^{-1}b\rVert\leq\lVert(1-E)^{-1}b\rVert\\
&= \sup_{2\leq i\leq n}\lVert((1-E)^{-1}b)_i\rVert\leq8\sqrt{2}n^2\lVert b\rVert= 8\sqrt{2}n^2\sup_{2\leq i\leq n}\lVert b_i\rVert,\qquad\forall b\in\mathcal{A}^{n-1}.
\end{aligned} ∥ R b ∥ = 1 ≤ i ≤ n sup ∥( R b ) i ∥ = ∥ L ( 1 − E ) − 1 b ∥ ≤ ∥ L ∥ ∥( 1 − E ) − 1 b ∥ ≤ ∥( 1 − E ) − 1 b ∥ = 2 ≤ i ≤ n sup ∥(( 1 − E ) − 1 b ) i ∥ ≤ 8 2 n 2 ∥ b ∥ = 8 2 n 2 2 ≤ i ≤ n sup ∥ b i ∥ , ∀ b ∈ A n − 1 . As given in Tao (2019) we try to shift from the systems of equations (6.6.1) and (6.6.2) to the solution of single equation. Let n ≥ 2 n\geq2 n ≥ 2 a : = ( 0 , … , n ) ∈ A n a := (0,\ldots,n)\in\mathcal{A}^n a := ( 0 , … , n ) ∈ A n
F : A n ∋ ( b i ) i = 1 n ↦ ( − 2 b 3 , … , − ( n − 1 ) b n , 0 ) ∈ A n − 1 F:\mathcal{A}^n\ni(b_i)_{i=1}^n\mapsto(-2b_3,\ldots,-(n-1)b_n,0)\in\mathcal{A}^{n-1} F : A n ∋ ( b i ) i = 1 n ↦ ( − 2 b 3 , … , − ( n − 1 ) b n , 0 ) ∈ A n − 1 and
G : A n × A n ∋ ( ( b i ) i = 1 n , ( c i ) i = 1 n ) ↦ ( − b 2 [ u , c n ] , … , − b n [ u , c n ] ) ∈ A n − 1 . G:\mathcal{A}^n\times\mathcal{A}^n\ni((b_i)_{i=1}^n,(c_i)_{i=1}^n)\mapsto(-b_2[u,c_n],\ldots,-b_n[u,c_n])\in\mathcal{A}^{n-1}. G : A n × A n ∋ (( b i ) i = 1 n , ( c i ) i = 1 n ) ↦ ( − b 2 [ u , c n ] , … , − b n [ u , c n ]) ∈ A n − 1 . We then have ∥ F ∥ ≤ n − 1 \lVert F\rVert\leq n-1 ∥ F ∥ ≤ n − 1 ∥ G ∥ ≤ 2 \lVert G\rVert\leq2 ∥ G ∥ ≤ 2
Proposition 6.6.7. Systems (6.6.1) and (6.6.2) have a solution b b b
T b = a + δ F ( b ) + δ G ( b , b ) . (60) Tb=a+\delta F(b)+\delta G(b,b). \tag*{(60)} T b = a + δ F ( b ) + δ G ( b , b ) . (60) Proof. Systems (6.6.1) and (6.6.2) have a solution b b b
[ ν , b i ] + [ u , b i − 1 ] = − i δ b i + 1 − δ b i [ u , b n ] , ∀ i = 2 , … , n − 1 [\nu,b_i]+[u,b_{i-1}]=-i\delta b_{i+1}-\delta b_i[u,b_n],\qquad\forall i=2,\ldots,n-1 [ ν , b i ] + [ u , b i − 1 ] = − i δ b i + 1 − δ b i [ u , b n ] , ∀ i = 2 , … , n − 1 and
[ ν , b n ] + [ u , b n − 1 ] = − δ b n [ u , b n ] + n ⋅ 1 M n ( A ) [\nu,b_n]+[u,b_{n-1}]=-\delta b_n[u,b_n]+n\cdot1_{M_n(\mathcal{A})} [ ν , b n ] + [ u , b n − 1 ] = − δ b n [ u , b n ] + n ⋅ 1 M n ( A ) if and only if
( [ ν , b i ] + [ u , b i − 1 ] ) i = 2 n = ( 0 , … , n ) + δ ( − 2 b 3 , … , − ( n − 1 ) b n , 0 ) + δ ( − b 2 [ u , b n ] , … , − b n [ u , b n ] ) \left([\nu,b_i]+[u,b_{i-1}]\right)_{i=2}^n=(0,\ldots,n)+\delta(-2b_3,\ldots,-(n-1)b_n,0)+\delta(-b_2[u,b_n],\ldots,-b_n[u,b_n]) ( [ ν , b i ] + [ u , b i − 1 ] ) i = 2 n = ( 0 , … , n ) + δ ( − 2 b 3 , … , − ( n − 1 ) b n , 0 ) + δ ( − b 2 [ u , b n ] , … , − b n [ u , b n ]) if and only if
T b = a + δ F ( b ) + δ G ( b , b ) . Tb=a+\delta F(b)+\delta G(b,b). T b = a + δ F ( b ) + δ G ( b , b ) . The above proposition reduces the work of solving systems (6.6.1) and (6.6.2) to a single operator equation. To solve (6.6.6) we need an abstract lemma from Tao (2019).
