Introduction
Let K K K Q Q Q O \mathcal{O} O π \pi π d ≥ 1 d \ge1 d ≥ 1 K d K^d K d μ ( O d ) = 1 \mu(\mathcal{O}^d)=1 μ ( O d ) = 1 ∥ ⋅ ∥ K \lVert\cdot\rVert_K ∥ ⋅ ∥ K K d K^d K d τ ≔ Q − d \tau\coloneqq Q^{-d} τ : = Q − d r ≔ τ 1 / 2 r \coloneqq\tau^{1/2} r : = τ 1/2 O d \mathcal{O}^d O d γ ∗ γ \gamma^*\gamma γ ∗ γ S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) [1 Section 6.1] . In [Arh26] we showed that this coincidence extends to an equivalence of energies between the radial Taibleson operators on O d \mathcal{O}^d O d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) K d K^d K d E r ( 2 ) E_r(2) E r ( 2 ) [41 ] , [43 ] .
Recall that E r ( 2 ) E_r(2) E r ( 2 ) v v v n n n v n v ∗ = r n vnv^*=rn v n v ∗ = r n { 0 } ∪ { λ ∈ C : ∣ λ ∣ ∈ r Z } \{0\}\cup\{\lambda\in\mathbb{C}:|\lambda|\in r^{\mathbb{Z}}\} { 0 } ∪ { λ ∈ C : ∣ λ ∣ ∈ r Z } x ↦ ∥ x ∥ K d x\mapsto\lVert x\rVert_K^d x ↦ ∥ x ∥ K d τ m \tau^m τ m m ∈ Z m\in\mathbb{Z} m ∈ Z K d ∖ { 0 } K^d\setminus\{0\} K d ∖ { 0 } h h h E r ( 2 ) E_r(2) E r ( 2 ) [31 ] gives, for any Borel function F : τ Z → [ 0 , ∞ ] F:\tau^{\mathbb{Z}}\to[0,\infty] F : τ Z → [ 0 , ∞ ]
∫ K d F ( ∥ x ∥ K d ) d x = ( 1 − τ ) ∑ m ∈ Z τ m F ( τ m ) = ( 1 − τ ) h ( F ( n ∗ n ) ) , (1) \int_{K^d}F(\lVert x\rVert_K^d)\,dx=(1-\tau)\sum_{m\in\mathbb{Z}}\tau^mF(\tau^m)=(1-\tau)h(F(n^*n)), \tag*{(1)} ∫ K d F (∥ x ∥ K d ) d x = ( 1 − τ ) m ∈ Z ∑ τ m F ( τ m ) = ( 1 − τ ) h ( F ( n ∗ n )) , (1) see Proposition 5.1. The restriction of (1) to the unit ball, that is to the spectral projection 1 [ 0 , 1 ] ( n ∗ n ) \mathbf{1}_{[0,1]}(n^*n) 1 [ 0 , 1 ] ( n ∗ n ) x ↦ v x v ∗ x\mapsto vxv^* x ↦ v x v ∗ E r ( 2 ) E_r(2) E r ( 2 ) S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) [43 ] , corresponds to the dilation x ↦ π x x\mapsto\pi x x ↦ π x K d K^d K d h ( v x v ∗ ) = τ − 1 h ( x ) h(vxv^*)=\tau^{-1}h(x) h ( v x v ∗ ) = τ − 1 h ( x ) τ − 1 μ ( A ) \tau^{-1}\mu(A) τ − 1 μ ( A ) E r ( 2 ) E_r(2) E r ( 2 ) K d K^d K d q q q L 2 ( K ) L^2(K) L 2 ( K ) ∣ x ∣ K |x|_K ∣ x ∣ K x ↦ π ± 1 x x\mapsto\pi^{\pm1}x x ↦ π ± 1 x [44 ] .
The analytic input is the Taibleson operator D α D^\alpha D α α > 0 \alpha>0 α > 0 K d K^d K d ∥ ξ ∥ K α \lVert\xi\rVert_K^\alpha ∥ ξ ∥ K α D r a d α D^\alpha_{\mathrm{rad}} D rad α ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) w m ≔ ( 1 − τ ) τ m w_m\coloneqq(1-\tau)\tau^m w m : = ( 1 − τ ) τ m ( g m ) m ∈ Z (g_m)_{m\in\mathbb{Z}} ( g m ) m ∈ Z ρ m + 1 \rho^{m+1} ρ m + 1 ρ ≔ Q α \rho\coloneqq Q^\alpha ρ : = Q α Proposition 2.1 ). On the other side, for 0 < η < 1 0<\eta<1 0 < η < 1
( L η ∞ f ) ( k ) = τ η − k − 1 ( f ( k ) − f ( k + 1 ) ) + η − k ( f ( k ) − f ( k − 1 ) ) , k ∈ Z , (2) (L_\eta^\infty f)(k)=\tau\eta^{-k-1}\bigl(f(k)-f(k+1)\bigr)+\eta^{-k}\bigl(f(k)-f(k-1)\bigr),\qquad k\in\mathbb{Z}, \tag*{(2)} ( L η ∞ f ) ( k ) = τ η − k − 1 ( f ( k ) − f ( k + 1 ) ) + η − k ( f ( k ) − f ( k − 1 ) ) , k ∈ Z , (2) which is symmetric on ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) η \eta η L η L_\eta L η ℓ N 0 2 ( w ) \ell_{N_0}^2(w) ℓ N 0 2 ( w ) N N N η N \eta^N η N N → ∞ N\to\infty N → ∞ η = τ \eta=\tau η = τ ( L τ ∞ f ) ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 ) ) (L_\tau^\infty f)(k)=\tau^{-k}(2f(k)-f(k+1)-f(k-1)) ( L τ ∞ f ) ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 )) τ \tau τ q q q τ = r 2 \tau=r^2 τ = r 2 [40 ] , [21 ] , see also [22 Section 1] .
Theorem 1.1 (Energy bridge on K d K^d K d α > 0 \alpha>0 α > 0 η ≔ Q − α \eta\coloneqq Q^{-\alpha} η : = Q − α σ α ≔ Q − d + α 2 \sigma_\alpha\coloneqq Q^{-\frac{d+\alpha}{2}} σ α : = Q − 2 d + α
The closed forms E D \mathcal{E}_D E D E η ∞ \mathcal{E}_\eta^\infty E η ∞ D r a d α D_{\mathrm{rad}}^\alpha D rad α L η ∞ L_\eta^\infty L η ∞
dom E D = dom E η ∞ = { f ∈ ℓ Z 2 ( w ) : ∑ m ∈ Z Q ( α − d ) m ∣ f ( m + 1 ) − f ( m ) ∣ 2 < ∞ } . \operatorname{dom}\mathcal{E}_D=\operatorname{dom}\mathcal{E}_\eta^\infty=\left\{f\in\ell_Z^2(w):\sum_{m\in\mathbb{Z}}Q^{(\alpha-d)m}\lvert f(m+1)-f(m)\rvert^2<\infty\right\}. dom E D = dom E η ∞ = { f ∈ ℓ Z 2 ( w ) : m ∈ Z ∑ Q ( α − d ) m ∣ f ( m + 1 ) − f ( m ) ∣ 2 < ∞ } . and for any f f f
( 1 + σ α ) − 2 E η ∞ ( f ) ≤ E D ( f ) ≤ ( 1 − σ α ) − 2 E η ∞ ( f ) . (3) (1+\sigma_\alpha)^{-2}\mathcal{E}_\eta^\infty(f)\leq\mathcal{E}_D(f)\leq(1-\sigma_\alpha)^{-2}\mathcal{E}_\eta^\infty(f). \tag*{(3)} ( 1 + σ α ) − 2 E η ∞ ( f ) ≤ E D ( f ) ≤ ( 1 − σ α ) − 2 E η ∞ ( f ) . (3) The constants are optimal.
The operators D r a d α D_{\mathrm{rad}}^\alpha D rad α L η ∞ L_\eta^\infty L η ∞ { Q α m : m ∈ Z } ∪ { 0 } \{Q^{\alpha m}:m\in\mathbb{Z}\}\cup\{0\} { Q α m : m ∈ Z } ∪ { 0 } 0 0 0 W W W ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) K d K^d K d W D r a d α W ∗ = L η ∞ W D_{\mathrm{rad}}^\alpha W^*=L_\eta^\infty W D rad α W ∗ = L η ∞ Q α m Q^{\alpha m} Q α m L η ∞ L_\eta^\infty L η ∞ k ↦ 1 φ 1 ( 0 ; τ ; η , η k + 1 − m ) k\mapsto{}_1\varphi_1(0;\tau;\eta,\eta^{k+1-m}) k ↦ 1 φ 1 ( 0 ; τ ; η , η k + 1 − m ) q q q d α − 1 \frac{d}{\alpha}-1 α d − 1 η \eta η
Let ϑ > 0 \vartheta>0 ϑ > 0 dom E D ⊂ dom ( L Q − ϑ ∞ ) α 2 ϑ \operatorname{dom}\mathcal{E}_D\subset\operatorname{dom}(L_{Q^{-\vartheta}}^\infty)^{\frac{\alpha}{2\vartheta}} dom E D ⊂ dom ( L Q − ϑ ∞ ) 2 ϑ α D r a d α D_{\mathrm{rad}}^\alpha D rad α ( L Q − ϑ ∞ ) α 2 ϑ (L_{Q^{-\vartheta}}^\infty)^{\frac{\alpha}{2\vartheta}} ( L Q − ϑ ∞ ) 2 ϑ α α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ
These assertions are proved in Theorem 4.2 , Theorem 3.6 and Theorem 4.6 . For α = d \alpha=d α = d ( 1 + τ ) − 2 (1+\tau)^{-2} ( 1 + τ ) − 2 ( 1 − τ ) − 2 (1-\tau)^{-2} ( 1 − τ ) − 2 D r a d d D_{\mathrm{rad}}^d D rad d 1 − τ 1-\tau 1 − τ ℓ Z 2 \ell_Z^2 ℓ Z 2 ( I − τ S ) − 1 (I-\tau S)^{-1} ( I − τ S ) − 1 S S S L η L_\eta L η q q q Theorem 4.2 ). The third part has the same threshold α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ ϑ = d \vartheta=d ϑ = d ( L τ ∞ ) α d (L_\tau^\infty)^{\frac{\alpha}{d}} ( L τ ∞ ) d α α < 3 d \alpha<3d α < 3 d
The proofs of the first two parts rest on two simple mechanisms. First, the coefficients of a radial function in the wavelet basis are, up to normalization, the differences between its value on a sphere and its mean on the next ball. This gives a martingale expression of E D \mathcal{E}_D E D Proposition 2.3 ), in which the mean on the ball is the image of the discrete gradient under a weighted Hardy operator ( I − τ S ) − 1 (I-\tau S)^{-1} ( I − τ S ) − 1 E η ∞ \mathcal{E}_\eta^\infty E η ∞ τ S \tau S τ S σ α \sigma_\alpha σ α (3) and its optimality. Secondly, both D r a d α D_{\mathrm{rad}}^\alpha D rad α L η ∞ L^\infty_\eta L η ∞ U U U K d K^d K d U − 1 A U = ρ − 1 A U^{-1}AU=\rho^{-1}A U − 1 A U = ρ − 1 A ρ \rho ρ L η ∞ L^\infty_\eta L η ∞ η \eta η L η ∞ L^\infty_\eta L η ∞ Theorem 3.6 ). The convergence of spectral projections and the covariance then give simple eigenvalues η z \eta^z η z z ∈ Z z\in\mathbb{Z} z ∈ Z q q q [22 Proposition 2.6] for these parameters, and gives the completeness of the corresponding system. The third part uses the dilation covariance once more: the transition coefficients between the two eigenbases depend only on the sum of the indices, so that the comparison of the forms reduces to a convolution operator on Z \mathbb{Z} Z Theorem 4.6 ).
On the quantum side, we work in the concrete model of E r ( 2 ) E_r(2) E r ( 2 ) ℓ Z 2 ⊗ L 2 ( T ) \ell^2_{\mathbb{Z}}\otimes L^2(\mathbb{T}) ℓ Z 2 ⊗ L 2 ( T ) L ∞ ( E r ( 2 ) ) = B ( ℓ Z 2 ) ⊗ ‾ L ∞ ( T ) L^\infty(E_r(2))=B(\ell^2_{\mathbb{Z}})\overline{\otimes}L^\infty(\mathbb{T}) L ∞ ( E r ( 2 )) = B ( ℓ Z 2 ) ⊗ L ∞ ( T ) h h h P m = 1 [ 0 , τ m ] ( n ∗ n ) P_m=\mathbf{1}_{[0,\tau^m]}(n^*n) P m = 1 [ 0 , τ m ] ( n ∗ n ) m ∈ Z m\in\mathbb{Z} m ∈ Z π m O d \pi^m\mathcal{O}^d π m O d Theorem 5.5 . Here J J J ∗ * ∗ F ( ∥ ⋅ ∥ K d ) ↦ F ( n ∗ n ) F(\lVert\cdot\rVert_K^d)\mapsto F(n^*n) F (∥ ⋅ ∥ K d ) ↦ F ( n ∗ n ) n ∗ n n^*n n ∗ n
Theorem 1.2 (Lifting to E r ( 2 ) E_r(2) E r ( 2 ) Suppose that r 2 = Q − d r^2=Q^{-d} r 2 = Q − d α > 0 \alpha>0 α > 0 ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 L ∞ ( E r ( 2 ) ) L^\infty(E_r(2)) L ∞ ( E r ( 2 )) σ \sigma σ T t J = J e − t D r a d α \mathcal{T}_tJ=Je^{-tD^\alpha_{\mathrm{rad}}} T t J = J e − t D rad α t ≥ 0 t\geq0 t ≥ 0
As in [Arh26, Theorem 7.2], this lifting is a probability mixture of conditional expectations, and does not use the coproduct of E r ( 2 ) E_r(2) E r ( 2 ) § 8 . It has radial generator L η ∞ L^\infty_\eta L η ∞ Theorem 1.2 .
Finally, we explain how the whole space appears in the random walk of [Arh26, Section 6]. Recall that the random walk on the dual of S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) [8 Section 4] , which is a Markov chain on { ( n , k ) ∈ N 2 : k ≤ n } \{(n,k)\in\mathbb{N}^2:k\leq n\} {( n , k ) ∈ N 2 : k ≤ n } k ∞ k_\infty k ∞ ν ( n , k ) ( j ) = P ( n , k ) ( k ∞ = j ) \nu_{(n,k)}(j)=\mathbf{P}_{(n,k)}(k_\infty=j) ν ( n , k ) ( j ) = P ( n , k ) ( k ∞ = j ) ϱ ∞ \varrho_\infty ϱ ∞ ϱ d \varrho_d ϱ d ϱ ∞ \varrho_\infty ϱ ∞ k k k O d \mathcal{O}^d O d k → ∞ k\to\infty k → ∞ O d \mathcal{O}^d O d K d K^d K d
Theorem 1.3 (Rescaled Poisson kernel). Let 0 < τ < 1 0<\tau<1 0 < τ < 1 λ ( θ ) ≔ 2 τ ( 1 − cos θ ) 1 − τ 2 \lambda(\theta)\coloneqq\frac{2\tau(1-\cos\theta)}{1-\tau^2} λ ( θ ) : = 1 − τ 2 2 τ ( 1 − c o s θ ) θ ∈ R \theta\in\mathbb{R} θ ∈ R d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 l ∈ Z l\in\mathbb{Z} l ∈ Z
lim k → ∞ ν ( d + k , k ) ( k + l ) = 1 2 π ∫ − π π ϱ d ϱ ∞ ( λ ( θ ) ) e i l θ d θ . \lim_{k\to\infty}\nu_{(d+k,k)}(k+l)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\varrho_d}{\varrho_\infty}(\lambda(\theta))e^{il\theta}\,\mathrm{d}\theta. k → ∞ lim ν ( d + k , k ) ( k + l ) = 2 π 1 ∫ − π π ϱ ∞ ϱ d ( λ ( θ )) e i l θ d θ . For d = 0 d=0 d = 0 ( τ ; τ ) ∞ 2 ∑ m ≥ 0 τ ∣ l ∣ + 2 m ( τ ; τ ) m ( τ ; τ ) m + ∣ l ∣ (\tau;\tau)^2_\infty\sum_{m\geq0}\frac{\tau^{|l|+2m}}{(\tau;\tau)_m(\tau;\tau)_{m+|l|}} ( τ ; τ ) ∞ 2 ∑ m ≥ 0 ( τ ; τ ) m ( τ ; τ ) m + ∣ l ∣ τ ∣ l ∣ + 2 m O ( τ k ) O(\tau^k) O ( τ k ) l l l
∑ l ∈ Z ( lim k → ∞ ν ( k , k ) ( k + l ) ) e i l θ = ( τ ; τ ) ∞ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 2 , θ ∈ R . \sum_{l\in\mathbb{Z}}\left(\lim_{k\to\infty}\nu_{(k,k)}(k+l)\right)e^{il\theta}
=\frac{(\tau;\tau)^2_\infty}{\lvert(\tau e^{i\theta};\tau)_\infty\rvert^2},\qquad\theta\in\mathbb{R}. l ∈ Z ∑ ( k → ∞ lim ν ( k , k ) ( k + l ) ) e i l θ = ∣( τ e i θ ; τ ) ∞ ∣ 2 ( τ ; τ ) ∞ 2 , θ ∈ R . This is proved in Theorem 6.1 . The limit operator is the translation-invariant horizontal network on Z \mathbb{Z} Z L τ ∞ L_{\tau}^{\infty} L τ ∞ z z z z + z − 1 = 2 − ( 1 − τ 2 ) λ τ z+z^{-1}=2-\frac{(1-\tau^{2})\lambda}{\tau} z + z − 1 = 2 − τ ( 1 − τ 2 ) λ e i θ e^{i\theta} e i θ
The correspondences in this paper have several distinct levels. The Haar identity is an identification of radial measures, and the energy bridge is an equivalence of closed forms with a common domain. In the spectral theorem the two radial operators are, more strongly, unitarily equivalent, whereas the Taibleson lifting and the quotient Casimir dynamics are different completely positive constructions. The contraction results in § 9 concern strong resolvent convergence on fixed angular bands, Mosco convergence of the full forms, and point-norm convergence of compressed completely positive maps on the quotient algebra. They do not assert global operator-norm convergence of the Hilbert-space operators.
Approach of the paper The paper follows the structure of [Arh26], and we refer to [Arh26] for the compact case. The two new phenomena are the following. Firstly, the whole space is invariant under dilations, and every operator we consider is covariant under the corresponding unitary U U U Z \mathbb{Z} Z L η ∞ L_{\eta}^{\infty} L η ∞ α > d 2 \alpha>\frac{d}{2} α > 2 d Proposition 3.1 ). At the infinity of K d K^{d} K d Lemma 4.4 ).
Structure of the paper Section 2 gives the radial analysis of the Taibleson operator on K d K^{d} K d § 3 studies the bilateral operators L η ∞ L_{\eta}^{\infty} L η ∞ § 4 proves Theorem 1.1 . § 6 recalls the quantum group E r ( 2 ) E_{r}(2) E r ( 2 ) (1) and Theorem 1.2 . § 7 proves Theorem 1.3 . § 8 identifies the quantum plane and the closed Casimir and constructs its covariant completely positive heat semigroup. § 9 proves the nonradial contraction, the Mosco limit and point-norm convergence of the quantum Markov dynamics.