Lemma 6.6.8. (Tao (2019)) Let X \mathcal{X} X Y \mathcal{Y} Y T , F : X → Y T,F:\mathcal{X}\to\mathcal{Y} T , F : X → Y G : X × X → Y G:\mathcal{X}\times\mathcal{X}\to\mathcal{Y} G : X × X → Y r > 0 r>0 r > 0 a ∈ Y a\in\mathcal{Y} a ∈ Y T T T R : Y → X R:\mathcal{Y}\to\mathcal{X} R : Y → X δ > 0 \delta>0 δ > 0
δ ( 2 ∥ F ∥ ∥ R ∥ + 4 r ∥ R ∥ 2 ∥ a ∥ ) < 1 , (61) \delta(2\lVert F\rVert\lVert R\rVert+4r\lVert R\rVert^{2}\lVert a\rVert)<1, \tag*{(61)} δ ( 2 ∥ F ∥ ∥ R ∥ + 4 r ∥ R ∥ 2 ∥ a ∥) < 1 , (61) then there exists b ∈ X b\in\mathcal{X} b ∈ X ∥ b ∥ ≤ 2 ∥ R ∥ ∥ a ∥ \lVert b\rVert\leq2\lVert R\rVert\lVert a\rVert ∥ b ∥ ≤ 2 ∥ R ∥ ∥ a ∥
T b = a + δ F ( b ) + δ G ( b , b ) . Tb=a+\delta F(b)+\delta G(b,b). T b = a + δ F ( b ) + δ G ( b , b ) . Theorem 6.6.9. For each n ≥ 2 n\geq2 n ≥ 2 b b b (60) such that ∥ b ∥ ≤ 16 2 n 3 \lVert b\rVert\leq16\sqrt{2}n^{3} ∥ b ∥ ≤ 16 2 n 3
Proof. We apply Lemma 6.6.8 for
δ ≔ 1 2000 n 5 . \delta\coloneqq\frac{1}{2000n^{5}}. δ : = 2000 n 5 1 . Then using Proposition 6.6.6, we get
δ ( 2 ∥ F ∥ ∥ R ∥ + 4 r ∥ R ∥ 2 ∥ a ∥ ) ≤ 1 2000 n 5 ( 2 ( n − 1 ) 8 2 n 2 + 4.2.128. n 4 . n ) ≤ 1 2000 n 5 ( 16 2 n 3 + 1024 n 5 ) < 1. \begin{aligned}
\delta(2\lVert F\rVert\lVert R\rVert+4r\lVert R\rVert^{2}\lVert a\rVert)
\leq\frac{1}{2000n^{5}}\left(2(n-1)8\sqrt{2}n^{2}+4.2.128.n^{4}.n\right) \\
&\leq\frac{1}{2000n^{5}}\left(16\sqrt{2}n^{3}+1024n^{5}\right)<1.