Radial analysis on the whole space
Parameters and radial functions
We use the conventions of [Arh26, Section 2.1]. Let K K K O \mathcal{O} O π \pi π Q ≔ # ( O / π O ) Q \coloneqq\#(\mathcal{O}/\pi\mathcal{O}) Q : = # ( O / π O ) ∣ π ∣ K = Q − 1 \lvert\pi\rvert_{K}=Q^{-1} ∣ π ∣ K = Q − 1 K d K^{d} K d d x \mathrm{d}x d x μ ( O d ) = 1 \mu(\mathcal{O}^{d})=1 μ ( O d ) = 1 χ \chi χ K K K O \mathcal{O} O O \mathcal{O} O f ^ ( ξ ) ≔ ∫ f ( x ) χ ( x ⋅ ξ ) d x \widehat{f}(\xi) \coloneqq\int f(x)\chi(x\cdot\xi)\,\mathrm{d}x f ( ξ ) : = ∫ f ( x ) χ ( x ⋅ ξ ) d x [36 Chapters I and II] . Throughout the paper, α > 0 \alpha>0 α > 0
τ ≔ Q − d , ρ ≔ Q α , η ≔ Q − α = ρ − 1 , r ≔ τ 1 2 , c ≔ ρ − 1 1 − τ ρ , σ α ≔ Q − d + α 2 = ( τ ρ ) 1 2 . (4) \tau\coloneqq Q^{-d},\qquad\rho\coloneqq Q^{\alpha},\qquad\eta\coloneqq Q^{-\alpha}=\rho^{-1},\qquad r \coloneqq\tau^{\frac{1}{2}},\qquad c \coloneqq\frac{\rho-1}{1-\frac{\tau}{\rho}},\qquad\sigma_{\alpha}\coloneqq Q^{-\frac{d+\alpha}{2}}=\left(\frac{\tau}{\rho}\right)^{\frac{1}{2}}. \tag*{(4)} τ : = Q − d , ρ : = Q α , η : = Q − α = ρ − 1 , r : = τ 2 1 , c : = 1 − ρ τ ρ − 1 , σ α : = Q − 2 d + α = ( ρ τ ) 2 1 . (4) In § 3 , the letters τ \tau τ η \eta η ( 0 , 1 ) (0,1) ( 0 , 1 ) m ∈ Z m\in\mathbb{Z} m ∈ Z B m ≔ π m O d B_{m}\coloneqq\pi^{m}\mathcal{O}^{d} B m : = π m O d S m ≔ B m ∖ B m + 1 S_{m}\coloneqq B_{m}\setminus B_{m+1} S m : = B m ∖ B m + 1
μ ( B m ) = τ m , μ ( S m ) = w m ≔ ( 1 − τ ) τ m , 1 ^ B m = τ m 1 B − m . (5) \mu(B_{m})=\tau^{m},\qquad\mu(S_{m})=w_{m}\coloneqq(1-\tau)\tau^{m},\qquad\widehat{\mathbf{1}}_{B_{m}}=\tau^{m}\mathbf{1}_{B_{-m}}. \tag*{(5)} μ ( B m ) = τ m , μ ( S m ) = w m : = ( 1 − τ ) τ m , 1 B m = τ m 1 B − m . (5) A radial function on K d K^d K d ∥ x ∥ K \lVert x\rVert_K ∥ x ∥ K S m S_m S m GL d ( O ) \operatorname{GL}_d(\mathcal{O}) GL d ( O ) f f f ( f m ) m ∈ Z (f_m)_{m\in\mathbb{Z}} ( f m ) m ∈ Z f ( m ) = f m f(m)=f_m f ( m ) = f m L r a d 2 ( K d ) L^2_{\mathrm{rad}}(K^d) L rad 2 ( K d ) ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) ⟨ f , g ⟩ = ∑ m ∈ Z w m f m g m ‾ \langle f,g\rangle=\sum_{m\in\mathbb{Z}}w_m f_m\overline{g_m} ⟨ f , g ⟩ = ∑ m ∈ Z w m f m g m ∑ m ≤ 0 w m = ∞ \sum_{m\leq0}w_m=\infty ∑ m ≤ 0 w m = ∞ ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) f : Z → C f:\mathbb{Z}\to\mathbb{C} f : Z → C
( Δ f ) m = d e f f m + 1 − f m , m ∈ Z . (\Delta f)_m \mathrel{\mathop{=}^{\mathrm{def}}} f_{m+1}-f_m,\qquad m\in\mathbb{Z}. ( Δ f ) m = def f m + 1 − f m , m ∈ Z . For any f ∈ ℓ Z 2 ( w ) f\in\ell^2_{\mathbb{Z}}(w) f ∈ ℓ Z 2 ( w ) m ∈ Z m\in\mathbb{Z} m ∈ Z f f f B m B_m B m
m m ( f ) = d e f 1 μ ( B m ) ∫ B m f = ( 1 − τ ) ∑ i ≥ 0 τ i f m + i . (6) m_m(f)\mathrel{\mathop{=}^{\mathrm{def}}}\frac{1}{\mu(B_m)}\int_{B_m}f=(1-\tau)\sum_{i\geq0}\tau^i f_{m+i}. \tag*{(6)} m m ( f ) = def μ ( B m ) 1 ∫ B m f = ( 1 − τ ) i ≥ 0 ∑ τ i f m + i . (6) The series converges absolutely: indeed w k ∣ f k ∣ 2 ≤ ∥ f ∥ 2 w_k|f_k|^2\leq\lVert f\rVert^2 w k ∣ f k ∣ 2 ≤ ∥ f ∥ 2 ∣ f k ∣ ≤ ( 1 − τ ) − 1 2 τ − k 2 ∥ f ∥ |f_k|\leq(1-\tau)^{-\frac{1}{2}}\tau^{-\frac{k}{2}}\lVert f\rVert ∣ f k ∣ ≤ ( 1 − τ ) − 2 1 τ − 2 k ∥ f ∥ τ i ∣ f m + i ∣ ≤ ( 1 − τ ) − 1 2 τ − m 2 τ i 2 ∥ f ∥ \tau^i|f_{m+i}|\leq(1-\tau)^{-\frac{1}{2}}\tau^{-\frac{m}{2}}\tau^{\frac{i}{2}}\lVert f\rVert τ i ∣ f m + i ∣ ≤ ( 1 − τ ) − 2 1 τ − 2 m τ 2 i ∥ f ∥
We also use the dilation. The operator U U U ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w )
( U f ) m = d e f τ 1 2 f m + 1 , m ∈ Z , (7) (Uf)_m\mathrel{\mathop{=}^{\mathrm{def}}}\tau^{\frac{1}{2}}f_{m+1},\qquad m\in\mathbb{Z}, \tag*{(7)} ( U f ) m = def τ 2 1 f m + 1 , m ∈ Z , (7) is unitary, since ∑ m w m τ ∣ f m + 1 ∣ 2 = ∑ m w m + 1 ∣ f m + 1 ∣ 2 \sum_m w_m\tau|f_{m+1}|^2=\sum_m w_{m+1}|f_{m+1}|^2 ∑ m w m τ ∣ f m + 1 ∣ 2 = ∑ m w m + 1 ∣ f m + 1 ∣ 2 π S m = S m + 1 \pi S_m=S_{m+1} π S m = S m + 1 F ↦ τ 1 2 F ( π ⋅ ) F\mapsto\tau^{\frac{1}{2}}F(\pi\cdot) F ↦ τ 2 1 F ( π ⋅ ) L 2 ( K d ) L^2(K^d) L 2 ( K d )
The radial Taibleson operator
The Taibleson operator D α D^\alpha D α α \alpha α D α = F − 1 M ∥ ξ ∥ K α F D^\alpha=\mathcal{F}^{-1}M_{\lVert\xi\rVert_K^\alpha}\mathcal{F} D α = F − 1 M ∥ ξ ∥ K α F L 2 ( K d ) L^2(K^d) L 2 ( K d )
D α f ( x ) = c ∫ K d f ( x ) − f ( y ) ∥ x − y ∥ K d + α d y . (8) D^\alpha f(x)=c\int_{K^d}\frac{f(x)-f(y)}{\lVert x-y\rVert_K^{d+\alpha}}\,dy. \tag*{(8)} D α f ( x ) = c ∫ K d ∥ x − y ∥ K d + α f ( x ) − f ( y ) d y . (8) with the constant c c c (4) , see [Arh26, Section 2.2], [3 (1.2)–(1.4)] and [20 Section 2.2] . For any integer m ∈ Z m\in\mathbb{Z} m ∈ Z
g m = d e f 1 B m + 1 − τ 1 B m s m , s m = d e f ( ( 1 − τ ) τ m + 1 ) 1 2 . (9) g_m\mathrel{\mathop{=}^{\mathrm{def}}}\frac{\mathbf{1}_{B_{m+1}}-\tau\mathbf{1}_{B_m}}{s_m},\qquad s_m\mathrel{\mathop{=}^{\mathrm{def}}}\left((1-\tau)\tau^{m+1}\right)^{\frac{1}{2}}. \tag*{(9)} g m = def s m 1 B m + 1 − τ 1 B m , s m = def ( ( 1 − τ ) τ m + 1 ) 2 1 . (9) It takes the values 0 0 0 S k S_k S k k < m k<m k < m − τ s m -\frac{\tau}{s_m} − s m τ S m S_m S m 1 − τ s m \frac{1-\tau}{s_m} s m 1 − τ B m + 1 B_{m+1} B m + 1 g n g_n g n n ∈ Z n\in\mathbb{Z} n ∈ Z
Proposition 2.1 The family ( g m ) m ∈ Z (g_m)_{m\in\mathbb{Z}} ( g m ) m ∈ Z ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) m ∈ Z m\in\mathbb{Z} m ∈ Z g m g_m g m dom D α \operatorname{dom}D^\alpha dom D α D α g m = ρ m + 1 g m D^\alpha g_m=\rho^{m+1}g_m D α g m = ρ m + 1 g m L r a d 2 ( K d ) L^2_{\mathrm{rad}}(K^d) L rad 2 ( K d ) D α D^\alpha D α D r a d α D^\alpha_{\mathrm{rad}} D rad α D α D^\alpha D α
D r a d α f = ∑ m ∈ Z ρ m + 1 ⟨ f , g m ⟩ g m , dom D r a d α = { f ∈ ℓ Z 2 ( w ) : ∑ m ∈ Z ρ 2 ( m + 1 ) ∣ ⟨ f , g m ⟩ ∣ 2 < ∞ } . D^\alpha_{\mathrm{rad}}f=\sum_{m\in\mathbb{Z}}\rho^{m+1}\langle f,g_m\rangle g_m,\qquad\operatorname{dom}D^\alpha_{\mathrm{rad}}=\left\{f\in\ell^2_{\mathbb{Z}}(w):\sum_{m\in\mathbb{Z}}\rho^{2(m+1)}|\langle f,g_m\rangle|^2<\infty\right\}. D rad α f = m ∈ Z ∑ ρ m + 1 ⟨ f , g m ⟩ g m , dom D rad α = { f ∈ ℓ Z 2 ( w ) : m ∈ Z ∑ ρ 2 ( m + 1 ) ∣ ⟨ f , g m ⟩ ∣ 2 < ∞ } . Its spectrum is { ρ m : m ∈ Z } ∪ { 0 } \{\rho^m:m\in\mathbb{Z}\}\cup\{0\} { ρ m : m ∈ Z } ∪ { 0 } ρ m \rho^m ρ m 0 0 0 U D r a d α U − 1 = ρ D r a d α UD^\alpha_{\mathrm{rad}}U^{-1}=\rho D^\alpha_{\mathrm{rad}} U D rad α U − 1 = ρ D rad α U g m = g m − 1 Ug_m=g_{m-1} U g m = g m − 1 m ∈ Z m\in\mathbb{Z} m ∈ Z
Proof. By (5) , the Fourier transform of the numerator of g m g_m g m τ m + 1 1 B − m − 1 − τ m 1 B − m = τ m + 1 1 S − m − 1 \tau^{m+1}1_{B_{-m-1}}-\tau^m1_{B_{-m}}=\tau^{m+1}1_{S_{-m-1}} τ m + 1 1 B − m − 1 − τ m 1 B − m = τ m + 1 1 S − m − 1 ∥ ξ ∥ K α \lVert\xi\rVert_K^\alpha ∥ ξ ∥ K α Q α ( m + 1 ) = ρ m + 1 Q^{\alpha(m+1)}=\rho^{m+1} Q α ( m + 1 ) = ρ m + 1 g m ∈ dom D α g_m\in\operatorname{dom}D^\alpha g m ∈ dom D α D α g m = ρ m + 1 g m D^\alpha g_m=\rho^{m+1}g_m D α g m = ρ m + 1 g m g m g_m g m ( 1 − τ ) τ m + 1 (1-\tau)\tau^{m+1} ( 1 − τ ) τ m + 1 m < m ′ m<m' m < m ′ g m g_m g m B m + 1 ⊃ B m ′ B_{m+1}\supset B_{m'} B m + 1 ⊃ B m ′ g m ′ g_{m'} g m ′
We show that it is complete. Let f ∈ ℓ Z 2 ( w ) f\in\ell^2_{\mathbb{Z}}(w) f ∈ ℓ Z 2 ( w )
⟨ f , g m ⟩ = 1 s m [ ( 1 − τ ) ∑ k ≥ m + 1 w k f k − τ w m f m ] = ( 1 − τ ) τ m + 1 s m ( m m + 1 ( f ) − f m ) = s m ( m m + 1 ( f ) − f m ) , (10) \begin{aligned}
\langle f,g_m\rangle=\frac{1}{s_m}\left[(1-\tau)\sum_{k\ge m+1}w_kf_k-\tau w_mf_m\right]=\frac{(1-\tau)\tau^{m+1}}{s_m}\bigl(m_{m+1}(f)-f_m\bigr) \\
&=s_m\bigl(m_{m+1}(f)-f_m\bigr), \tag*{(10)}
\end{aligned} ⟨ f , g m ⟩ = s m 1 [ ( 1 − τ ) k ≥ m + 1 ∑ w k f k − τ w m f m ] = s m ( 1 − τ ) τ m + 1 ( m m + 1 ( f ) − f m ) = s m ( m m + 1 ( f ) − f m ) , (10) where we used ∑ k ≥ m + 1 w k f k = τ m + 1 m m + 1 ( f ) \sum_{k\ge m+1}w_kf_k=\tau^{m+1}m_{m+1}(f) ∑ k ≥ m + 1 w k f k = τ m + 1 m m + 1 ( f ) (6) . Moreover, the absolute convergence of the series (6) gives, for any m ∈ Z m\in\mathbb{Z} m ∈ Z
m m + 1 ( f ) − f m = ( 1 − τ ) ∑ i ≥ 1 τ i − 1 f m + i − f m = ∑ i ≥ 0 τ i ( Δ f ) m + i . (11) m_{m+1}(f)-f_m=(1-\tau)\sum_{i\ge1}\tau^{i-1}f_{m+i}-f_m=\sum_{i\ge0}\tau^i(\Delta f)_{m+i}. \tag*{(11)} m m + 1 ( f ) − f m = ( 1 − τ ) i ≥ 1 ∑ τ i − 1 f m + i − f m = i ≥ 0 ∑ τ i ( Δ f ) m + i . (11) Indeed, the series ∑ i ≥ 0 τ i ( f m + i + 1 − f m + i ) \sum_{i\ge0}\tau^i(f_{m+i+1}-f_{m+i}) ∑ i ≥ 0 τ i ( f m + i + 1 − f m + i ) ∑ i ≥ 0 τ i f m + i + 1 − ∑ i ≥ 0 τ i f m + i = ∑ i ≥ 1 ( τ i − 1 − τ i ) f m + i − f m \sum_{i\ge0}\tau^if_{m+i+1}-\sum_{i\ge0}\tau^if_{m+i}=\sum_{i\ge1}(\tau^{i-1}-\tau^i)f_{m+i}-f_m ∑ i ≥ 0 τ i f m + i + 1 − ∑ i ≥ 0 τ i f m + i = ∑ i ≥ 1 ( τ i − 1 − τ i ) f m + i − f m f f f g m g_m g m h m : = m m + 1 ( f ) − f m h_m\mathrel{\mathop:}=m_{m+1}(f)-f_m h m : = m m + 1 ( f ) − f m h = 0 h=0 h = 0 (10) . On the other hand, (11) gives h m − τ h m + 1 = ( Δ f ) m h_m-\tau h_{m+1}=(\Delta f)_m h m − τ h m + 1 = ( Δ f ) m m ∈ Z m\in\mathbb{Z} m ∈ Z Δ f = 0 \Delta f=0 Δ f = 0 f f f f = 0 f=0 f = 0
Each g m g_m g m D α D^\alpha D α E ⊂ R E\subset\mathbb{R} E ⊂ R 1 E ( D α ) 1_E(D^\alpha) 1 E ( D α ) g m g_m g m 1 E ( ρ m + 1 ) g m 1_E(\rho^{m+1})g_m 1 E ( ρ m + 1 ) g m L r a d 2 ( K d ) L^2_{\mathrm{rad}}(K^d) L rad 2 ( K d ) D α D^\alpha D α D r a d α D^\alpha_{\mathrm{rad}} D rad α ( g m ) (g_m) ( g m ) ρ m + 1 \rho^{m+1} ρ m + 1 ρ m + 1 → 0 \rho^{m+1}\to0 ρ m + 1 → 0 m → − ∞ m\to-\infty m → − ∞ { ρ m : m ∈ Z } ∪ { 0 } \{\rho^m:m\in\mathbb{Z}\}\cup\{0\} { ρ m : m ∈ Z } ∪ { 0 } 0 0 0 ( g m ) (g_m) ( g m ) k ∈ Z k\in\mathbb{Z} k ∈ Z
( U g m ) k = τ 1 2 g m ( k + 1 ) = τ 1 2 s m ( 1 k ≥ m − τ 1 k ≥ m − 1 ) = τ 1 2 s m − 1 s m g m − 1 ( k ) = g m − 1 ( k ) , (Ug_m)_k=\tau^{\frac12}g_m(k+1)=\frac{\tau^{\frac12}}{s_m}(1_{k\ge m}-\tau1_{k\ge m-1})=\frac{\tau^{\frac12}s_{m-1}}{s_m}g_{m-1}(k)=g_{m-1}(k), ( U g m ) k = τ 2 1 g m ( k + 1 ) = s m τ 2 1 ( 1 k ≥ m − τ 1 k ≥ m − 1 ) = s m τ 2 1 s m − 1 g m − 1 ( k ) = g m − 1 ( k ) , since s m − 1 = τ − 1 2 s m s_{m-1}=\tau^{-\frac12}s_m s m − 1 = τ − 2 1 s m U D r a d α U − 1 g m − 1 = U D r a d α g m = ρ m + 1 g m − 1 = ρ D r a d α g m − 1 UD^\alpha_{\mathrm{rad}}U^{-1}g_{m-1}=UD^\alpha_{\mathrm{rad}}g_m=\rho^{m+1}g_{m-1}=\rho D^\alpha_{\mathrm{rad}}g_{m-1} U D rad α U − 1 g m − 1 = U D rad α g m = ρ m + 1 g m − 1 = ρ D rad α g m − 1 U D r a d α U − 1 = ρ D r a d α UD^\alpha_{\mathrm{rad}}U^{-1}=\rho D^\alpha_{\mathrm{rad}} U D rad α U − 1 = ρ D rad α
Remark 2.2 The functions g m g_m g m p p p [23 ] , and the spectral decomposition of isotropic Markov generators on ultrametric spaces in such bases is studied in detail in [BGP+14]. On the unit ball, the analogous statement is [Arh26, Theorem 2.3], where the eigenvalues of the regional operator are ρ m + 1 − κ \rho^{m+1}-\kappa ρ m + 1 − κ
A martingale formula for the energy
We denote by E D \mathcal{E}_D E D D r a d α D^\alpha_{\mathrm{rad}} D rad α E D ( f ) = ∥ ( D r a d α ) 1 2 f ∥ 2 = ∑ m ρ m + 1 ∣ ⟨ f , g m ⟩ ∣ 2 \mathcal{E}_D(f)=\lVert(D^\alpha_{\mathrm{rad}})^{\frac12}f\rVert^2=\sum_m\rho^{m+1}|\langle f,g_m\rangle|^2 E D ( f ) = ∥( D rad α ) 2 1 f ∥ 2 = ∑ m ρ m + 1 ∣ ⟨ f , g m ⟩ ∣ 2 dom E D = dom ( D r a d α ) 1 2 \operatorname{dom}\mathcal{E}_D=\operatorname{dom}(D^\alpha_{\mathrm{rad}})^{\frac12} dom E D = dom ( D rad α ) 2 1 E D \mathcal{E}_D E D ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
w ‾ m : = ( ρ τ ) m + 1 = Q ( α − d ) ( m + 1 ) , m ∈ Z , (12) \overline{w}_m\mathrel{\mathop:}=(\rho\tau)^{m+1}=Q^{(\alpha-d)(m+1)},\qquad m\in\mathbb{Z}, \tag*{(12)} w m : = ( ρ τ ) m + 1 = Q ( α − d ) ( m + 1 ) , m ∈ Z , (12) and the shift S S S ( S u ) m ≔ u m + 1 (Su)_m \coloneqq u_{m+1} ( S u ) m : = u m + 1
Proposition 2.3 For any f ∈ ℓ Z 2 ( w ) f \in\ell^2_{\mathbb{Z}}(w) f ∈ ℓ Z 2 ( w ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
E D ( f ) = ( 1 − τ ) ∑ m ∈ Z ϖ m ∣ f m − m m + 1 ( f ) ∣ 2 = ( 1 − τ ) ∑ m ∈ Z ϖ m ∣ ∑ i ≥ 0 τ i ( Δ f ) m + i ∣ 2 (13) \mathcal{E}_{D}(f)=(1-\tau)\sum_{m\in\mathbb{Z}}\varpi_m\lvert f_m-m_{m+1}(f)\rvert^2=(1-\tau)\sum_{m\in\mathbb{Z}}\varpi_m\left\lvert\sum_{i\geq0}\tau^i(\Delta f)_{m+i}\right\rvert^2 \tag*{(13)} E D ( f ) = ( 1 − τ ) m ∈ Z ∑ ϖ m ∣ f m − m m + 1 ( f ) ∣ 2 = ( 1 − τ ) m ∈ Z ∑ ϖ m i ≥ 0 ∑ τ i ( Δ f ) m + i 2 (13) Proof. By (10) , we have ρ m + 1 ∣ ⟨ f , g m ⟩ ∣ 2 = ρ m + 1 ( 1 − τ ) τ m + 1 ∣ m m + 1 ( f ) − f m ∣ 2 = ( 1 − τ ) ϖ m ∣ m m + 1 ( f ) − f m ∣ 2 \rho^{m+1}\lvert\langle f,g_m\rangle\rvert^2=\rho^{m+1}(1-\tau)\tau^{m+1}\lvert m_{m+1}(f)-f_m\rvert^2=(1-\tau)\varpi_m\lvert m_{m+1}(f)-f_m\rvert^2 ρ m + 1 ∣⟨ f , g m ⟩ ∣ 2 = ρ m + 1 ( 1 − τ ) τ m + 1 ∣ m m + 1 ( f ) − f m ∣ 2 = ( 1 − τ ) ϖ m ∣ m m + 1 ( f ) − f m ∣ 2 m m m (11) gives the second one.