\end{aligned} δ ( 2 ∥ F ∥ ∥ R ∥ + 4 r ∥ R ∥ 2 ∥ a ∥) ≤ 2000 n 5 1 ( 2 ( n − 1 ) 8 2 n 2 + 4.2.128. n 4 . n ) ≤ 2000 n 5 1 ( 16 2 n 3 + 1024 n 5 ) < 1. Lemma 6.6.8 now says that there exists a b b b
Theorem 6.6.10. For each n ≥ 2 n\geq2 n ≥ 2 b b b (60) and ∥ b ∥ ≤ 16 2 n 3 \lVert b\rVert\leq16\sqrt{2}n^{3} ∥ b ∥ ≤ 16 2 n 3 μ = 1 2 \mu=\frac{1}{2} μ = 2 1 D μ , X μ ∈ M n ( A ) D_{\mu},X_{\mu}\in M_{n}(\mathcal{A}) D μ , X μ ∈ M n ( A )
∥ D μ ∥ = O ( n 5 ) , ∥ X μ ∥ = O ( 1 ) , ∥ [ D μ , X μ ] − 1 M n ( A ) ∥ = O ( n 3 2 − n ) . \lVert D_{\mu}\rVert=O(n^{5}),\quad\lVert X_{\mu}\rVert=O(1),\quad\lVert[D_{\mu},X_{\mu}]-1_{M_{n}(\mathcal{A})}\rVert=O(n^{3}2^{-n}). ∥ D μ ∥ = O ( n 5 ) , ∥ X μ ∥ = O ( 1 ) , ∥[ D μ , X μ ] − 1 M n ( A ) ∥ = O ( n 3 2 − n ) . Proof. Let D μ , X μ ∈ M n ( A ) D_{\mu},X_{\mu}\in M_{n}(\mathcal{A}) D μ , X μ ∈ M n ( A )
∥ D μ ∥ ≤ 4.2000 n 5 ∥ u ∥ + 2.2000 n 5 ∥ v ∥ + ( n − 1 ) + 1 2000 n 5 ∑ i = 1 n 1 2 n − i − 1 16 2 n 3 ∥ u ∥ = O ( n 5 ) , \begin{aligned}
\lVert D_{\mu}\rVert\leq4.2000n^{5}\lVert u\rVert+2.2000n^{5}\lVert v\rVert+(n-1)+\frac{1}{2000n^{5}}\sum_{i=1}^{n}\frac{1}{2^{n-i-1}}16\sqrt{2}n^{3}\lVert u\rVert\\
&=O(n^{5}),
\end{aligned} ∥ D μ ∥ ≤ 4.2000 n 5 ∥ u ∥ + 2.2000 n 5 ∥ v ∥ + ( n − 1 ) + 2000 n 5 1 i = 1 ∑ n 2 n − i − 1 1 16 2 n 3 ∥ u ∥ = O ( n 5 ) , ∥ X μ ∥ ≤ 1 + 1 2000 n 5 ∑ i = 1 n 1 2 n − i − 1 16 2 n 3 = O ( 1 ) , \lVert X_{\mu}\rVert\leq1+\frac{1}{2000n^{5}}\sum_{i=1}^{n}\frac{1}{2^{n-i-1}}16\sqrt{2}n^{3}=O(1), ∥ X μ ∥ ≤ 1 + 2000 n 5 1 i = 1 ∑ n 2 n − i − 1 1 16 2 n 3 = O ( 1 ) , ∥ [ D μ , X μ ] − 1 M n ( A ) ∥ ≤ 2 μ n − 1 ( ∥ v ∥ ∥ b 1 ∥ + δ ∥ b 2 ∥ + δ ∥ b 1 ∥ ∥ u ∥ ∥ b n ∥ ) \left\|[D_\mu,X_\mu]-1_{M_n(\mathcal{A})}\right\| \leq2\mu^{n-1}\left(\|v\|\|b_1\|+\delta\|b_2\|+\delta\|b_1\|\|u\|\|b_n\|\right) [ D μ , X μ ] − 1 M n ( A ) ≤ 2 μ n − 1 ( ∥ v ∥∥ b 1 ∥ + δ ∥ b 2 ∥ + δ ∥ b 1 ∥∥ u ∥∥ b n ∥ ) ≤ 2 1 2 n − 1 ( ∥ u ∥ 16 2 n 3 + 1 2000 n 5 16 2 n 3 + 1 2000 n 5 16 2 n 3 ∥ u ∥ 16 2 n 3 ) \leq2\frac{1}{2^{n-1}}\left(\|u\|16\sqrt{2}n^3+\frac{1}{2000n^5}16\sqrt{2}n^3+\frac{1}{2000n^5}16\sqrt{2}n^3\|u\|16\sqrt{2}n^3\right) ≤ 2 2 n − 1 1 ( ∥ u ∥16 2 n 3 + 2000 n 5 1 16 2 n 3 + 2000 n 5 1 16 2 n 3 ∥ u ∥16 2 n 3 ) ≤ 2 n 3 2 n − 1 ( ∥ v ∥ 16 2 + 1 2000 n 5 16 2 + 1 2000 n 5 16 2 n 3 ∥ u ∥ 16 2 ) = O ( n 3 2 − n ) . \leq2\frac{n^3}{2^{n-1}}\left(\|v\|16\sqrt{2}+\frac{1}{2000n^5}16\sqrt{2}+\frac{1}{2000n^5}16\sqrt{2}n^3\|u\|16\sqrt{2}\right)=O(n^3 2^{-n}). ≤ 2 2 n − 1 n 3 ( ∥ v ∥16 2 + 2000 n 5 1 16 2 + 2000 n 5 1 16 2 n 3 ∥ u ∥16 2 ) = O ( n 3 2 − n ) . Theorem 6.6.11. Let 0 < ε ≤ 1 / 2 0<\varepsilon\leq1/2 0 < ε ≤ 1/2 n n n D , X ∈ M n ( A ) D,X\in M_n(\mathcal{A}) D , X ∈ M n ( A )