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Remark 2.4 The form E D \mathcal{E}_{D} E D
E D ( f ) = c 2 ∑ m ≠ k w m w k Q ( d + α ) min ( m , k ) ∣ f m − f k ∣ 2 = c ( 1 − τ ) 2 ∑ j ≥ 1 τ j ∑ m ∈ Z ( ρ τ ) m ∣ f m + j − f m ∣ 2 . \mathcal{E}_{D}(f)=\frac{c}{2}\sum_{m\ne k}w_mw_kQ^{(d+\alpha)\min(m,k)}\lvert f_m-f_k\rvert^2=c(1-\tau)^2\sum_{j\geq1}\tau^j\sum_{m\in\mathbb{Z}}(\rho\tau)^m\lvert f_{m+j}-f_m\rvert^2. E D ( f ) = 2 c m = k ∑ w m w k Q ( d + α ) m i n ( m , k ) ∣ f m − f k ∣ 2 = c ( 1 − τ ) 2 j ≥ 1 ∑ τ j m ∈ Z ∑ ( ρ τ ) m ∣ f m + j − f m ∣ 2 . for any f ∈ D f\in\mathcal{D} f ∈ D § 3 . This is the bilateral version of [Arh26, (2.11)]. Indeed, such a function f f f (8) and a symmetrization give ⟨ D α f , f ⟩ = c 2 ∬ ∣ f ( x ) − f ( y ) ∣ 2 ∥ x − y ∥ K d + α d x d y \langle D^\alpha f,f\rangle=\frac{c}{2}\iint\frac{\lvert f(x)-f(y)\rvert^2}{\lVert x-y\rVert_K^{d+\alpha}}\,dx\,dy ⟨ D α f , f ⟩ = 2 c ∬ ∥ x − y ∥ K d + α ∣ f ( x ) − f ( y ) ∣ 2 d x d y x ∈ S m x\in S_m x ∈ S m y ∈ S k y\in S_k y ∈ S k m ≠ k m\ne k m = k ∥ x − y ∥ K = Q − min ( m , k ) \lVert x-y\rVert_K=Q^{-\min(m,k)} ∥ x − y ∥ K = Q − m i n ( m , k ) S m S_m S m w m w k Q ( d + α ) m = ( 1 − τ ) 2 ρ m τ k w_mw_kQ^{(d+\alpha)m}=(1-\tau)^2\rho^m\tau^k w m w k Q ( d + α ) m = ( 1 − τ ) 2 ρ m τ k m < k m<k m < k (13) .
Bilateral birth–death operators and Hahn–Exton q q q
In this section, 0 < τ < 1 0<\tau<1 0 < τ < 1 0 < η < 1 0<\eta<1 0 < η < 1 w k = ( 1 − τ ) τ k w_k=(1-\tau)\tau^k w k = ( 1 − τ ) τ k k ∈ Z k\in\mathbb{Z} k ∈ Z ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) ( a ; η ) n ≔ ∏ i = 0 n − 1 ( 1 − a η i ) (a;\eta)_n\coloneqq\prod_{i=0}^{n-1}(1-a\eta^i) ( a ; η ) n : = ∏ i = 0 n − 1 ( 1 − a η i ) n ∈ N 0 ∪ { ∞ } n\in\mathbb{N}_0\cup\{\infty\} n ∈ N 0 ∪ { ∞ }
1 φ 1 ( 0 ; b ; η , x ) ≔ ∑ j ≥ 0 ( − 1 ) j η j ( j − 1 ) 2 ( η ; η ) j ( b ; η ) j x j , x ∈ C , (14) {}_1\varphi_1(0;b;\eta,x)\coloneqq\sum_{j\geq0}\frac{(-1)^j\eta^{\frac{j(j-1)}{2}}}{(\eta;\eta)_j(b;\eta)_j}x^j,\qquad x\in\mathbb{C}, \tag*{(14)} 1 φ 1 ( 0 ; b ; η , x ) : = j ≥ 0 ∑ ( η ; η ) j ( b ; η ) j ( − 1 ) j η 2 j ( j − 1 ) x j , x ∈ C , (14) of [15 Section 1.2] , see also [22 (2.1)] and [12 (17.4.1)] , which defines an entire function of x x x b ∈ C ∖ { η − i : i ∈ N 0 } b\in\mathbb{C}\setminus\{\eta^{-i}:i\in\mathbb{N}_0\} b ∈ C ∖ { η − i : i ∈ N 0 }
The closed form and the operator
For any k ∈ Z k\in\mathbb{Z} k ∈ Z
β k ≔ τ η − k − 1 , δ k ≔ η − k , c k ≔ w k β k = w k + 1 δ k + 1 = ( 1 − τ ) ( τ η ) k + 1 . (15) \beta_k\coloneqq\tau\eta^{-k-1},\qquad\delta_k\coloneqq\eta^{-k},\qquad c_k\coloneqq w_k\beta_k=w_{k+1}\delta_{k+1}=(1-\tau)\left(\frac{\tau}{\eta}\right)^{k+1}. \tag*{(15)} β k : = τ η − k − 1 , δ k : = η − k , c k : = w k β k = w k + 1 δ k + 1 = ( 1 − τ ) ( η τ ) k + 1 . (15) For any function f : Z → C f:\mathbb{Z}\to\mathbb{C} f : Z → C L f : Z → C Lf:\mathbb{Z}\to\mathbb{C} L f : Z → C
( L f ) ( k ) ≔ β k ( f ( k ) − f ( k + 1 ) ) + δ k ( f ( k ) − f ( k − 1 ) ) , k ∈ Z . (16) (Lf)(k)\coloneqq\beta_k\bigl(f(k)-f(k+1)\bigr)+\delta_k\bigl(f(k)-f(k-1)\bigr),\qquad k\in\mathbb{Z}. \tag*{(16)} ( L f ) ( k ) : = β k ( f ( k ) − f ( k + 1 ) ) + δ k ( f ( k ) − f ( k − 1 ) ) , k ∈ Z . (16) Equivalently, w k ( L f ) ( k ) = c k − 1 ( Δ f ) k − 1 − c k ( Δ f ) k w_k(Lf)(k)=c_{k-1}(\Delta f)_{k-1}-c_k(\Delta f)_k w k ( L f ) ( k ) = c k − 1 ( Δ f ) k − 1 − c k ( Δ f ) k f , g : Z → C f,g:\mathbb{Z}\to\mathbb{C} f , g : Z → C g g g
∑ k ∈ Z w k ( L g ) ( k ) f ( k ) ‾ = ∑ k ∈ Z c k ( Δ g ) k ( Δ f ) k ‾ = ∑ k ∈ Z w k g ( k ) ( L f ) ( k ) ‾ . (17) \sum_{k\in\mathbb{Z}}w_k(Lg)(k)\overline{f(k)}
=\sum_{k\in\mathbb{Z}}c_k(\Delta g)_k\overline{(\Delta f)_k}
=\sum_{k\in\mathbb{Z}}w_kg(k)\overline{(Lf)(k)}. \tag*{(17)} k ∈ Z ∑ w k ( Lg ) ( k ) f ( k ) = k ∈ Z ∑ c k ( Δ g ) k ( Δ f ) k = k ∈ Z ∑ w k g ( k ) ( L f ) ( k ) . (17) For any f ∈ ℓ Z 2 ( w ) f\in\ell_Z^2(w) f ∈ ℓ Z 2 ( w )
E η ∞ ( f ) ≔ d e f ∑ k ∈ Z c k ∣ f ( k + 1 ) − f ( k ) ∣ 2 ∈ [ 0 , ∞ ] , dom E η ∞ ≔ d e f { f ∈ ℓ Z 2 ( w ) : E η ∞ ( f ) < ∞ } . (18) \mathcal{E}^{\infty}_{\eta}(f)\mathrel{\mathop{\coloneqq}^{\mathrm{def}}}\sum_{k\in\mathbb{Z}}c_k\lvert f(k+1)-f(k)\rvert^2\in[0,\infty],\qquad
\operatorname{dom}\mathcal{E}^{\infty}_{\eta}\mathrel{\mathop{\coloneqq}^{\mathrm{def}}}\left\{f\in\ell_Z^2(w):\mathcal{E}^{\infty}_{\eta}(f)<\infty\right\}. \tag*{(18)} E η ∞ ( f ) : = def k ∈ Z ∑ c k ∣ f ( k + 1 ) − f ( k ) ∣ 2 ∈ [ 0 , ∞ ] , dom E η ∞ : = def { f ∈ ℓ Z 2 ( w ) : E η ∞ ( f ) < ∞ } . (18) and we denote by E η ∞ ( f , g ) \mathcal{E}^{\infty}_{\eta}(f,g) E η ∞ ( f , g ) D \mathcal{D} D f : Z → C f:\mathbb{Z}\to\mathbb{C} f : Z → C M ≥ 0 M\geq0 M ≥ 0 f ( k ) = 0 f(k)=0 f ( k ) = 0 k ≤ − M k\leq-M k ≤ − M f f f { k ≥ M } \{k\geq M\} { k ≥ M } D \mathcal{D} D j ∈ Z j\in\mathbb{Z} j ∈ Z 1 ≥ j ≔ d e f 1 { k ≥ j } \mathbf{1}_{\geq j}\mathrel{\mathop{\coloneqq}^{\mathrm{def}}}\mathbf{1}_{\{k\geq j\}} 1 ≥ j : = def 1 { k ≥ j } D \mathcal{D} D
Proposition 3.1 1. The space D \mathcal{D} D ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) dom E η ∞ \operatorname{dom}\mathcal{E}^{\infty}_{\eta} dom E η ∞ g ∈ D g\in\mathcal{D} g ∈ D L g Lg Lg f ∈ dom E η ∞ f\in\operatorname{dom}\mathcal{E}^{\infty}_{\eta} f ∈ dom E η ∞
E η ∞ ( f , g ) = ⟨ f , L g ⟩ ℓ Z 2 ( w ) . (19) \mathcal{E}^{\infty}_{\eta}(f,g)=\langle f,Lg\rangle_{\ell_Z^2(w)}. \tag*{(19)} E η ∞ ( f , g ) = ⟨ f , Lg ⟩ ℓ Z 2 ( w ) . (19) The form E η ∞ \mathcal{E}^{\infty}_{\eta} E η ∞ D \mathcal{D} D
The restriction of L L L D \mathcal{D} D ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) L η ∞ L^{\infty}_{\eta} L η ∞ E η ∞ \mathcal{E}^{\infty}_{\eta} E η ∞
For any f ∈ dom L η ∞ f\in\operatorname{dom}L^{\infty}_{\eta} f ∈ dom L η ∞ L η ∞ f = L f L^{\infty}_{\eta}f=Lf L η ∞ f = L f lim K → + ∞ c K ( f ( K + 1 ) − f ( K ) ) = 0 \lim_{K\to+\infty}c_K\bigl(f(K+1)-f(K)\bigr)=0 lim K → + ∞ c K ( f ( K + 1 ) − f ( K ) ) = 0
The kernel of L η ∞ L^{\infty}_{\eta} L η ∞ { 0 } \{0\} { 0 }
The unitary U U U (7) satisfies U D = D U\mathcal{D}=\mathcal{D} U D = D E η ∞ ( U f ) = η E η ∞ ( f ) \mathcal{E}^{\infty}_{\eta}(Uf)=\eta\mathcal{E}^{\infty}_{\eta}(f) E η ∞ ( U f ) = η E η ∞ ( f ) f ∈ ℓ Z 2 ( w ) f\in\ell_Z^2(w) f ∈ ℓ Z 2 ( w ) U − 1 L η ∞ U = η L η ∞ U^{-1}L^{\infty}_{\eta}U=\eta L^{\infty}_{\eta} U − 1 L η ∞ U = η L η ∞
Proof: 1. The finitely supported functions belong to D \mathcal{D} D ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) g ∈ D g\in\mathcal{D} g ∈ D M M M Δ g \Delta g Δ g [ − M − 1 , M − 1 ] [-M-1,M-1] [ − M − 1 , M − 1 ] E η ∞ ( g ) < ∞ \mathcal{E}^{\infty}_{\eta}(g)<\infty E η ∞ ( g ) < ∞ L g Lg Lg [ − M − 1 , M ] [-M-1,M] [ − M − 1 , M ] w k ( L g ) ( k ) = c k − 1 ( Δ g ) k − 1 − c k ( Δ g ) k w_k(Lg)(k)=c_{k-1}(\Delta g)_{k-1}-c_k(\Delta g)_k w k ( Lg ) ( k ) = c k − 1 ( Δ g ) k − 1 − c k ( Δ g ) k f ∈ dom E η ∞ f\in\operatorname{dom}\mathcal{E}^{\infty}_{\eta} f ∈ dom E η ∞ a ≤ − M − 2 a\leq-M-2 a ≤ − M − 2 b ≥ M + 1 b\geq M+1 b ≥ M + 1
∑ k = a b − 1 c k ( Δ f ) k ( Δ g ) k ‾ = ∑ k = a + 1 b − 1 f ( k ) ( c k − 1 ( Δ g ) k − 1 − c k ( Δ g ) k ) ‾ + c b − 1 f ( b ) ( Δ g ) b − 1 ‾ − c a f ( a ) ( Δ g ) a ‾ . \sum_{k=a}^{b-1}c_k(\Delta f)_k\overline{(\Delta g)_k}
=
\sum_{k=a+1}^{b-1}f(k)\overline{\left(c_{k-1}(\Delta g)_{k-1}-c_k(\Delta g)_k\right)}
+c_{b-1}f(b)\overline{(\Delta g)_{b-1}}-c_af(a)\overline{(\Delta g)_a}. k = a ∑ b − 1 c k ( Δ f ) k ( Δ g ) k = k = a + 1 ∑ b − 1 f ( k ) ( c k − 1 ( Δ g ) k − 1 − c k ( Δ g ) k ) + c b − 1 f ( b ) ( Δ g ) b − 1 − c a f ( a ) ( Δ g ) a . The boundary terms vanish, since ( Δ g ) a = ( Δ g ) b − 1 = 0 (\Delta g)_a=(\Delta g)_{b-1}=0 ( Δ g ) a = ( Δ g ) b − 1 = 0 E η ∞ ( f , g ) \mathcal{E}^{\infty}_{\eta}(f,g) E η ∞ ( f , g ) ∑ k w k f ( k ) ( L g ) ( k ) ‾ = ⟨ f , L g ⟩ \sum_k w_kf(k)\overline{(Lg)(k)}=\langle f,Lg\rangle ∑ k w k f ( k ) ( Lg ) ( k ) = ⟨ f , Lg ⟩ [ a , b ] [a,b] [ a , b ] (19) .
Let ( f i ) (f_i) ( f i ) dom E η ∞ \operatorname{dom}\mathcal{E}^{\infty}_{\eta} dom E η ∞ f f f ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) E η ∞ ( f i − f j ) → 0 \mathcal{E}^{\infty}_{\eta}(f_i-f_j)\to0 E η ∞ ( f i − f j ) → 0 i , j → ∞ i,j\to\infty i , j → ∞ ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) E η ∞ ( f − f j ) ≤ lim inf i E η ∞ ( f i − f j ) \mathcal{E}^{\infty}_{\eta}(f-f_j)\leq\liminf_i\mathcal{E}^{\infty}_{\eta}(f_i-f_j) E η ∞ ( f − f j ) ≤ lim inf i E η ∞ ( f i − f j ) j → ∞ j\to\infty j → ∞
Now, let f ∈ dom E η ∞ f\in\operatorname{dom}\mathcal{E}^{\infty}_{\eta} f ∈ dom E η ∞ M , N ≥ 1 M,N\geq1 M , N ≥ 1 f ( M , N ) ∈ D f^{(M,N)}\in\mathcal{D} f ( M , N ) ∈ D f ( M , N ) ( k ) = 0 f^{(M,N)}(k)=0 f ( M , N ) ( k ) = 0 k < − M k<-M k < − M f ( M , N ) ( k ) = f ( k ) f^{(M,N)}(k)=f(k) f ( M , N ) ( k ) = f ( k ) − M ≤ k ≤ N -M\leq k\leq N − M ≤ k ≤ N f ( M , N ) ( k ) = f ( N ) f^{(M,N)}(k)=f(N) f ( M , N ) ( k ) = f ( N ) k > N k>N k > N u = f − f ( M , N ) u=f-f^{(M,N)} u = f − f ( M , N ) ( Δ u ) k = ( Δ f ) k (\Delta u)_k=(\Delta f)_k ( Δ u ) k = ( Δ f ) k k ≤ − M − 2 k\le-M-2 k ≤ − M − 2 k ≥ N k\ge N k ≥ N ( Δ u ) − M − 1 = − f ( − M − 1 ) (\Delta u)_{-M-1}=-f(-M-1) ( Δ u ) − M − 1 = − f ( − M − 1 ) ( Δ u ) k = 0 (\Delta u)_k=0 ( Δ u ) k = 0
E η ∞ ( u ) = ∑ k ≤ − M − 2 c k ∣ ( Δ f ) k ∣ 2 + c − M − 1 ∣ f ( − M − 1 ) ∣ 2 + ∑ k ≥ N c k ∣ ( Δ f ) k ∣ 2 . \mathcal{E}^{\infty}_{\eta}(u)=\sum_{k\le-M-2}c_k|(\Delta f)_k|^2+c_{-M-1}|f(-M-1)|^2+\sum_{k\ge N}c_k|(\Delta f)_k|^2. E η ∞ ( u ) = k ≤ − M − 2 ∑ c k ∣ ( Δ f ) k ∣ 2 + c − M − 1 ∣ f ( − M − 1 ) ∣ 2 + k ≥ N ∑ c k ∣ ( Δ f ) k ∣ 2 . The first and third terms are tails of a convergent series. For the second one, we have
c − M − 1 ∣ f ( − M − 1 ) ∣ 2 = β − M − 1 w − M − 1 ∣ f ( − M − 1 ) ∣ 2 ≤ τ η M ∥ f ∥ 2 , c_{-M-1}|f(-M-1)|^2=\beta_{-M-1}w_{-M-1}|f(-M-1)|^2\le\tau\eta^M\|f\|^2, c − M − 1 ∣ f ( − M − 1 ) ∣ 2 = β − M − 1 w − M − 1 ∣ f ( − M − 1 ) ∣ 2 ≤ τ η M ∥ f ∥ 2 , which tends to zero. On the other hand, ∥ u ∥ 2 = ∑ k < − M w k ∣ f ( k ) ∣ 2 + ∑ k > N w k ∣ f ( k ) − f ( N ) ∣ 2 \|u\|^2=\sum_{k<-M}w_k|f(k)|^2+\sum_{k>N}w_k|f(k)-f(N)|^2 ∥ u ∥ 2 = ∑ k < − M w k ∣ f ( k ) ∣ 2 + ∑ k > N w k ∣ f ( k ) − f ( N ) ∣ 2 k > N k>N k > N
∣ f ( k ) − f ( N ) ∣ 2 ≤ ( ∑ i = N k − 1 c i ∣ ( Δ f ) i ∣ 2 ) ( ∑ i = N k − 1 c i − 1 ) ≤ ε N ∑ i = N k − 1 c i − 1 , ε N : = ∑ i ≥ N c i ∣ ( Δ f ) i ∣ 2 . |f(k)-f(N)|^2\le\left(\sum_{i=N}^{k-1}c_i|(\Delta f)_i|^2\right)\left(\sum_{i=N}^{k-1}c_i^{-1}\right)\le\varepsilon_N\sum_{i=N}^{k-1}c_i^{-1},\qquad\varepsilon_N\mathrel{\mathop:}=\sum_{i\ge N}c_i|(\Delta f)_i|^2. ∣ f ( k ) − f ( N ) ∣ 2 ≤ ( i = N ∑ k − 1 c i ∣ ( Δ f ) i ∣ 2 ) ( i = N ∑ k − 1 c i − 1 ) ≤ ε N i = N ∑ k − 1 c i − 1 , ε N : = i ≥ N ∑ c i ∣ ( Δ f ) i ∣ 2 . Tonelli’s theorem gives the exact estimate
∑ k > N w k ∑ i = N k − 1 c i − 1 = ∑ i = N ∞ c i − 1 ∑ k > i w k = 1 1 − τ ∑ i = N ∞ η i + 1 = η N + 1 ( 1 − τ ) ( 1 − η ) . \sum_{k>N}w_k\sum_{i=N}^{k-1}c_i^{-1}=\sum_{i=N}^{\infty}c_i^{-1}\sum_{k>i}w_k=\frac{1}{1-\tau}\sum_{i=N}^{\infty}\eta^{i+1}=\frac{\eta^{N+1}}{(1-\tau)(1-\eta)}. k > N ∑ w k i = N ∑ k − 1 c i − 1 = i = N ∑ ∞ c i − 1 k > i ∑ w k = 1 − τ 1 i = N ∑ ∞ η i + 1 = ( 1 − τ ) ( 1 − η ) η N + 1 . Consequently ∑ k > N w k ∣ f ( k ) − f ( N ) ∣ 2 ≤ ε N η N + 1 / ( ( 1 − τ ) ( 1 − η ) ) → 0 \sum_{k>N}w_k|f(k)-f(N)|^2\le\varepsilon_N\eta^{N+1}/((1-\tau)(1-\eta))\to0 ∑ k > N w k ∣ f ( k ) − f ( N ) ∣ 2 ≤ ε N η N + 1 / (( 1 − τ ) ( 1 − η )) → 0 M M M N N N f ( M , N ) → f f^{(M,N)}\to f f ( M , N ) → f ( E η ∞ ( ⋅ ) + ∥ ⋅ ∥ 2 ) 1 / 2 (\mathcal{E}^{\infty}_{\eta}(\cdot)+\|\cdot\|^2)^{1/2} ( E η ∞ ( ⋅ ) + ∥ ⋅ ∥ 2 ) 1/2 D \mathcal{D} D
By (19) , for any f , g ∈ D f,g\in\mathcal{D} f , g ∈ D ⟨ f , L g ⟩ = E η ∞ ( f , g ) = E η ∞ ( g , f ) ‾ = ⟨ L f , g ⟩ \langle f,Lg\rangle=\mathcal{E}^{\infty}_{\eta}(f,g)=\overline{\mathcal{E}^{\infty}_{\eta}(g,f)}=\langle Lf,g\rangle ⟨ f , Lg ⟩ = E η ∞ ( f , g ) = E η ∞ ( g , f ) = ⟨ L f , g ⟩ ⟨ L f , f ⟩ = E η ∞ ( f ) ≥ 0 \langle Lf,f\rangle=\mathcal{E}^{\infty}_{\eta}(f)\ge0 ⟨ L f , f ⟩ = E η ∞ ( f ) ≥ 0 L ∣ D L|_{\mathcal{D}} L ∣ D L ∣ D + I L|_{\mathcal{D}}+I L ∣ D + I h ∈ ℓ z 2 ( w ) h\in\ell^2_z(w) h ∈ ℓ z 2 ( w ) ( L + I ) D (L+I)\mathcal{D} ( L + I ) D L L L h h h h h h 1 { j } ∈ D \mathbf{1}_{\{j\}}\in\mathcal{D} 1 { j } ∈ D (17) , we obtain w j ( ( L + I ) h ) ( j ) = 0 w_j((L+I)h)(j)=0 w j (( L + I ) h ) ( j ) = 0
( L h ) ( k ) = − h ( k ) , k ∈ Z . (20) (Lh)(k)=-h(k),\qquad k\in\mathbb{Z}. \tag*{(20)} ( L h ) ( k ) = − h ( k ) , k ∈ Z . (20) Applying the orthogonality to 1 ≥ j \mathbf{1}_{\ge j} 1 ≥ j L 1 ≥ j = − β j − 1 1 { j − 1 } + δ j 1 { j } L\mathbf{1}_{\ge j}=-\beta_{j-1}\mathbf{1}_{\{j-1\}}+\delta_j\mathbf{1}_{\{j\}} L 1 ≥ j = − β j − 1 1 { j − 1 } + δ j 1 { j } w j − 1 β j − 1 = w j δ j = c j − 1 w_{j-1}\beta_{j-1}=w_j\delta_j=c_{j-1} w j − 1 β j − 1 = w j δ j = c j − 1 ∑ k ≥ j w k h ( k ) + c j − 1 ( h ( j ) − h ( j − 1 ) ) = 0 \sum_{k\ge j}w_kh(k)+c_{j-1}(h(j)-h(j-1))=0 ∑ k ≥ j w k h ( k ) + c j − 1 ( h ( j ) − h ( j − 1 )) = 0
c j − 1 ( Δ h ) j − 1 = − R j , R j : = ∑ k ≥ j w k h ( k ) , j ∈ Z . (21) c_{j-1}(\Delta h)_{j-1}=-R_j,\qquad R_j\mathrel{\mathop:}=\sum_{k\ge j}w_kh(k),\qquad j\in\mathbb{Z}. \tag*{(21)} c j − 1 ( Δ h ) j − 1 = − R j , R j : = k ≥ j ∑ w k h ( k ) , j ∈ Z . (21) By the Cauchy–Schwarz inequality, ∣ R j ∣ ≤ ( ∑ k ≥ j w k ) 1 / 2 ∥ h ∥ = τ − 1 / 2 ∥ h ∥ |R_j|\le\left(\sum_{k\ge j}w_k\right)^{1/2}\|h\|=\tau^{-1/2}\|h\| ∣ R j ∣ ≤ ( ∑ k ≥ j w k ) 1/2 ∥ h ∥ = τ − 1/2 ∥ h ∥ w k ∣ h ( k ) ∣ 2 → 0 w_k|h(k)|^2\to0 w k ∣ h ( k ) ∣ 2 → 0 k → ± ∞ k\to\pm\infty k → ± ∞ ∣ h ( k ) ∣ = o ( τ − k / 2 ) |h(k)|=o(\tau^{-k/2}) ∣ h ( k ) ∣ = o ( τ − k /2 ) a < b a<b a < b (21) and R k − R k + 1 = w k h ( k ) R_k-R_{k+1}=w_kh(k) R k − R k + 1 = w k h ( k )
∑ k = a b c k ( Δ h ) k 2 = (3.8) − ∑ k = a b R k + 1 ( h ( k + 1 ) − h ( k ) ) = R a + 1 h ( a ) − R b + 1 h ( b + 1 ) − ∑ k = a + 1 b w k h ( k ) 2 . \sum_{k=a}^{b}c_k(\Delta h)_k^2\mathrel{\mathop=^{\text{(3.8)}}}-\sum_{k=a}^{b}R_{k+1}(h(k+1)-h(k))=R_{a+1}h(a)-R_{b+1}h(b+1)-\sum_{k=a+1}^{b}w_kh(k)^2. k = a ∑ b c k ( Δ h ) k 2 = (3.8) − k = a ∑ b R k + 1 ( h ( k + 1 ) − h ( k )) = R a + 1 h ( a ) − R b + 1 h ( b + 1 ) − k = a + 1 ∑ b w k h ( k ) 2 . Since ∣ R b + 1 h ( b + 1 ) ∣ = O ( τ − 1 / 2 ) o ( τ − b / 2 ) → 0 |R_{b+1}h(b+1)|=O(\tau^{-1/2})o(\tau^{-b/2})\to0 ∣ R b + 1 h ( b + 1 ) ∣ = O ( τ − 1/2 ) o ( τ − b /2 ) → 0 b → + ∞ b\to+\infty b → + ∞
∑ k ≥ a c k ( Δ h ) k 2 + ∑ k > a w k h ( k ) 2 = R a + 1 h ( a ) , a ∈ Z . \sum_{k\ge a}c_k(\Delta h)_k^2+\sum_{k>a}w_kh(k)^2=R_{a+1}h(a),\qquad a\in\mathbb{Z}. k ≥ a ∑ c k ( Δ h ) k 2 + k > a ∑ w k h ( k ) 2 = R a + 1 h ( a ) , a ∈ Z . The left-hand side is nonnegative and nonincreasing in a a a O ( τ a / 2 ) o ( τ − a / 2 ) = o ( 1 ) O(\tau^{a/2})o(\tau^{-a/2})=o(1) O ( τ a /2 ) o ( τ − a /2 ) = o ( 1 ) a → − ∞ a\to-\infty a → − ∞ a a a h = 0 h=0 h = 0 L ∣ D + I L|_D+I L ∣ D + I
Let L ‾ \overline{L} L L ∣ D L|_D L ∣ D ∥ ( L ‾ + I ) f ∥ ≥ ∥ f ∥ \|(\overline{L}+I)f\|\ge\|f\| ∥ ( L + I ) f ∥ ≥ ∥ f ∥ f ∈ dom L ‾ f\in\operatorname{dom}\overline{L} f ∈ dom L L ‾ + I \overline{L}+I L + I ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) g ∈ dom L η ∗ g\in\operatorname{dom}L_\eta^* g ∈ dom L η ∗ f ∈ dom L ‾ f\in\operatorname{dom}\overline{L} f ∈ dom L ( L ‾ + I ) f = ( L η ∗ + I ) g (\overline{L}+I)f=(L_\eta^*+I)g ( L + I ) f = ( L η ∗ + I ) g g − f g-f g − f Ker ( L η ∗ + I ) = Ran ( L ‾ + I ) ⊥ = { 0 } \operatorname{Ker}(L_\eta^*+I)=\operatorname{Ran}(\overline{L}+I)^\perp=\{0\} Ker ( L η ∗ + I ) = Ran ( L + I ) ⊥ = { 0 } L η ∗ = L ‾ L_\eta^*=\overline{L} L η ∗ = L L ‾ \overline{L} L f ↦ ⟨ L ‾ f , f ⟩ f\mapsto\langle\overline{L}f,f\rangle f ↦ ⟨ L f , f ⟩ D D D E η ∞ \mathcal{E}_\eta^\infty E η ∞ D D D (19) . Since D D D E η ∞ \mathcal{E}_\eta^\infty E η ∞ L ‾ \overline{L} L E η ∞ \mathcal{E}_\eta^\infty E η ∞
Let f ∈ dom L η ∞ f\in\operatorname{dom}L_\eta^\infty f ∈ dom L η ∞ j ∈ Z j\in\mathbb{Z} j ∈ Z 1 { j } ∈ D \mathbf{1}_{\{j\}}\in D 1 { j } ∈ D w j ( L η ∞ f ) ( j ) = ⟨ L η ∞ f , 1 { j } ⟩ = ⟨ f , L 1 { j } ⟩ = w j ( L f ) ( j ) w_j(L_\eta^\infty f)(j)=\langle L_\eta^\infty f,\mathbf{1}_{\{j\}}\rangle=\langle f,L\mathbf{1}_{\{j\}}\rangle=w_j(Lf)(j) w j ( L η ∞ f ) ( j ) = ⟨ L η ∞ f , 1 { j } ⟩ = ⟨ f , L 1 { j } ⟩ = w j ( L f ) ( j ) (17) . Hence L η ∞ f = L f L_\eta^\infty f=Lf L η ∞ f = L f ⟨ f , L 1 ≥ j ⟩ = c j − 1 ( f ( j ) − f ( j − 1 ) ) \langle f,L\mathbf{1}_{\geq j}\rangle=c_{j-1}(f(j)-f(j-1)) ⟨ f , L 1 ≥ j ⟩ = c j − 1 ( f ( j ) − f ( j − 1 ))
∣ ⟨ L η ∞ f , 1 ≥ j ⟩ ∣ ≤ ( ∑ k ≥ j w k ) 1 / 2 ∥ L η ∞ f ∥ = τ − j / 2 ∥ L η ∞ f ∥ . \left|\left\langle L_\eta^\infty f,\mathbf{1}_{\geq j}\right\rangle\right|\leq\left(\sum_{k\geq j}w_k\right)^{1/2}\|L_\eta^\infty f\|=\tau^{-j/2}\|L_\eta^\infty f\|. ⟨ L η ∞ f , 1 ≥ j ⟩ ≤ k ≥ j ∑ w k 1/2 ∥ L η ∞ f ∥ = τ − j /2 ∥ L η ∞ f ∥. Since these two quantities are equal, we obtain c j − 1 ( f ( j ) − f ( j − 1 ) ) → 0 c_{j-1}(f(j)-f(j-1))\to0 c j − 1 ( f ( j ) − f ( j − 1 )) → 0 j → + ∞ j\to+\infty j → + ∞
If L η ∞ f = 0 L_\eta^\infty f=0 L η ∞ f = 0 E η ∞ ( f ) = 0 \mathcal{E}_\eta^\infty(f)=0 E η ∞ ( f ) = 0 Δ f = 0 \Delta f=0 Δ f = 0 f f f f ∈ ℓ Z 2 ( w ) f\in\ell^2_{\mathbb{Z}}(w) f ∈ ℓ Z 2 ( w ) f = 0 f=0 f = 0
The first assertion is clear. For any f ∈ ℓ Z 2 ( w ) f\in\ell^2_{\mathbb{Z}}(w) f ∈ ℓ Z 2 ( w )
E η ∞ ( U f ) = τ ∑ k ∈ Z c k ∣ f ( k + 2 ) − f ( k + 1 ) ∣ 2 = τ ∑ k ∈ Z c k − 1 ∣ ( Δ f ) k ∣ 2 = η E η ∞ ( f ) . (22) \mathcal{E}_{\eta}^{\infty}(Uf)=\tau\sum_{k\in\mathbb{Z}}c_k|f(k+2)-f(k+1)|^2=\tau\sum_{k\in\mathbb{Z}}c_{k-1}|(\Delta f)_k|^2=\eta\mathcal{E}_{\eta}^{\infty}(f). \tag*{(22)} E η ∞ ( U f ) = τ k ∈ Z ∑ c k ∣ f ( k + 2 ) − f ( k + 1 ) ∣ 2 = τ k ∈ Z ∑ c k − 1 ∣ ( Δ f ) k ∣ 2 = η E η ∞ ( f ) . (22) Since U U U U − 1 L η ∞ U U^{-1}L_\eta^\infty U U − 1 L η ∞ U η E η ∞ \eta\mathcal{E}_\eta^\infty η E η ∞ η L η ∞ \eta L_\eta^\infty η L η ∞
Remark 3.2 The solutions of L f = 0 Lf=0 L f = 0 f f f c k ( Δ f ) k c_k(\Delta f)_k c k ( Δ f ) k k k k − ∞ -\infty − ∞ K d K^d K d (16) is in the limit point case, in the sense of the Weyl alternative for Jacobi operators [38 Chapter 2] , since the constant function 1 1 1 − ∞ -\infty − ∞ + ∞ +\infty + ∞ K d K^d K d ( Δ f ) k = C c k − 1 (\Delta f)_k=Cc_k^{-1} ( Δ f ) k = C c k − 1 C ≠ 0 C\ne0 C = 0 ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) + ∞ +\infty + ∞ η 2 < τ \eta^2<\tau η 2 < τ τ = Q − d \tau=Q^{-d} τ = Q − d η = Q − α \eta=Q^{-\alpha} η = Q − α α > d / 2 \alpha>d/2 α > d /2 D D D + ∞ +\infty + ∞ η < τ \eta<\tau η < τ α > d \alpha>d α > d
Eigenfunctions
Proposition 3.3 Let λ ∈ C \lambda\in\mathbb{C} λ ∈ C f λ : Z → C f_\lambda:\mathbb{Z}\to\mathbb{C} f λ : Z → C
f λ ( k ) : = 1 φ 1 ( 0 ; τ ; η , λ η k + 1 ) , k ∈ Z . (23) f_\lambda(k)\mathrel{\mathop:}= {}_1\varphi_1(0;\tau;\eta,\lambda\eta^{k+1}),\qquad k\in\mathbb{Z}. \tag*{(23)} f λ ( k ) : = 1 φ 1 ( 0 ; τ ; η , λ η k + 1 ) , k ∈ Z . (23) We have L f λ = λ f λ Lf_\lambda=\lambda f_\lambda L f λ = λ f λ f λ ( k ) → 1 f_\lambda(k)\to1 f λ ( k ) → 1 c k ( Δ f λ ) k = O ( τ − k ) c_k(\Delta f_\lambda)_k=O(\tau^{-k}) c k ( Δ f λ ) k = O ( τ − k ) k → + ∞ k\to+\infty k → + ∞
Let g : Z → C g:\mathbb{Z}\to\mathbb{C} g : Z → C L g = λ g Lg=\lambda g Lg = λ g c k ( Δ g ) k → 0 c_k(\Delta g)_k\to0 c k ( Δ g ) k → 0 k → + ∞ k\to+\infty k → + ∞ g ( k ) = o ( τ − k / 2 ) g(k)=o(\tau^{-k/2}) g ( k ) = o ( τ − k /2 ) k → + ∞ k\to+\infty k → + ∞ g g g f λ f_\lambda f λ