∥ [ D , X ] − 1 M n ( A ) ∥ ≤ ε \left\|[D,X]-1_{M_n(\mathcal{A})}\right\|\leq\varepsilon [ D , X ] − 1 M n ( A ) ≤ ε such that
∥ D ∥ ∥ X ∥ = O ( log 5 1 ε ) . \|D\|\|X\|=O\left(\log^5\frac{1}{\varepsilon}\right). ∥ D ∥∥ X ∥ = O ( log 5 ε 1 ) . Proof. Let D μ , X μ ∈ M n ( A ) D_\mu,X_\mu\in M_n(\mathcal{A}) D μ , X μ ∈ M n ( A ) α , β , γ > 0 \alpha,\beta,\gamma>0 α , β , γ > 0
∥ D μ ∥ ≤ α n 5 , ∥ X μ ∥ ≤ β , ∥ [ D μ , X μ ] − 1 M n ( A ) ∥ ≤ γ n 3 2 − n . \|D_\mu\|\leq\alpha n^5,\qquad\|X_\mu\|\leq\beta,\qquad\left\|[D_\mu,X_\mu]-1_{M_n(\mathcal{A})}\right\|\leq\gamma n^3 2^{-n}. ∥ D μ ∥ ≤ α n 5 , ∥ X μ ∥ ≤ β , [ D μ , X μ ] − 1 M n ( A ) ≤ γ n 3 2 − n . Since 2 n > n 4 2^n>n^4 2 n > n 4 n n n γ n 3 2 − n < ε \gamma n^3 2^{-n}<\varepsilon γ n 3 2 − n < ε n n n c c c n = c log 1 ε n=c\log\frac{1}{\varepsilon} n = c log ε 1 γ n 3 2 − n < ε \gamma n^3 2^{-n}<\varepsilon γ n 3 2 − n < ε ∥ D μ ∥ = O ( log 5 ( 1 ε ) ) \|D_\mu\|=O\left(\log^5\left(\frac{1}{\varepsilon}\right)\right) ∥ D μ ∥ = O ( log 5 ( ε 1 ) ) ∥ [ D μ , X μ ] − 1 M n ( A ) ∥ ≤ ε \left\|[D_\mu,X_\mu]-1_{M_n(\mathcal{A})}\right\|\leq\varepsilon [ D μ , X μ ] − 1 M n ( A ) ≤ ε
Theorem 6.6.11 and Theorem 6.6.5 easily give the following.
Theorem 6.6.12. Let A \mathcal{A} A u , v ∈ A u,v\in\mathcal{A} u , v ∈ A (57) holds. Then for each 0 < ε ≤ 1 / 2 0<\varepsilon\leq1/2 0 < ε ≤ 1/2 d , x ∈ A d,x\in\mathcal{A} d , x ∈ A
∥ [ d , x ] − 1 A ∥ ≤ ε \left\|[d,x]-1_{\mathcal{A}}\right\|\leq\varepsilon ∥ [ d , x ] − 1 A ∥ ≤ ε such that
∥ d ∥ ∥ x ∥ = O ( log 5 1 ε ) . \|d\|\|x\|=O\left(\log^5\frac{1}{\varepsilon}\right). ∥ d ∥∥ x ∥ = O ( log 5 ε 1 ) . Remark 6.6.13. Let A \mathcal{A} A [80 ] we have
A ≅ M n 1 ( C ) ⊕ ⋯ ⊕ M n r ( C ) , (62) \mathcal{A}\cong M_{n_1}(\mathbb{C})\oplus\cdots\oplus M_{n_r}(\mathbb{C}), \tag*{(62)} A ≅ M n 1 ( C ) ⊕ ⋯ ⊕ M n r ( C ) , (62) for unique (upto permutation) natural numbers n 1 , … , n r n_1,\ldots,n_r n 1 , … , n r Tr \operatorname{Tr} Tr Tr ( 1 A ) = 1 \operatorname{Tr}(1_{\mathcal{A}})=1 Tr ( 1 A ) = 1 A \mathcal{A} A
(i) A \mathcal{A} A (57) . Suppose that there are such isometries. Then
1 = Tr ( u u ∗ + v v ∗ ) = Tr ( u u ∗ ) + Tr ( v v ∗ ) = Tr ( u ∗ u ) + Tr ( v ∗ v ) = 2 1=\operatorname{Tr}(uu^{*}+vv^{*})=\operatorname{Tr}(uu^{*})+\operatorname{Tr}(vv^{*})=\operatorname{Tr}(u^{*}u)+\operatorname{Tr}(v^{*}v)=2 1 = Tr ( u u ∗ + v v ∗ ) = Tr ( u u ∗ ) + Tr ( v v ∗ ) = Tr ( u ∗ u ) + Tr ( v ∗ v ) = 2 which is impossible.