Proof. 1. Put y ≔ η k y \coloneqq\eta^k y : = η k j ≥ 0 j \ge0 j ≥ 0 k ↦ y j = η j k k \mapsto y^j=\eta^{jk} k ↦ y j = η j k
L ( y j ) = τ η − k − 1 η j k ( 1 − η j ) + η − k η j k ( 1 − η − j ) = − η − j ( 1 − η j ) ( 1 − τ η j − 1 ) y j − 1 , L(y^j)=\tau\eta^{-k-1}\eta^{jk}(1-\eta^j)+\eta^{-k}\eta^{jk}(1-\eta^{-j})=-\eta^{-j}(1-\eta^j)(1-\tau\eta^{j-1})y^{j-1}, L ( y j ) = τ η − k − 1 η j k ( 1 − η j ) + η − k η j k ( 1 − η − j ) = − η − j ( 1 − η j ) ( 1 − τ η j − 1 ) y j − 1 , which vanishes for j = 0 j=0 j = 0 f λ ( k ) = ∑ j ≥ 0 a j ( λ η ) j y j f_\lambda(k)=\sum_{j\ge0}a_j(\lambda\eta)^jy^j f λ ( k ) = ∑ j ≥ 0 a j ( λ η ) j y j a j = ( − 1 ) j η j ( j − 1 ) / 2 ( η ; η ) j ( τ ; η ) j a_j=\frac{(-1)^j\eta^{j(j-1)/2}}{(\eta;\eta)_j(\tau;\eta)_j} a j = ( η ; η ) j ( τ ; η ) j ( − 1 ) j η j ( j − 1 ) /2 y y y L L L L L L k − 1 k-1 k − 1 k k k k + 1 k+1 k + 1
( L f λ ) ( k ) = − ∑ j ≥ 1 a j ( λ η ) j η − j ( 1 − η j ) ( 1 − τ η j − 1 ) y j − 1 = λ ∑ j ≥ 1 a j − 1 ( λ η ) j − 1 y j − 1 = λ f λ ( k ) , (Lf_\lambda)(k)=-\sum_{j\ge1}a_j(\lambda\eta)^j\eta^{-j}(1-\eta^j)(1-\tau\eta^{j-1})y^{j-1}=\lambda\sum_{j\ge1}a_{j-1}(\lambda\eta)^{j-1}y^{j-1}=\lambda f_\lambda(k), ( L f λ ) ( k ) = − j ≥ 1 ∑ a j ( λ η ) j η − j ( 1 − η j ) ( 1 − τ η j − 1 ) y j − 1 = λ j ≥ 1 ∑ a j − 1 ( λ η ) j − 1 y j − 1 = λ f λ ( k ) , where we used a j ( 1 − η j ) ( 1 − τ η j − 1 ) = − η j − 1 a j − 1 a_j(1-\eta^j)(1-\tau\eta^{j-1})=-\eta^{j-1}a_{j-1} a j ( 1 − η j ) ( 1 − τ η j − 1 ) = − η j − 1 a j − 1 k → + ∞ k\to+\infty k → + ∞ y → 0 y\to0 y → 0 f λ ( k ) = 1 + O ( η k ) f_\lambda(k)=1+O(\eta^k) f λ ( k ) = 1 + O ( η k ) ( Δ f λ ) k = O ( η k ) (\Delta f_\lambda)_k=O(\eta^k) ( Δ f λ ) k = O ( η k ) c k ( Δ f λ ) k = O ( ( τ η ) k η k ) = O ( τ k ) c_k(\Delta f_\lambda)_k=O((\frac{\tau}{\eta})^k\eta^k)=O(\tau^k) c k ( Δ f λ ) k = O (( η τ ) k η k ) = O ( τ k )
For two solutions f , g f,g f , g L f = λ f Lf=\lambda f L f = λ f L g = λ g Lg=\lambda g Lg = λ g [38 Chapter 1] ) W ( k ) ≔ c k ( g ( k ) ( Δ f ) k − f ( k ) ( Δ g ) k ) W(k)\coloneqq c_k(g(k)(\Delta f)_k-f(k)(\Delta g)_k) W ( k ) : = c k ( g ( k ) ( Δ f ) k − f ( k ) ( Δ g ) k ) k k k c k − 1 ( Δ f ) k − 1 − c k ( Δ f ) k = λ w k f ( k ) c_{k-1}(\Delta f)_{k-1}-c_k(\Delta f)_k=\lambda w_kf(k) c k − 1 ( Δ f ) k − 1 − c k ( Δ f ) k = λ w k f ( k ) g ( k ) g(k) g ( k ) g g g f ( k ) f(k) f ( k ) W ( k ) = c k − 1 ( g ( k ) ( Δ f ) k − 1 − f ( k ) ( Δ g ) k − 1 ) W(k)=c_{k-1}(g(k)(\Delta f)_{k-1}-f(k)(\Delta g)_{k-1}) W ( k ) = c k − 1 ( g ( k ) ( Δ f ) k − 1 − f ( k ) ( Δ g ) k − 1 ) g ( k ) ( Δ f ) k − 1 − f ( k ) ( Δ g ) k − 1 = g ( k − 1 ) f ( k ) − f ( k − 1 ) g ( k ) = g ( k − 1 ) ( Δ f ) k − 1 − f ( k − 1 ) ( Δ g ) k − 1 g(k)(\Delta f)_{k-1}-f(k)(\Delta g)_{k-1}=g(k-1)f(k)-f(k-1)g(k)=g(k-1)(\Delta f)_{k-1}-f(k-1)(\Delta g)_{k-1} g ( k ) ( Δ f ) k − 1 − f ( k ) ( Δ g ) k − 1 = g ( k − 1 ) f ( k ) − f ( k − 1 ) g ( k ) = g ( k − 1 ) ( Δ f ) k − 1 − f ( k − 1 ) ( Δ g ) k − 1 f = f λ f=f_\lambda f = f λ g g g g ( k ) c k ( Δ f ) k = o ( τ − k / 2 ) O ( τ k ) → 0 g(k)c_k(\Delta f)_k=o(\tau^{-k/2})O(\tau^k)\to0 g ( k ) c k ( Δ f ) k = o ( τ − k /2 ) O ( τ k ) → 0 f ( k ) c k ( Δ g ) k → 0 f(k)c_k(\Delta g)_k\to0 f ( k ) c k ( Δ g ) k → 0 k → + ∞ k\to+\infty k → + ∞ W = 0 W=0 W = 0 g ( k ) f ( k + 1 ) = f ( k ) g ( k + 1 ) g(k)f(k+1)=f(k)g(k+1) g ( k ) f ( k + 1 ) = f ( k ) g ( k + 1 ) k k k f ( k ) → 1 f(k)\to1 f ( k ) → 1 k 0 k_0 k 0 f ( k ) ≠ 0 f(k)\ne0 f ( k ) = 0 k ≥ k 0 k\ge k_0 k ≥ k 0 g = C f g=Cf g = C f { k ≥ k 0 } \{k\ge k_0\} { k ≥ k 0 } C ∈ C C\in\mathbb{C} C ∈ C L g = λ g Lg=\lambda g Lg = λ g g ( k − 1 ) g(k-1) g ( k − 1 ) g ( k ) g(k) g ( k ) g ( k + 1 ) g(k+1) g ( k + 1 ) δ k ≠ 0 \delta_k\ne0 δ k = 0 g = C f g=Cf g = C f Z \mathbb{Z} Z
For any z ∈ Z z\in\mathbb{Z} z ∈ Z f z ≔ f η z f_z\coloneqq f_{\eta^z} f z : = f η z f z ( k ) = 1 φ 1 ( 0 ; τ ; η , η z + k + 1 ) f_z(k)={}_1\varphi_1(0;\tau;\eta,\eta^{z+k+1}) f z ( k ) = 1 φ 1 ( 0 ; τ ; η , η z + k + 1 )
f z ( k + 1 ) = f z + 1 ( k ) , that is U f z = τ 1 2 f z + 1 , z , k ∈ Z , (24) f_z(k+1)=f_{z+1}(k),\qquad\text{that is}\qquad Uf_z=\tau^{\frac12}f_{z+1},\qquad z,k\in\mathbb{Z}, \tag*{(24)} f z ( k + 1 ) = f z + 1 ( k ) , that is U f z = τ 2 1 f z + 1 , z , k ∈ Z , (24) where the second equality holds as soon as f z ∈ ℓ Z 2 ( w ) f_z\in\ell^2_\mathbb{Z}(w) f z ∈ ℓ Z 2 ( w )
Remark 3.4. Let ν > − 1 \nu>-1 ν > − 1 τ = η ν + 1 \tau=\eta^{\nu+1} τ = η ν + 1 q q q ν \nu ν η \eta η J ν ( x ; η ) = x ν ( η ν + 1 ; η ) ∞ ( η ; η ) ∞ 1 φ 1 ( 0 ; η ν + 1 ; η , η x 2 ) J_\nu(x;\eta)=x^\nu\frac{(\eta^{\nu+1};\eta)_\infty}{(\eta;\eta)_\infty}{}_1\varphi_1(0;\eta^{\nu+1};\eta,\eta x^2) J ν ( x ; η ) = x ν ( η ; η ) ∞ ( η ν + 1 ; η ) ∞ 1 φ 1 ( 0 ; η ν + 1 ; η , η x 2 ) [22 (3.3)] for the corresponding normalization in base q 2 q^2 q 2
f z ( k ) = ( η ; η ) ∞ ( τ ; η ) ∞ x − ν J ν ( x ; η ) , x = η z + k 2 . f_z(k)=\frac{(\eta;\eta)_\infty}{(\tau;\eta)_\infty}x^{-\nu}J_\nu(x;\eta),\qquad x=\eta^{\frac{z+k}{2}}. f z ( k ) = ( τ ; η ) ∞ ( η ; η ) ∞ x − ν J ν ( x ; η ) , x = η 2 z + k . If τ = Q − d \tau=Q^{-d} τ = Q − d η = Q − α \eta=Q^{-\alpha} η = Q − α ν = d α − 1 \nu=\frac{d}{\alpha}-1 ν = α d − 1 α = d \alpha=d α = d q q q 0 0 0 τ = r 2 \tau=r^2 τ = r 2 [22 Proposition 2.6, (2.11)] , which reads, after replacing their parameters z z z q q q τ 1 2 \tau^{\frac12} τ 2 1 η \eta η
∑ k ∈ Z τ k f z ( k ) f z ′ ( k ) = δ z z ′ τ − z ( η ; η ) ∞ 2 ( τ ; η ) ∞ 2 , z , z ′ ∈ Z . (25) \sum_{k\in\mathbb{Z}}\tau^k f_z(k)f_{z'}(k)=\delta_{zz'}\tau^{-z}\frac{(\eta;\eta)_\infty^2}{(\tau;\eta)_\infty^2},\qquad z,z'\in\mathbb{Z}. \tag*{(25)} k ∈ Z ∑ τ k f z ( k ) f z ′ ( k ) = δ z z ′ τ − z ( τ ; η ) ∞ 2 ( η ; η ) ∞ 2 , z , z ′ ∈ Z . (25) The orthogonality and completeness are already contained in [22 Proposition 2.6 and Section 3] : after normalization, the kernel in (25) is symmetric in the lattice and spectral indices, and the corresponding transform has the same inverse. Theorem 3.6 gives an independent operator-theoretic proof, up to the value of the norm, and identifies this transform with the particular self-adjoint realization of Proposition 3.1 .
The spectrum by a limit transition
We now relate L ∞ ∞ L_{\infty}^{\infty} L ∞ ∞ η \eta η L η L_{\eta} L η L η L_{\eta} L η ℓ N 0 2 ( w ) \ell^{2}_{\mathbb{N}_{0}}(w) ℓ N 0 2 ( w ) E η ( f ) = ∑ k ≥ 0 w k β k η ∣ f ( k + 1 ) − f ( k ) ∣ 2 \mathcal{E}_{\eta}(f)=\sum_{k\geq0}w_{k}\beta_{k}^{\eta}|f(k+1)-f(k)|^{2} E η ( f ) = ∑ k ≥ 0 w k β k η ∣ f ( k + 1 ) − f ( k ) ∣ 2 β k η = τ η − k − 1 ( 1 − η k + 1 ) \beta_{k}^{\eta}=\tau\eta^{-k-1}(1-\eta^{k+1}) β k η = τ η − k − 1 ( 1 − η k + 1 ) δ k η = η − k ( 1 − η k ) \delta_{k}^{\eta}=\eta^{-k}(1-\eta^{k}) δ k η = η − k ( 1 − η k ) L η L_{\eta} L η ( L η f ) ( k ) = β k η ( f ( k ) − f ( k + 1 ) ) + δ k η ( f ( k ) − f ( k − 1 ) ) (L_{\eta}f)(k)=\beta_{k}^{\eta}(f(k)-f(k+1))+\delta_{k}^{\eta}(f(k)-f(k-1)) ( L η f ) ( k ) = β k η ( f ( k ) − f ( k + 1 )) + δ k η ( f ( k ) − f ( k − 1 )) η k \eta^{k} η k L η L_{\eta} L η
Λ n ( η ) = η − n ( 1 − η n ) ( 1 − τ η n ) , n ∈ N 0 , (26) \Lambda_{n}(\eta)=\eta^{-n}(1-\eta^{n})(1-\tau\eta^{n}),\qquad n\in\mathbb{N}_{0}, \tag*{(26)} Λ n ( η ) = η − n ( 1 − η n ) ( 1 − τ η n ) , n ∈ N 0 , (26) which form a strictly increasing sequence, and the corresponding eigenvectors are the little η \eta η p n ( η k ; τ η − 1 , 1 ; η ) p_{n}(\eta^{k};\tau\eta^{-1},1;\eta) p n ( η k ; τ η − 1 , 1 ; η ) N ≥ 0 N\geq0 N ≥ 0 V N : ℓ N 0 2 ( w ) → ℓ Z 2 ( w ) V_{N}:\ell^{2}_{\mathbb{N}_{0}}(w)\to\ell^{2}_{\mathbb{Z}}(w) V N : ℓ N 0 2 ( w ) → ℓ Z 2 ( w )
( V N f ) ( k ) ≔ τ N 2 f ( k + N ) if k ≥ − N , ( V N f ) ( k ) ≔ 0 if k < − N . (V_{N}f)(k)\coloneqq\tau^{\frac{N}{2}}f(k+N)\quad\text{if }k\geq-N,\qquad(V_{N}f)(k)\coloneqq0\quad\text{if }k<-N. ( V N f ) ( k ) : = τ 2 N f ( k + N ) if k ≥ − N , ( V N f ) ( k ) : = 0 if k < − N . Its range is the space H N H_{N} H N { k < − N } \{k<-N\} { k < − N }
A ~ N ≔ η N V N L η V N ∗ ⊕ 0 H N ⊥ , (27) \widetilde{A}_{N}\coloneqq\eta^{N}V_{N}L_{\eta}V_{N}^{*}\oplus0_{H_{N}^{\perp}}, \tag*{(27)} A N : = η N V N L η V N ∗ ⊕ 0 H N ⊥ , (27) which is a nonnegative self-adjoint operator on ℓ Z 2 ( w ) \ell^{2}_{\mathbb{Z}}(w) ℓ Z 2 ( w ) H N H_{N} H N η N L η \eta^{N}L_{\eta} η N L η η N Λ n ( η ) \eta^{N}\Lambda_{n}(\eta) η N Λ n ( η ) n ≥ 1 n\geq1 n ≥ 1 0 0 0 V N V_{N} V N O d \mathcal{O}^{d} O d B − N = π − N O d B_{-N}=\pi^{-N}\mathcal{O}^{d} B − N = π − N O d L 2 L^{2} L 2
Lemma 3.5 Let g ∈ D g\in\mathcal{D} g ∈ D M M M D \mathcal{D} D N > M N>M N > M g g g dom A ~ N \operatorname{dom}\widetilde{A}_{N} dom A N
( ( A ~ N − L ) g ) ( k ) = − η N [ τ ( g ( k ) − g ( k + 1 ) ) + ( g ( k ) − g ( k − 1 ) ) ] , k ∈ Z . ((\widetilde{A}_{N}-L)g)(k)=-\eta^{N}\left[\tau(g(k)-g(k+1))+(g(k)-g(k-1))\right],\qquad k\in\mathbb{Z}. (( A N − L ) g ) ( k ) = − η N [ τ ( g ( k ) − g ( k + 1 )) + ( g ( k ) − g ( k − 1 )) ] , k ∈ Z . In particular, ∥ ( A ~ N − L ∞ ) g ∥ → 0 \|(\widetilde{A}_{N}-L_{\infty})g\|\to0 ∥ ( A N − L ∞ ) g ∥ → 0 N → ∞ N\to\infty N → ∞
Proof : Since N > M N>M N > M g ∈ H N g\in H_{N} g ∈ H N u = V N ∗ g u=V_{N}^{*}g u = V N ∗ g u ( m ) = τ − N 2 g ( m − N ) u(m)=\tau^{-\frac{N}{2}}g(m-N) u ( m ) = τ − 2 N g ( m − N ) { m ≥ M + N } \{m\geq M+N\} { m ≥ M + N } u ∈ dom L η u\in\operatorname{dom}L_{\eta} u ∈ dom L η L η u L_{\eta}u L η u L η L_{\eta} L η u u u dom E η \operatorname{dom}\mathcal{E}_{\eta} dom E η Δ u \Delta u Δ u f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_{\eta} f ∈ dom E η b ≥ M + N + 1 b\geq M+N+1 b ≥ M + N + 1
∑ k = 0 b − 1 w k β k η ( Δ f ) k ( Δ u ) k = ∑ k = 0 b − 1 w k f ( k ) ( L η u ) ( k ) + w b − 1 β b − 1 η f ( b ) ( Δ u ) b − 1 . \sum_{k=0}^{b-1}w_{k}\beta_{k}^{\eta}(\Delta f)_{k}(\Delta u)_{k}
=\sum_{k=0}^{b-1}w_{k}f(k)(L_{\eta}u)(k)+w_{b-1}\beta_{b-1}^{\eta}f(b)(\Delta u)_{b-1}. k = 0 ∑ b − 1 w k β k η ( Δ f ) k ( Δ u ) k = k = 0 ∑ b − 1 w k f ( k ) ( L η u ) ( k ) + w b − 1 β b − 1 η f ( b ) ( Δ u ) b − 1 . where L η u L_{\eta}u L η u ( Δ u ) b − 1 = 0 (\Delta u)_{b-1}=0 ( Δ u ) b − 1 = 0 L η u L_{\eta}u L η u { k ≥ M + N + 1 } \{k\geq M+N+1\} { k ≥ M + N + 1 } E η ( f , u ) = ⟨ f , L η u ⟩ \mathcal{E}_{\eta}(f,u)=\langle f,L_{\eta}u\rangle E η ( f , u ) = ⟨ f , L η u ⟩ f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_{\eta} f ∈ dom E η u ∈ dom L η u\in\operatorname{dom}L_{\eta} u ∈ dom L η L η u L_{\eta}u L η u g = V N u ∈ dom A ~ N g=V_{N}u\in\operatorname{dom}\widetilde{A}_{N} g = V N u ∈ dom A N k ≥ − N k\geq-N k ≥ − N
( A ~ N g ) ( k ) = η N τ − N 2 ( L η u ) ( k + N ) = τ η − k − 1 ( 1 − η k + N + 1 ) ( g ( k ) − g ( k + 1 ) ) + η − k ( 1 − η k + N ) ( g ( k ) − g ( k − 1 ) ) , (28) \begin{aligned}
(\widetilde{A}_{N}g)(k)=\eta^{N}\tau^{-\frac{N}{2}}(L_{\eta}u)(k+N) \\
&=\tau\eta^{-k-1}(1-\eta^{k+N+1})(g(k)-g(k+1))+\eta^{-k}(1-\eta^{k+N})(g(k)-g(k-1)), \tag*{(28)}
\end{aligned} ( A N g ) ( k ) = η N τ − 2 N ( L η u ) ( k + N ) = τ η − k − 1 ( 1 − η k + N + 1 ) ( g ( k ) − g ( k + 1 )) + η − k ( 1 − η k + N ) ( g ( k ) − g ( k − 1 )) , (28) and ( A ~ N g ) ( k ) = 0 (\widetilde{A}_N g)(k)=0 ( A N g ) ( k ) = 0 k < − N k<-N k < − N (16) , and using g ( k ) = 0 g(k)=0 g ( k ) = 0 k ≤ − M k\leq-M k ≤ − M [ − M − 1 , M + 1 ] [-M-1,M+1] [ − M − 1 , M + 1 ] 4 η N max ∣ g ∣ 4\eta^N\max|g| 4 η N max ∣ g ∣ L η ∞ g = L g L^\infty_\eta g=Lg L η ∞ g = Lg
We say that a sequence ( A N ) (A_N) ( A N ) A A A ( A N − z ) − 1 → ( A − z ) − 1 (A_N-z)^{-1}\to(A-z)^{-1} ( A N − z ) − 1 → ( A − z ) − 1 z ∈ C ∖ R z\in\mathbb{C}\setminus\mathbb{R} z ∈ C ∖ R
Theorem 3.6 Let 0 < τ < 1 0<\tau<1 0 < τ < 1 0 < η < 1 0<\eta<1 0 < η < 1
The operators A ~ N \widetilde{A}_N A N L η ∞ L^\infty_\eta L η ∞ N → ∞ N\to\infty N → ∞
The spectrum of L η ∞ L^\infty_\eta L η ∞ { η z : z ∈ Z } ∪ { 0 } \{\eta^z:z\in\mathbb{Z}\}\cup\{0\} { η z : z ∈ Z } ∪ { 0 } η z \eta^z η z 0 0 0
For any z ∈ Z z\in\mathbb{Z} z ∈ Z f z f_z f z ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) L η ∞ L^\infty_\eta L η ∞ η z \eta^z η z ( ϕ z ) z ∈ Z (\phi_z)_{z\in\mathbb{Z}} ( ϕ z ) z ∈ Z ϕ z : = ∥ f z ∥ − 1 f z \phi_z\mathrel{\mathop:}=\|f_z\|^{-1}f_z ϕ z : = ∥ f z ∥ − 1 f z ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) U ϕ z = ϕ z + 1 U\phi_z=\phi_{z+1} U ϕ z = ϕ z + 1
For any z ∈ Z z\in\mathbb{Z} z ∈ Z e N − z η e_{N-z}^\eta e N − z η L N L_N L N Λ N − z ( η ) \Lambda_{N-z}(\eta) Λ N − z ( η ) C V N e N − z η \mathbb{C}V_Ne_{N-z}^\eta C V N e N − z η C ϕ z \mathbb{C}\phi_z C ϕ z N → ∞ N\to\infty N → ∞
Proof : 1. Let z ∈ C ∖ R z\in\mathbb{C}\setminus\mathbb{R} z ∈ C ∖ R g ∈ D g\in\mathcal{D} g ∈ D N N N g ∈ dom A ~ N g\in\operatorname{dom}\widetilde{A}_N g ∈ dom A N
( A ~ N − z ) − 1 ( L η ∞ − z ) g − ( L η ∞ − z ) − 1 ( L η ∞ − z ) g = ( A ~ N − z ) − 1 ( L η ∞ − A ~ N ) g . (\widetilde{A}_N-z)^{-1}(L^\infty_\eta-z)g-(L^\infty_\eta-z)^{-1}(L^\infty_\eta-z)g=(\widetilde{A}_N-z)^{-1}(L^\infty_\eta-\widetilde{A}_N)g. ( A N − z ) − 1 ( L η ∞ − z ) g − ( L η ∞ − z ) − 1 ( L η ∞ − z ) g = ( A N − z ) − 1 ( L η ∞ − A N ) g . whose norm is bounded by ∣ Im z ∣ − 1 ∥ ( L η ∞ − A ~ N ) g ∥ |\operatorname{Im}z|^{-1}\|(L^\infty_\eta-\widetilde{A}_N)g\| ∣ Im z ∣ − 1 ∥ ( L η ∞ − A N ) g ∥ D \mathcal{D} D L η ∞ L^\infty_\eta L η ∞ ( L η ∞ − z ) D (L^\infty_\eta-z)\mathcal{D} ( L η ∞ − z ) D ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) ∥ ( A ~ N − z ) − 1 ∥ ≤ ∣ Im z ∣ − 1 \|(\widetilde{A}_N-z)^{-1}\|\leq|\operatorname{Im}z|^{-1} ∥ ( A N − z ) − 1 ∥ ≤ ∣ Im z ∣ − 1 N N N ( A ~ N − z ) − 1 → ( L η ∞ − z ) − 1 (\widetilde{A}_N-z)^{-1}\to(L^\infty_\eta-z)^{-1} ( A N − z ) − 1 → ( L η ∞ − z ) − 1
Let 0 < a < b 0<a<b 0 < a < b η Z \eta^{\mathbb{Z}} η Z L η ∞ L^\infty_\eta L η ∞ Z a , b : = { z ∈ Z : a < η z < b } Z_{a,b}\mathrel{\mathop:}=\{z\in\mathbb{Z}:a<\eta^z<b\} Z a , b : = { z ∈ Z : a < η z < b } N N N A ~ N \widetilde{A}_N A N ( a , b ) (a,b) ( a , b ) η N Λ N − z ( η ) \eta^N\Lambda_{N-z}(\eta) η N Λ N − z ( η ) z ∈ Z a , b z\in Z_{a,b} z ∈ Z a , b z ∈ Z z\in\mathbb{Z} z ∈ Z
η N Λ N − z ( η ) = η z ( 1 − η N − z ) ( 1 − τ η N − z ) ⟶ N → ∞ η z . (29) \eta^N\Lambda_{N-z}(\eta)=\eta^z(1-\eta^{N-z})(1-\tau\eta^{N-z})\longrightarrow_{N\to\infty}\eta^z. \tag*{(29)} η N Λ N − z ( η ) = η z ( 1 − η N − z ) ( 1 − τ η N − z ) ⟶ N → ∞ η z . (29) Let z + z_+ z + η z + ≥ b \eta^{z_+}\geq b η z + ≥ b z − z_- z − η z − ≤ a \eta^{z_-}\leq a η z − ≤ a N N N η N Λ N − z + ( η ) > b \eta^N\Lambda_{N-z_+}(\eta)>b η N Λ N − z + ( η ) > b η N Λ N − z − ( η ) < a \eta^N\Lambda_{N-z_-}(\eta)<a η N Λ N − z − ( η ) < a η N Λ N − z ( η ) ∈ ( a , b ) \eta^N\Lambda_{N-z}(\eta)\in(a,b) η N Λ N − z ( η ) ∈ ( a , b ) z ∈ Z a , b z\in Z_{a,b} z ∈ Z a , b n ↦ η N Λ n ( η ) n\mapsto\eta^N\Lambda_n(\eta) n ↦ η N Λ n ( η ) η N Λ n ( η ) \eta^N\Lambda_n(\eta) η N Λ n ( η ) n ≥ N − z + n\geq N-z_+ n ≥ N − z + b b b n ≤ N − z − n\leq N-z_- n ≤ N − z − a a a Z a , b = { z : z + < z < z − } Z_{a,b}=\{z:z_+<z<z_-\} Z a , b = { z : z + < z < z − } 1 ( a , b ) ( A ~ N ) \mathbf{1}_{(a,b)}(\widetilde{A}_N) 1 ( a , b ) ( A N ) card Z a , b \operatorname{card}Z_{a,b} card Z a , b N N N [33 Theorem VIII.24] , we have 1 ( a , b ) ( A ~ N ) → 1 ( a , b ) ( L η ∞ ) \mathbf{1}_{(a,b)}(\widetilde{A}_N)\to\mathbf{1}_{(a,b)}(L^\infty_\eta) 1 ( a , b ) ( A N ) → 1 ( a , b ) ( L η ∞ ) p p p p p p u 1 , … , u p + 1 u_1,\ldots,u_{p+1} u 1 , … , u p + 1 P P P P u 1 , … , P u p + 1 Pu_1,\ldots,Pu_{p+1} P u 1 , … , P u p + 1 rank 1 ( a , b ) ( L η ∞ ) ≤ card Z a , b \operatorname{rank}\mathbf{1}_{(a,b)}(L^\infty_\eta)\leq\operatorname{card}Z_{a,b} rank 1 ( a , b ) ( L η ∞ ) ≤ card Z a , b
Now, let λ ∈ ( 0 , ∞ ) ∖ η Z \lambda\in(0,\infty)\setminus\eta^{\mathbb{Z}} λ ∈ ( 0 , ∞ ) ∖ η Z L η ∞ L^\infty_\eta L η ∞ a < λ < b a<\lambda<b a < λ < b [ a , b ] ∩ η Z = ∅ [a,b]\cap\eta^{\mathbb{Z}}=\varnothing [ a , b ] ∩ η Z = ∅ Z a , b = ∅ Z_{a,b}=\varnothing Z a , b = ∅ 1 ( a , b ) ( L η ∞ ) = 0 \mathbf{1}_{(a,b)}(L^\infty_\eta)=0 1 ( a , b ) ( L η ∞ ) = 0 λ \lambda λ L η ∞ ≥ 0 L^\infty_\eta\ge0 L η ∞ ≥ 0 η Z ∪ { 0 } \eta^{\mathbb Z} \cup\{0\} η Z ∪ { 0 } z ∈ Z z \in\mathbb{Z} z ∈ Z ( a , b ) (a,b) ( a , b ) Z a , b = { z } Z_{a,b} = \{z\} Z a , b = { z } 1 { η z } ( L η ∞ ) \mathbf{1}_{\{\eta^z\}}(L^\infty_\eta) 1 { η z } ( L η ∞ ) 0 0 0 Proposition 3.1 , the spectral measure of L η ∞ L^\infty_\eta L η ∞ η Z \eta^{\mathbb Z} η Z L η ∞ = ∑ z ∈ Z η z 1 { η z } ( L η ∞ ) L^\infty_\eta= \sum_{z \in\mathbb{Z}} \eta^z \mathbf{1}_{\{\eta^z\}}(L^\infty_\eta) L η ∞ = ∑ z ∈ Z η z 1 { η z } ( L η ∞ ) Proposition 3.1 , if L η ∞ f = λ f L^\infty_\eta f = \lambda f L η ∞ f = λ f L η ∞ U f = η λ U f L^\infty_\eta Uf = \eta\lambda Uf L η ∞ U f = η λ U f U U U λ \lambda λ η λ \eta\lambda η λ Ker ( L η ∞ − η z ) \operatorname{Ker}(L^\infty_\eta-\eta^z) Ker ( L η ∞ − η z ) z ∈ Z z \in\mathbb{Z} z ∈ Z L η ∞ ≠ 0 L^\infty_\eta\ne0 L η ∞ = 0
Let ϕ \phi ϕ η z \eta^z η z Proposition 3.1 , we have L ϕ = η z ϕ L\phi= \eta^z\phi L ϕ = η z ϕ c k ( Δ ϕ ) k → 0 c_k(\Delta\phi)_k \to0 c k ( Δ ϕ ) k → 0 k → + ∞ k \to+\infty k → + ∞ w k ∣ ϕ ( k ) ∣ 2 ≤ ∥ ϕ ∥ 2 w_k|\phi(k)|^2 \le\|\phi\|^2 w k ∣ ϕ ( k ) ∣ 2 ≤ ∥ ϕ ∥ 2 ϕ ( k ) = O ( τ − k / 2 ) \phi(k) = O(\tau^{-k/2}) ϕ ( k ) = O ( τ − k /2 ) ϕ ( k ) = o ( τ − k / 2 ) \phi(k) = o(\tau^{-k/2}) ϕ ( k ) = o ( τ − k /2 ) w k ∣ ϕ ( k ) ∣ 2 → 0 w_k|\phi(k)|^2 \to0 w k ∣ ϕ ( k ) ∣ 2 → 0 Proposition 3.3 , ϕ \phi ϕ f z f_z f z f z ∈ ℓ 2 ( w ) f_z \in\ell^2(w) f z ∈ ℓ 2 ( w ) f z f_z f z ( ϕ z ) (\phi_z) ( ϕ z ) (24) gives ∥ f z + 1 ∥ = τ − 1 / 2 ∥ f z ∥ \|f_{z+1}\| = \tau^{-1/2}\|f_z\| ∥ f z + 1 ∥ = τ − 1/2 ∥ f z ∥ U ϕ z = ϕ z + 1 U\phi_z = \phi_{z+1} U ϕ z = ϕ z + 1
Choose 0 < a < b 0 < a < b 0 < a < b Z a , b = { z } Z_{a,b} = \{z\} Z a , b = { z } N N N 1 ( a , b ) ( A ^ N ) \mathbf{1}_{(a,b)}(\widehat A_N) 1 ( a , b ) ( A N ) C V N e N − z \mathbb{C}V_Ne_{N-z} C V N e N − z 1 ( a , b ) ( L η ∞ ) \mathbf{1}_{(a,b)}(L^\infty_\eta) 1 ( a , b ) ( L η ∞ ) C ϕ z \mathbb{C}\phi_z C ϕ z
Remark 3.7 Part 4 is a Hilbert space version of the limit transition from little q q q q q q [22 Proposition A.1] , which states that 2 ϕ 1 ( q − n , q n + 1 a b ; q a ; q , q n x ) → 1 ϕ 1 ( 0 ; q a ; q , x ) {}_2\phi_1(q^{-n},q^{n+1}ab;qa;q,q^nx) \to{}_1\phi_1(0;qa;q,x) 2 ϕ 1 ( q − n , q n + 1 ab ; q a ; q , q n x ) → 1 ϕ 1 ( 0 ; q a ; q , x ) q = η q=\eta q = η a = τ η − 1 a=\tau\eta^{-1} a = τ η − 1 b = 1 b=1 b = 1 V N e N − z V_Ne_{N-z} V N e N − z k ↦ p N − z ( η k + N ; τ η − 1 , 1 ; η ) = 2 ϕ 1 ( η − ( N − z ) , η N − z ; τ ; η , η N − z η k + z + 1 ) k \mapsto p_{N-z}(\eta^{k+N};\tau\eta^{-1},1;\eta) = {}_2\phi_1(\eta^{-(N-z)},\eta^{N-z};\tau;\eta,\eta^{N-z}\eta^{k+z+1}) k ↦ p N − z ( η k + N ; τ η − 1 , 1 ; η ) = 2 ϕ 1 ( η − ( N − z ) , η N − z ; τ ; η , η N − z η k + z + 1 ) f z ( k ) f_z(k) f z ( k ) Theorem 3.6 upgrades this to the convergence of the corresponding operators and spectral projections.
The energy bridge on the whole space
We return to the local field, so that τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0 ρ = Q α \rho=Q^\alpha ρ = Q α η = Q − α \eta=Q^{-\alpha} η = Q − α η τ = ρ τ \frac{\eta}{\tau}=\rho\tau τ η = ρ τ c k = ( 1 − τ ) ϖ k c_k=(1-\tau)\varpi_k c k = ( 1 − τ ) ϖ k ϖ \varpi ϖ (12) . We denote by ℓ Z 2 ( ϖ ) \ell^2_{\mathbb Z}(\varpi) ℓ Z 2 ( ϖ ) ∥ u ∥ ϖ = ( ∑ m ϖ m ∣ u m ∣ 2 ) 1 / 2 \|u\|_\varpi= \left(\sum_m \varpi_m|u_m|^2\right)^{1/2} ∥ u ∥ ϖ = ( ∑ m ϖ m ∣ u m ∣ 2 ) 1/2
Optimal constants
Lemma 4.1 The shift S S S ℓ Z 2 ( ϖ ) \ell^2_{\mathbb Z}(\varpi) ℓ Z 2 ( ϖ ) τ S = σ α S ~ \tau S=\sigma_\alpha\widetilde S τ S = σ α S S ~ \widetilde S S ℓ Z 2 ( ϖ ) \ell^2_{\mathbb Z}(\varpi) ℓ Z 2 ( ϖ ) I − τ S I-\tau S I − τ S ℓ Z 2 ( ϖ ) \ell^2_{\mathbb Z}(\varpi) ℓ Z 2 ( ϖ ) ( I − τ S ) − 1 = ∑ i ≥ 0 τ i S i (I-\tau S)^{-1}=\sum_{i\ge0}\tau^iS^i ( I − τ S ) − 1 = ∑ i ≥ 0 τ i S i u ∈ ℓ Z 2 ( ϖ ) u\in\ell^2_{\mathbb Z}(\varpi) u ∈ ℓ Z 2 ( ϖ )
( 1 + σ α ) − 1 ∥ u ∥ ϖ ≤ ∥ ( I − τ S ) − 1 u ∥ ϖ ≤ ( 1 − σ α ) − 1 ∥ u ∥ ϖ . (30) (1+\sigma_\alpha)^{-1}\|u\|_\varpi\le\|(I-\tau S)^{-1}u\|_\varpi\le(1-\sigma_\alpha)^{-1}\|u\|_\varpi. \tag*{(30)} ( 1 + σ α ) − 1 ∥ u ∥ ϖ ≤ ∥ ( I − τ S ) − 1 u ∥ ϖ ≤ ( 1 − σ α ) − 1 ∥ u ∥ ϖ . (30) Both constants are optimal, also when u u u
Proof. Put S ~ ≔ ( ρ τ ) 1 / 2 S \widetilde S \coloneqq(\rho\tau)^{1/2}S S : = ( ρ τ ) 1/2 S u ∈ ℓ Z 2 ( ϖ ) u\in\ell^2_{\mathbb Z}(\varpi) u ∈ ℓ Z 2 ( ϖ ) ∥ S ~ u ∥ ϖ 2 = ∑ m ϖ m ρ τ ∣ u m + 1 ∣ 2 = ∑ m ϖ m + 1 ∣ u m + 1 ∣ 2 = ∥ u ∥ ϖ 2 \|\widetilde Su\|_\varpi^2=\sum_m\varpi_m\rho\tau|u_{m+1}|^2=\sum_m\varpi_{m+1}|u_{m+1}|^2=\|u\|_\varpi^2 ∥ S u ∥ ϖ 2 = ∑ m ϖ m ρ τ ∣ u m + 1 ∣ 2 = ∑ m ϖ m + 1 ∣ u m + 1 ∣ 2 = ∥ u ∥ ϖ 2 S ~ \widetilde S S S ~ \widetilde S S τ ( ρ τ ) − 1 / 2 = σ α \tau(\rho\tau)^{-1/2}=\sigma_\alpha τ ( ρ τ ) − 1/2 = σ α u ↦ ( ϖ m 1 / 2 u m ) m u\mapsto(\varpi_m^{1/2}u_m)_m u ↦ ( ϖ m 1/2 u m ) m ℓ Z 2 ( ϖ ) \ell^2_{\mathbb Z}(\varpi) ℓ Z 2 ( ϖ ) ℓ Z 2 \ell^2_{\mathbb Z} ℓ Z 2 S ~ \widetilde S S e − i θ e^{-i\theta} e − i θ L 2 ( T ) L^2(\mathbb T) L 2 ( T ) σ α < 1 \sigma_\alpha<1 σ α < 1 I − τ S = I − σ α S ~ I-\tau S=I-\sigma_\alpha\widetilde S I − τ S = I − σ α S v v v 1 − σ α ≤ ∥ ( I − σ α S ~ ) v ∥ ≤ 1 + σ α 1-\sigma_\alpha\le\|(I-\sigma_\alpha\widetilde S)v\|\le1+\sigma_\alpha 1 − σ α ≤ ∥ ( I − σ α S ) v ∥ ≤ 1 + σ α (30) . For the optimality, let ζ ∈ { − 1 , 1 } \zeta\in\{-1,1\} ζ ∈ { − 1 , 1 } θ ↦ ∣ 1 − σ α e − i θ ∣ − 1 \theta\mapsto\lvert1-\sigma_\alpha e^{-i\theta}\rvert^{-1} θ ↦ ∣ 1 − σ α e − i θ ∣ − 1 T \mathbb{T} T ∣ 1 − σ α ζ ∣ − 1 \lvert1-\sigma_\alpha\zeta\rvert^{-1} ∣ 1 − σ α ζ ∣ − 1 e − i θ = ζ e^{-i\theta}=\zeta e − i θ = ζ v j v_j v j
∥ ( I − σ α S ~ ) − 1 v j ∥ ϖ → ∣ 1 − σ α ζ ∣ − 1 . \left\|(I-\sigma_\alpha\widetilde{S})^{-1}v_j\right\|_{\varpi}\to\lvert1-\sigma_\alpha\zeta\rvert^{-1}. ( I − σ α S ) − 1 v j ϖ → ∣ 1 − σ α ζ ∣ − 1 . The cases ζ = 1 \zeta=1 ζ = 1 ζ = − 1 \zeta=-1 ζ = − 1
Theorem 4.2 Let α > 0 \alpha>0 α > 0 η = Q − α \eta=Q^{-\alpha} η = Q − α
For any f ∈ ℓ Z 2 ( w ) f\in\ell_Z^2(w) f ∈ ℓ Z 2 ( w ) f ∈ dom E D f\in\operatorname{dom}\mathcal{E}_D f ∈ dom E D Δ f ∈ ℓ Z 2 ( ϖ ) \Delta f\in\ell_Z^2(\varpi) Δ f ∈ ℓ Z 2 ( ϖ ) f ∈ dom E η ∞ f\in\operatorname{dom}\mathcal{E}_\eta^\infty f ∈ dom E η ∞
E D ( f ) = ( 1 − τ ) ∥ ( I − τ S ) − 1 Δ f ∥ ϖ 2 , E η ∞ ( f ) = ( 1 − τ ) ∥ Δ f ∥ ϖ 2 . (31) \mathcal{E}_D(f)=(1-\tau)\left\|(I-\tau S)^{-1}\Delta f\right\|_{\varpi}^{2},\qquad\mathcal{E}_\eta^\infty(f)=(1-\tau)\left\|\Delta f\right\|_{\varpi}^{2}. \tag*{(31)} E D ( f ) = ( 1 − τ ) ( I − τ S ) − 1 Δ f ϖ 2 , E η ∞ ( f ) = ( 1 − τ ) ∥ Δ f ∥ ϖ 2 . (31) For any f ∈ dom E D f\in\operatorname{dom}\mathcal{E}_D f ∈ dom E D
( 1 + σ α ) − 2 E η ∞ ( f ) ≤ E D ( f ) ≤ ( 1 − σ α ) − 2 E η ∞ ( f ) , (1+\sigma_\alpha)^{-2}\mathcal{E}_\eta^\infty(f)\leq\mathcal{E}_D(f)\leq(1-\sigma_\alpha)^{-2}\mathcal{E}_\eta^\infty(f), ( 1 + σ α ) − 2 E η ∞ ( f ) ≤ E D ( f ) ≤ ( 1 − σ α ) − 2 E η ∞ ( f ) , and these constants are optimal.