(ii) In (Tao (2019)), Tao observed that if H \mathcal{H} H D , X ∈ B ( H ) D,X\in\mathcal{B}(\mathcal{H}) D , X ∈ B ( H ) ∥ [ D , X ] − 1 B ( H ) ∥ < 1 \lVert[D,X]-1_{\mathcal{B}(\mathcal{H})}\rVert<1 ∥[ D , X ] − 1 B ( H ) ∥ < 1 A \mathcal{A} A d , x ∈ A d,x\in\mathcal{A} d , x ∈ A ∥ [ d , x ] − 1 A ∥ < 1 \lVert[d,x]-1_{\mathcal{A}}\rVert<1 ∥[ d , x ] − 1 A ∥ < 1 Theorem 6.6.12 fails for every finite dimensional unital C*-algebra. Let d , x ∈ A d,x\in\mathcal{A} d , x ∈ A d d d D D D x x x X X X D , X ∈ M n ( C ) D,X\in M_n(\mathbb{C}) D , X ∈ M n ( C ) n n n Tr ( [ D , X ] ) = 0 \operatorname{Tr}([D,X])=0 Tr ([ D , X ]) = 0 λ 1 , … , λ n \lambda_1,\ldots,\lambda_n λ 1 , … , λ n [ D , X ] [D,X] [ D , X ] ∑ j = 1 n λ j = Tr ( [ D , X ] ) = 0 \sum_{j=1}^{n}\lambda_j=\operatorname{Tr}([D,X])=0 ∑ j = 1 n λ j = Tr ([ D , X ]) = 0
n = ∣ ∑ j = 1 n ( λ j − 1 ) ∣ ≤ ∑ j = 1 n ∣ λ j − 1 ∣ . n=\left|\sum_{j=1}^{n}(\lambda_j-1)\right|\leq\sum_{j=1}^{n}|\lambda_j-1|. n = j = 1 ∑ n ( λ j − 1 ) ≤ j = 1 ∑ n ∣ λ j − 1∣. Previous inequality says that there is atleast one j j j ∣ λ j − 1 ∣ ≥ 1 |\lambda_j-1|\geq1 ∣ λ j − 1∣ ≥ 1 [ D , X ] − 1 M n ( C ) [D,X]-1_{M_n(\mathbb{C})} [ D , X ] − 1 M n ( C ) λ 1 − 1 , … , λ n − 1 \lambda_1-1,\ldots,\lambda_n-1 λ 1 − 1 , … , λ n − 1
∥ [ d , x ] − 1 A ∥ = ∥ [ D , X ] − 1 M n ( C ) ∥ ≥ sup 1 ≤ j ≤ n ∣ λ j − 1 ∣ ≥ 1. \lVert[d,x]-1_{\mathcal{A}}\rVert=\lVert[D,X]-1_{M_n(\mathbb{C})}\rVert\geq\sup_{1\leq j\leq n}|\lambda_j-1|\geq1. ∥[ d , x ] − 1 A ∥ = ∥[ D , X ] − 1 M n ( C ) ∥ ≥ 1 ≤ j ≤ n sup ∣ λ j − 1∣ ≥ 1. Conclusion and future work : In this appendix we showed that the result of Tao’s is valid in more general spaces. One of the future objectives is to improve the bounds in Theorem 6.6.12 and Theorem 6.5.8 .
BIBIOGRAPHY 