The operators D r a d α D_{\mathrm{rad}}^\alpha D rad α L η ∞ L_\eta^\infty L η ∞ { ρ m : m ∈ Z } ∪ { 0 } \{\rho^m:m\in\mathbb{Z}\}\cup\{0\} { ρ m : m ∈ Z } ∪ { 0 } ρ m \rho^m ρ m 0 0 0 W W W ℓ Z 2 ( w ) \ell_Z^2(w) ℓ Z 2 ( w ) W g m ≔ ϕ − m − 1 Wg_m\coloneqq\phi_{-m-1} W g m : = ϕ − m − 1 m ∈ Z m\in\mathbb{Z} m ∈ Z W D r a d α W ∗ = L η ∞ WD_{\mathrm{rad}}^\alpha W^*=L_\eta^\infty W D rad α W ∗ = L η ∞ W U = U W WU=UW W U = U W
The operators D r a d α D_{\mathrm{rad}}^\alpha D rad α L η ∞ L_\eta^\infty L η ∞
Proof: 1. Let f ∈ ℓ Z 2 ( w ) f\in\ell_Z^2(w) f ∈ ℓ Z 2 ( w ) h m ≔ m m + 1 ( f ) − f m h_m\coloneqq m_{m+1}(f)-f_m h m : = m m + 1 ( f ) − f m u ≔ Δ f u\coloneqq\Delta f u : = Δ f (11) , h m = ∑ i ≥ 0 τ i u m + i h_m=\sum_{i\geq0}\tau^i u_{m+i} h m = ∑ i ≥ 0 τ i u m + i h m − τ h m + 1 = u m h_m-\tau h_{m+1}=u_m h m − τ h m + 1 = u m m m m u = ( I − τ S ) h u=(I-\tau S)h u = ( I − τ S ) h Proposition 2.3 , we have E D ( f ) = ( 1 − τ ) ∥ h ∥ ϖ 2 \mathcal{E}_D(f)=(1-\tau)\lVert h\rVert_{\varpi}^2 E D ( f ) = ( 1 − τ ) ∥ h ∥ ϖ 2 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] h ∈ ℓ Z 2 ( ϖ ) h\in\ell_Z^2(\varpi) h ∈ ℓ Z 2 ( ϖ ) u = ( I − τ S ) h ∈ ℓ Z 2 ( ϖ ) u=(I-\tau S)h\in\ell_Z^2(\varpi) u = ( I − τ S ) h ∈ ℓ Z 2 ( ϖ ) u ∈ ℓ Z 2 ( ϖ ) u\in\ell_Z^2(\varpi) u ∈ ℓ Z 2 ( ϖ ) ( I − τ S ) − 1 u = ∑ i ≥ 0 τ i S i u (I-\tau S)^{-1}u=\sum_{i\geq0}\tau^iS^iu ( I − τ S ) − 1 u = ∑ i ≥ 0 τ i S i u ℓ Z 2 ( ϖ ) \ell_Z^2(\varpi) ℓ Z 2 ( ϖ ) ∑ i ≥ 0 τ i u m + i = h m \sum_{i\geq0}\tau^iu_{m+i}=h_m ∑ i ≥ 0 τ i u m + i = h m h = ( I − τ S ) − 1 u ∈ ℓ Z 2 ( ϖ ) h=(I-\tau S)^{-1}u\in\ell_Z^2(\varpi) h = ( I − τ S ) − 1 u ∈ ℓ Z 2 ( ϖ ) E η ∞ ( f ) = ∑ k c k ∣ u k ∣ 2 = ( 1 − τ ) ∥ u ∥ ϖ 2 \mathcal{E}_\eta^\infty(f)=\sum_k c_k\lvert u_k\rvert^2=(1-\tau)\lVert u\rVert_{\varpi}^2 E η ∞ ( f ) = ∑ k c k ∣ u k ∣ 2 = ( 1 − τ ) ∥ u ∥ ϖ 2
The inequalities follow from (31) and (30) . Let v v v f f f f m ≔ ∑ i < m v i f_m\coloneqq\sum_{i<m}v_i f m : = ∑ i < m v i − ∞ -\infty − ∞ + ∞ +\infty + ∞ D ⊂ ℓ Z 2 ( w ) D\subset\ell_Z^2(w) D ⊂ ℓ Z 2 ( w ) Δ f = v \Delta f=v Δ f = v dom E D \operatorname{dom}\mathcal{E}_D dom E D
By Proposition 2.1 , D r a d α D_{\mathrm{rad}}^\alpha D rad α ( g m ) (g_m) ( g m ) ρ m + 1 \rho^{m+1} ρ m + 1 Theorem 3.6 , L η ∞ L_\eta^\infty L η ∞ ( ϕ z ) (\phi_z) ( ϕ z ) η z = ρ − z \eta^z=\rho^{-z} η z = ρ − z W W W g m g_m g m D r a d α D_{\mathrm{rad}}^\alpha D rad α ρ m + 1 \rho^{m+1} ρ m + 1 ϕ − m − 1 \phi_{-m-1} ϕ − m − 1 L η ∞ L_\eta^\infty L η ∞ W D r a d α W ∗ = L η ∞ WD_{\mathrm{rad}}^\alpha W^*=L_\eta^\infty W D rad α W ∗ = L η ∞ U g m = g m − 1 Ug_m=g_{m-1} U g m = g m − 1 U ϕ z = ϕ z + 1 U\phi_z=\phi_{z+1} U ϕ z = ϕ z + 1 W U g m = ϕ − m = U W g m WUg_m=\phi_{-m}=UWg_m W U g m = ϕ − m = U W g m m m m
Since the eigenvalues are simple, a common eigenvector would be a multiple of some g m g_m g m f z f_z f z g m g_m g m { k ≥ m + 1 } \{k\geq m+1\} { k ≥ m + 1 } f z f_z f z ( L f z ) ( k ) = 0 (Lf_z)(k)=0 ( L f z ) ( k ) = 0 k ≥ m + 2 k\geq m+2 k ≥ m + 2 η z f z ( k ) = 0 \eta^zf_z(k)=0 η z f z ( k ) = 0 k ≥ m + 2 k\geq m+2 k ≥ m + 2 f z ( k ) → 1 f_z(k)\to1 f z ( k ) → 1 k → + ∞ k\to+\infty k → + ∞ Proposition 3.3 ).
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For α = d \alpha=d α = d ϖ = 1 \varpi=1 ϖ = 1 σ d = τ \sigma_d=\tau σ d = τ c k = 1 − τ c_k=1-\tau c k = 1 − τ L τ ∞ f ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 ) ) L_\tau^\infty f(k)=\tau^{-k}(2f(k)-f(k+1)-f(k-1)) L τ ∞ f ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 )) E τ ∞ ( f ) = ( 1 − τ ) ∥ Δ f ∥ ℓ Z 2 2 \mathcal{E}_\tau^\infty(f)=(1-\tau)\lVert\Delta f\rVert_{\ell_Z^2}^2 E τ ∞ ( f ) = ( 1 − τ ) ∥ Δ f ∥ ℓ Z 2 2 E D = \mathcal{E}_D= E D = ( 1 − τ ) ∥ ( I − τ S ) − 1 Δ f ∥ ℓ ϖ 2 2 (1-\tau)\|(I-\tau S)^{-1}\Delta f\|_{\ell^2_\varpi}^2 ( 1 − τ ) ∥ ( I − τ S ) − 1 Δ f ∥ ℓ ϖ 2 2 ℓ Z 2 \ell^2_{\mathbb{Z}} ℓ Z 2 ∣ 1 − τ e i θ ∣ − 2 |1-\tau e^{i\theta}|^{-2} ∣1 − τ e i θ ∣ − 2 ( 1 + τ ) − 2 (1+\tau)^{-2} ( 1 + τ ) − 2 ( 1 − τ ) − 2 (1-\tau)^{-2} ( 1 − τ ) − 2 c 1 = 1 − τ 1 + τ c_1=\frac{1-\tau}{1+\tau} c 1 = 1 + τ 1 − τ c 2 = ( 1 − τ ) − 3 c_2=(1-\tau)^{-3} c 2 = ( 1 − τ ) − 3 R r a d d R_{\mathrm{rad}}^d R rad d L τ L_\tau L τ (10) for m ≥ 0 m\geq0 m ≥ 0 ρ m + 1 − κ \rho^{m+1}-\kappa ρ m + 1 − κ R r a d α R_{\mathrm{rad}}^\alpha R rad α κ = 1 − τ 1 − τ ρ − 1 \kappa=\frac{1-\tau}{1-\tau\rho^{-1}} κ = 1 − τ ρ − 1 1 − τ α > 0 \alpha>0 α > 0 c 1 ′ = ρ − 1 ρ − 1 ( 1 + σ α ) − 2 c'_1=\frac{\rho-1}{\rho-1}(1+\sigma_\alpha)^{-2} c 1 ′ = ρ − 1 ρ − 1 ( 1 + σ α ) − 2 c 2 ′ = ρ ρ − 1 ( 1 − σ α ) − 2 c'_2=\frac{\rho}{\rho-1}(1-\sigma_\alpha)^{-2} c 2 ′ = ρ − 1 ρ ( 1 − σ α ) − 2 1 − κ ρ ≤ 1 − κ ρ − m − 1 ≤ 1 1-\frac{\kappa}{\rho}\leq1-\kappa\rho^{-m-1}\leq1 1 − ρ κ ≤ 1 − κ ρ − m − 1 ≤ 1 1 − η ≤ 1 − η m + 1 ≤ 1 1-\eta\leq1-\eta^{m+1}\leq1 1 − η ≤ 1 − η m + 1 ≤ 1 ∥ τ S ∥ ℓ η 2 ( ϖ ) → ℓ η 2 ( ϖ ) ≤ σ α \|\tau S\|_{\ell^2_{\eta}(\varpi)\to\ell^2_{\eta}(\varpi)}\leq\sigma_\alpha ∥ τ S ∥ ℓ η 2 ( ϖ ) → ℓ η 2 ( ϖ ) ≤ σ α α = d \alpha=d α = d c 1 ′ = ( 1 + τ ) − 3 c'_1=(1+\tau)^{-3} c 1 ′ = ( 1 + τ ) − 3 c 2 ′ = ( 1 − τ ) − 3 c'_2=(1-\tau)^{-3} c 2 ′ = ( 1 − τ ) − 3 c 2 ′ c'_2 c 2 ′ α → d − \alpha\to d^- α → d −
Powers of the bilateral operators
In this subsection, we fix ϑ > 0 \vartheta>0 ϑ > 0 η ′ : = Q − ϑ \eta'\mathrel{\vcentcolon=}Q^{-\vartheta} η ′ : = Q − ϑ s : = ϑ δ s\mathrel{\vcentcolon=}\frac{\vartheta}{\delta} s : = δ ϑ ( φ z ) z ∈ Z (\varphi_z)_{z\in\mathbb{Z}} ( φ z ) z ∈ Z L η ′ ∞ L_{\eta'}^\infty L η ′ ∞ Theorem 3.6 . Then ( L η ′ ∞ ) s φ z = η ′ z s φ z = ρ − z φ z (L_{\eta'}^\infty)^s\varphi_z=\eta'^zs\varphi_z=\rho^{-z}\varphi_z ( L η ′ ∞ ) s φ z = η ′ z s φ z = ρ − z φ z ( L η ′ ∞ ) s (L_{\eta'}^\infty)^s ( L η ′ ∞ ) s D r a d α D_{\mathrm{rad}}^\alpha D rad α ϑ > 0 \vartheta>0 ϑ > 0
Lemma 4.4 Let λ > 0 \lambda>0 λ > 0 φ ∈ ℓ η 2 ( w ) \varphi\in\ell^2_\eta(w) φ ∈ ℓ η 2 ( w ) L φ = λ φ L\varphi=\lambda\varphi L φ = λ φ L L L (16) with parameters τ \tau τ η ′ \eta' η ′ 0 < θ < 1 0<\theta<1 0 < θ < 1 φ ( k ) = O ( θ ∣ k ∣ ) \varphi(k)=O(\theta^{|k|}) φ ( k ) = O ( θ ∣ k ∣ ) k → − ∞ k\to-\infty k → − ∞
Proof. We can assume that φ ≠ 0 \varphi\ne0 φ = 0 ( L φ ) ( k ) = λ φ ( k ) (L\varphi)(k)=\lambda\varphi(k) ( L φ ) ( k ) = λ φ ( k ) δ k = η ′ − k \delta_k=\eta'^{-k} δ k = η ′ − k
φ ( k − 1 ) = − A k φ ( k ) − τ η ′ φ ( k + 1 ) , A k : = λ η ′ k − 1 − τ η ′ , k ∈ Z . (32) \varphi(k-1)=-A_k\varphi(k)-\frac{\tau}{\eta'}\varphi(k+1),\qquad A_k\mathrel{\vcentcolon=}\lambda\eta'^k-1-\frac{\tau}{\eta'},\qquad k\in\mathbb{Z}. \tag*{(32)} φ ( k − 1 ) = − A k φ ( k ) − η ′ τ φ ( k + 1 ) , A k : = λ η ′ k − 1 − η ′ τ , k ∈ Z . (32) Since A k → + ∞ A_k\to+\infty A k → + ∞ k → − ∞ k\to-\infty k → − ∞ k 0 k_0 k 0 A k ≥ 2 + τ η ′ A_k\geq2+\frac{\tau}{\eta'} A k ≥ 2 + η ′ τ k ≤ k 0 k\leq k_0 k ≤ k 0 ∣ φ ( k ) ∣ ≥ ∣ φ ( k + 1 ) ∣ |\varphi(k)|\geq|\varphi(k+1)| ∣ φ ( k ) ∣ ≥ ∣ φ ( k + 1 ) ∣ k ≤ k 0 k\leq k_0 k ≤ k 0 (32) gives ∣ φ ( k − 1 ) ∣ ≥ ( A k − τ η ′ ) ∣ φ ( k ) ∣ ≥ 2 ∣ φ ( k ) ∣ |\varphi(k-1)|\geq(A_k-\frac{\tau}{\eta'})|\varphi(k)|\geq2|\varphi(k)| ∣ φ ( k − 1 ) ∣ ≥ ( A k − η ′ τ ) ∣ φ ( k ) ∣ ≥ 2∣ φ ( k ) ∣ ∣ φ ( j ) ∣ ≥ 2 k − j ∣ φ ( k ) ∣ |\varphi(j)|\geq2^{k-j}|\varphi(k)| ∣ φ ( j ) ∣ ≥ 2 k − j ∣ φ ( k ) ∣ j ≤ k j\leq k j ≤ k w j → ∞ w_j\to\infty w j → ∞ j → − ∞ j\to-\infty j → − ∞ φ ∈ ℓ η 2 ( w ) \varphi\in\ell^2_\eta(w) φ ∈ ℓ η 2 ( w ) φ ( j ) → 0 \varphi(j)\to0 φ ( j ) → 0 φ ( k ) = 0 \varphi(k)=0 φ ( k ) = 0 φ ( k + 1 ) = 0 \varphi(k+1)=0 φ ( k + 1 ) = 0 φ = 0 \varphi=0 φ = 0 ∣ φ ( k ) ∣ < ∣ φ ( k + 1 ) ∣ |\varphi(k)|<|\varphi(k+1)| ∣ φ ( k ) ∣ < ∣ φ ( k + 1 ) ∣ k ≤ k 0 k\leq k_0 k ≤ k 0 (32) gives
A k ∣ φ ( k ) ∣ ≤ ∣ φ ( k − 1 ) ∣ + τ η ′ ∣ φ ( k + 1 ) ∣ ≤ ( 1 + τ η ′ ) ∣ φ ( k + 1 ) ∣ , k ≤ k 0 . A_k|\varphi(k)|\leq|\varphi(k-1)|+\frac{\tau}{\eta'}|\varphi(k+1)|\leq\left(1+\frac{\tau}{\eta'}\right)|\varphi(k+1)|,\qquad k\leq k_0. A k ∣ φ ( k ) ∣ ≤ ∣ φ ( k − 1 ) ∣ + η ′ τ ∣ φ ( k + 1 ) ∣ ≤ ( 1 + η ′ τ ) ∣ φ ( k + 1 ) ∣ , k ≤ k 0 . Since A k − 1 ( 1 + τ η ′ ) → 0 A_k^{-1}(1+\frac{\tau}{\eta'})\to0 A k − 1 ( 1 + η ′ τ ) → 0 k → − ∞ k\to-\infty k → − ∞ ∣ φ ( k ) ∣ ∣ φ ( k + 1 ) ∣ \frac{|\varphi(k)|}{|\varphi(k+1)|} ∣ φ ( k + 1 ) ∣ ∣ φ ( k ) ∣ θ \theta θ
By the dilation covariance, the transition coefficients between the two eigenbases depend only on the sum of the indices. Indeed, U g n = g n − 1 Ug_n=g_{n-1} U g n = g n − 1 U φ z = φ z + 1 U\varphi_z=\varphi_{z+1} U φ z = φ z + 1 ⟨ φ z + 1 , g n − 1 ⟩ = ⟨ U φ z , U g n ⟩ = ⟨ φ z , g n ⟩ \langle\varphi_{z+1},g_{n-1}\rangle=\langle U\varphi_z,Ug_n\rangle=\langle\varphi_z,g_n\rangle ⟨ φ z + 1 , g n − 1 ⟩ = ⟨ U φ z , U g n ⟩ = ⟨ φ z , g n ⟩ ( t ( p ) ) p ∈ Z (t(p))_{p\in\mathbb{Z}} ( t ( p ) ) p ∈ Z
⟨ φ z , g n ⟩ = t ( n + z ) , n , z ∈ Z , ∑ p ∈ Z t ( p ) 2 = ∑ n ∈ Z ∣ ⟨ φ 0 , g n ⟩ ∣ 2 = 1. (33) \langle\varphi_z,g_n\rangle=t(n+z),\qquad n,z\in\mathbb{Z},\qquad\sum_{p\in\mathbb{Z}}t(p)^2=\sum_{n\in\mathbb{Z}}|\langle\varphi_0,g_n\rangle|^2=1. \tag*{(33)} ⟨ φ z , g n ⟩ = t ( n + z ) , n , z ∈ Z , p ∈ Z ∑ t ( p ) 2 = n ∈ Z ∑ ∣ ⟨ φ 0 , g n ⟩ ∣ 2 = 1. (33) Lemma 4.5 1. There exist constants 0 < C 1 ≤ C 2 0<C_1\leq C_2 0 < C 1 ≤ C 2 C 1 τ ρ 2 η ′ p ≤ ∣ t ( p ) ∣ ≤ C 2 τ ρ 2 η ′ p C_1\tau^{\frac{\rho}{2}}\eta'^p\leq|t(p)|\leq C_2\tau^{\frac{\rho}{2}}\eta'^p C 1 τ 2 ρ η ′ p ≤ ∣ t ( p ) ∣ ≤ C 2 τ 2 ρ η ′ p p p p
For any 0 < θ < 1 0<\theta<1 0 < θ < 1 t ( p ) = O ( θ ∣ p ∣ ) t(p)=O(\theta^{|p|}) t ( p ) = O ( θ ∣ p ∣ ) p → − ∞ p\to-\infty p → − ∞
Proof: 1. By (10) , we have t ( p ) = ⟨ ϕ 0 , g p ⟩ = s p ( m p + 1 ( ϕ 0 ) − ϕ 0 ( p ) ) t(p)=\langle\phi_0,g_p\rangle=s_p(m_{p+1}(\phi_0)-\phi_0(p)) t ( p ) = ⟨ ϕ 0 , g p ⟩ = s p ( m p + 1 ( ϕ 0 ) − ϕ 0 ( p )) ϕ 0 = ∥ f 0 ∥ − 1 f 0 \phi_0=\lVert f_0\rVert^{-1}f_0 ϕ 0 = ∥ f 0 ∥ − 1 f 0 f 0 ( k ) = 1 ϕ 1 ( 0 ; τ ; η ′ , η ′ k + 1 ) f_0(k)={}_1\phi_1(0;\tau;\eta',\eta'^{k+1}) f 0 ( k ) = 1 ϕ 1 ( 0 ; τ ; η ′ , η ′ k + 1 ) f 0 ( k ) = ∑ j ≥ 0 a j η ′ j ( k + 1 ) f_0(k)=\sum_{j\geq0}a_j\eta'^{j(k+1)} f 0 ( k ) = ∑ j ≥ 0 a j η ′ j ( k + 1 ) a j = ( − 1 ) j η ′ j ( j − 1 ) / 2 ( η ′ ; η ′ ) j ( τ ; η ′ ) j a_j=\frac{(-1)^j\eta'^{j(j-1)/2}}{(\eta';\eta')_j(\tau;\eta')_j} a j = ( η ′ ; η ′ ) j ( τ ; η ′ ) j ( − 1 ) j η ′ j ( j − 1 ) /2 j ≥ 0 j\geq0 j ≥ 0 k ↦ η ′ j k k\mapsto\eta'^{jk} k ↦ η ′ j k m p + 1 ( η ′ j ⋅ ) = ( 1 − τ ) ∑ i ≥ 1 τ i − 1 η ′ j ( p + i ) = ( 1 − τ ) η ′ j ( p + 1 ) 1 − τ η ′ j m_{p+1}(\eta'^{j\cdot})=(1-\tau)\sum_{i\geq1}\tau^{i-1}\eta'^{j(p+i)}=\frac{(1-\tau)\eta'^{j(p+1)}}{1-\tau\eta'^j} m p + 1 ( η ′ j ⋅ ) = ( 1 − τ ) ∑ i ≥ 1 τ i − 1 η ′ j ( p + i ) = 1 − τ η ′ j ( 1 − τ ) η ′ j ( p + 1 )
m p + 1 ( f 0 ) − f 0 ( p ) = ∑ j ≥ 1 a j η ′ j ( p + 1 ) ( 1 − τ 1 − τ η ′ j − 1 ) = ∑ j ≥ 1 a j η ′ j ( p + 1 ) τ ( η ′ j − 1 ) 1 − τ η ′ j . m_{p+1}(f_0)-f_0(p)=\sum_{j\geq1}a_j\eta'^{j(p+1)}\left(\frac{1-\tau}{1-\tau\eta'^j}-1\right)=\sum_{j\geq1}a_j\eta'^{j(p+1)}\frac{\tau(\eta'^j-1)}{1-\tau\eta'^j}. m p + 1 ( f 0 ) − f 0 ( p ) = j ≥ 1 ∑ a j η ′ j ( p + 1 ) ( 1 − τ η ′ j 1 − τ − 1 ) = j ≥ 1 ∑ a j η ′ j ( p + 1 ) 1 − τ η ′ j τ ( η ′ j − 1 ) . The term j = 1 j=1 j = 1 η ′ p + 1 ( 1 − τ ) ( 1 − τ η ′ ) \frac{\eta'^{p+1}}{(1-\tau)(1-\tau\eta')} ( 1 − τ ) ( 1 − τ η ′ ) η ′ p + 1 O ( η ′ 2 p ) O(\eta'^{2p}) O ( η ′ 2 p ) p → + ∞ p\to+\infty p → + ∞ s p = ( ( 1 − τ ) τ p + 1 ) 1 2 s_p=((1-\tau)\tau^{p+1})^{\frac12} s p = (( 1 − τ ) τ p + 1 ) 2 1 ϕ 0 = ∥ f 0 ∥ − 1 f 0 \phi_0=\lVert f_0\rVert^{-1}f_0 ϕ 0 = ∥ f 0 ∥ − 1 f 0
Let p ∈ Z p\in\mathbb{Z} p ∈ Z g p ∈ D ⊂ dom L η ′ ∞ g_p\in D\subset\operatorname{dom}L_{\eta'}^\infty g p ∈ D ⊂ dom L η ′ ∞ L η ′ ∞ ϕ 0 = ϕ 0 L_{\eta'}^\infty\phi_0=\phi_0 L η ′ ∞ ϕ 0 = ϕ 0 t ( p ) = ⟨ L η ′ ∞ ϕ 0 , g p ⟩ = ⟨ ϕ 0 , L g p ⟩ t(p)=\langle L_{\eta'}^\infty\phi_0,g_p\rangle=\langle\phi_0,Lg_p\rangle t ( p ) = ⟨ L η ′ ∞ ϕ 0 , g p ⟩ = ⟨ ϕ 0 , L g p ⟩ Δ g p \Delta g_p Δ g p { p − 1 , p } \{p-1,p\} { p − 1 , p } ∣ ( Δ g p ) p − 1 ∣ = 1 s p \lvert(\Delta g_p)_{p-1}\rvert=\frac{1}{s_p} ∣( Δ g p ) p − 1 ∣ = s p 1 ( Δ g p ) p = 1 s p (\Delta g_p)_p=\frac{1}{s_p} ( Δ g p ) p = s p 1 w k ( L g p ) ( k ) = c k − 1 ( Δ g p ) k − 1 − c k ( Δ g p ) k w_k(Lg_p)(k)=c_{k-1}(\Delta g_p)_{k-1}-c_k(\Delta g_p)_k w k ( L g p ) ( k ) = c k − 1 ( Δ g p ) k − 1 − c k ( Δ g p ) k { p − 1 , p , p + 1 } \{p-1,p,p+1\} { p − 1 , p , p + 1 } 2 max ( c p − 1 , c p ) s p \frac{2\max(c_{p-1},c_p)}{s_p} s p 2 m a x ( c p − 1 , c p ) c k = ( 1 − τ ) ( τ η ′ ) k + 1 c_k=(1-\tau)\left(\frac{\tau}{\eta'}\right)^{k+1} c k = ( 1 − τ ) ( η ′ τ ) k + 1 C C C p p p
∣ t ( p ) ∣ ≤ 6 max ( c p − 1 , c p ) s p max ∣ i ∣ ≤ 1 ∣ ϕ 0 ( p + i ) ∣ ≤ C ( τ 1 2 η ′ ) p max ∣ i ∣ ≤ 1 ∣ ϕ 0 ( p + i ) ∣ . \lvert t(p)\rvert\leq\frac{6\max(c_{p-1},c_p)}{s_p}\max_{\lvert i\rvert\leq1}\lvert\phi_0(p+i)\rvert\leq C\left(\frac{\tau^{\frac12}}{\eta'}\right)^p\max_{\lvert i\rvert\leq1}\lvert\phi_0(p+i)\rvert. ∣ t ( p )∣ ≤ s p 6 max ( c p − 1 , c p ) ∣ i ∣ ≤ 1 max ∣ ϕ 0 ( p + i )∣ ≤ C ( η ′ τ 2 1 ) p ∣ i ∣ ≤ 1 max ∣ ϕ 0 ( p + i )∣ . By Lemma 4.4 , ϕ 0 \phi_0 ϕ 0 − ∞ -\infty − ∞
Theorem 4.6 Let α > 0 \alpha>0 α > 0 ϑ > 0 \vartheta>0 ϑ > 0 s ≔ α ϑ s\coloneqq\frac{\alpha}{\vartheta} s : = ϑ α
There exists a constant C > 0 C>0 C > 0 ∥ ( L Q − ϑ ∞ ) s 2 f ∥ 2 ≤ C E D ( f ) \left\lVert(L_{Q^{-\vartheta}}^\infty)^{\frac{s}{2}}f\right\rVert^2\leq C\mathcal{E}_D(f) ( L Q − ϑ ∞ ) 2 s f 2 ≤ C E D ( f ) f ∈ dom E D f\in\operatorname{dom}\mathcal{E}_D f ∈ dom E D dom E D ⊂ dom ( L Q − ϑ ∞ ) s 2 \operatorname{dom}\mathcal{E}_D\subset\operatorname{dom}(L_{Q^{-\vartheta}}^\infty)^{\frac{s}{2}} dom E D ⊂ dom ( L Q − ϑ ∞ ) 2 s
The closed forms of D r a d α D^\alpha_{\mathrm{rad}} D rad α ( L Q − ϑ ∞ ) s (L_{Q^{-\vartheta}}^\infty)^s ( L Q − ϑ ∞ ) s α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ
If α ≥ d + 2 ϑ \alpha\geq d+2\vartheta α ≥ d + 2 ϑ L Q − ϑ ∞ L_{Q^{-\vartheta}}^\infty L Q − ϑ ∞ dom E D \operatorname{dom}\mathcal{E}_D dom E D
Proof: Let f ∈ ℓ 2 ( w ) f\in\ell^2(w) f ∈ ℓ 2 ( w ) a z ≔ ⟨ f , ϕ z ⟩ a_z\coloneqq\langle f,\phi_z\rangle a z : = ⟨ f , ϕ z ⟩ c n ≔ ⟨ f , g n ⟩ c_n\coloneqq\langle f,g_n\rangle c n : = ⟨ f , g n ⟩ ∥ ( L η ′ ∞ ) s 2 f ∥ 2 = ∑ z ρ − z ∣ a z ∣ 2 \left\lVert(L_{\eta'}^\infty)^{\frac{s}{2}}f\right\rVert^2=\sum_z\rho^{-z}\lvert a_z\rvert^2 ( L η ′ ∞ ) 2 s f 2 = ∑ z ρ − z ∣ a z ∣ 2 E D ( f ) = ∑ n ρ n + 1 ∣ c n ∣ 2 \mathcal{E}_D(f)=\sum_n\rho^{n+1}\lvert c_n\rvert^2 E D ( f ) = ∑ n ρ n + 1 ∣ c n ∣ 2 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] ( g n ) (g_n) ( g n ) ( ϕ z ) (\phi_z) ( ϕ z ) (33) are real, Parseval’s identity gives
c n = ∑ z ∈ Z a z t ( n + z ) , a z = ∑ n ∈ Z c n t ( n + z ) , c_n=\sum_{z\in\mathbb{Z}}a_zt(n+z),\qquad a_z=\sum_{n\in\mathbb{Z}}c_nt(n+z), c n = z ∈ Z ∑ a z t ( n + z ) , a z = n ∈ Z ∑ c n t ( n + z ) , the series being absolutely convergent by the Cauchy–Schwarz inequality and (33) . Put B m ≔ ρ m + 1 2 a − m − 1 B_m\coloneqq\rho^{\frac{m+1}{2}}a_{-m-1} B m : = ρ 2 m + 1 a − m − 1 C n ≔ ρ n + 1 2 c n C_n\coloneqq\rho^{\frac{n+1}{2}}c_n C n : = ρ 2 n + 1 c n ∑ m ∣ B m ∣ 2 = ∥ ( L η ′ ∞ ) s 2 f ∥ 2 \sum_m\lvert B_m\rvert^2=\left\lVert(L_{\eta'}^\infty)^{\frac{s}{2}}f\right\rVert^2 ∑ m ∣ B m ∣ 2 = ( L η ′ ∞ ) 2 s f 2 ∑ n ∣ C n ∣ 2 = E D ( f ) \sum_n\lvert C_n\rvert^2=\mathcal{E}_D(f) ∑ n ∣ C n ∣ 2 = E D ( f )
C n = ∑ m ∈ Z κ 1 ( n − m ) B m , B m = ∑ n ∈ Z κ 2 ( m − n ) C n , (34) C_n=\sum_{m\in\mathbb{Z}}\kappa_1(n-m)B_m,\qquad B_m=\sum_{n\in\mathbb{Z}}\kappa_2(m-n)C_n, \tag*{(34)} C n = m ∈ Z ∑ κ 1 ( n − m ) B m , B m = n ∈ Z ∑ κ 2 ( m − n ) C n , (34) where κ 1 ( j ) ≔ ρ j 2 t ( j − 1 ) \kappa_1(j)\coloneqq\rho^{\frac{j}{2}}t(j-1) κ 1 ( j ) : = ρ 2 j t ( j − 1 ) κ 2 ( j ) ≔ ρ j 2 t ( − j − 1 ) \kappa_2(j)\coloneqq\rho^{\frac{j}{2}}t(-j-1) κ 2 ( j ) : = ρ 2 j t ( − j − 1 )
By part 2 of Lemma 4.5 , the sequence ( κ 2 ( j ) ) j ≥ 0 (\kappa_2(j))_{j\geq0} ( κ 2 ( j ) ) j ≥ 0 ∣ t ∣ ≤ 1 \lvert t\rvert\leq1 ∣ t ∣ ≤ 1 ( κ 2 ( j ) ) j < 0 (\kappa_2(j))_{j<0} ( κ 2 ( j ) ) j < 0 ρ j 2 \rho^{\frac{j}{2}} ρ 2 j
is bounded on ℓ Z 2 \ell^2_{\mathbb Z} ℓ Z 2 ∥ κ 2 ∥ ℓ Z 2 \|\kappa_2\|_{\ell^2_{\mathbb Z}} ∥ κ 2 ∥ ℓ Z 2 f ∈ dom E D f \in\operatorname{dom}\mathcal{E}_D f ∈ dom E D C ∈ ℓ Z 2 C \in\ell^2_{\mathbb Z} C ∈ ℓ Z 2 B ∈ ℓ Z 2 B \in\ell^2_{\mathbb Z} B ∈ ℓ Z 2 (34) and ∥ ( L η ′ ∞ ) ϑ 2 f ∥ 2 ≤ ∥ κ 2 ∥ ℓ Z 2 2 E D ( f ) \|(L^\infty_{\eta'})^{\frac{\vartheta}{2}}f\|^2 \leq\|\kappa_2\|_{\ell^2_{\mathbb Z}}^2\mathcal{E}_D(f) ∥ ( L η ′ ∞ ) 2 ϑ f ∥ 2 ≤ ∥ κ 2 ∥ ℓ Z 2 2 E D ( f )
and 3. Suppose that α < d + 2 ϑ \alpha< d+2\vartheta α < d + 2 ϑ Lemma 4.5 , we have ∣ κ 1 ( j ) ∣ ≤ C ρ j 2 τ η ′ j = C Q j ( α − d − 2 ϑ ) 2 |\kappa_1(j)| \leq C\rho^{\frac{j}{2}}\tau^{\eta'j}=CQ^{\frac{j(\alpha-d-2\vartheta)}{2}} ∣ κ 1 ( j ) ∣ ≤ C ρ 2 j τ η ′ j = C Q 2 j ( α − d − 2 ϑ ) j j j ∣ κ 1 ( j ) ∣ ≤ ρ j 2 |\kappa_1(j)| \leq\rho^{\frac{j}{2}} ∣ κ 1 ( j ) ∣ ≤ ρ 2 j j < 0 j<0 j < 0 κ 1 ∈ ℓ Z 2 \kappa_1 \in\ell^2_{\mathbb Z} κ 1 ∈ ℓ Z 2 (34) gives E D ( f ) ≤ ∥ κ 1 ∥ ℓ Z 2 2 ∥ ( L η ′ ∞ ) ϑ 2 f ∥ 2 \mathcal{E}_D(f) \leq\|\kappa_1\|_{\ell^2_{\mathbb Z}}^2\|(L^\infty_{\eta'})^{\frac{\vartheta}{2}}f\|^2 E D ( f ) ≤ ∥ κ 1 ∥ ℓ Z 2 2 ∥ ( L η ′ ∞ ) 2 ϑ f ∥ 2 f ∈ dom ( L η ′ ∞ ) ϑ 2 f \in\operatorname{dom}(L^\infty_{\eta'})^{\frac{\vartheta}{2}} f ∈ dom ( L η ′ ∞ ) 2 ϑ α ≥ d + 2 ϑ \alpha\geq d+2\vartheta α ≥ d + 2 ϑ z ∈ Z z \in\mathbb{Z} z ∈ Z f = φ z f=\varphi_z f = φ z c n = t ( n + z ) c_n=t(n+z) c n = t ( n + z ) Lemma 4.5
E D ( φ z ) ≥ ∑ n ≥ n 0 ρ n + 1 C 1 2 τ n + z η ′ 2 ( n + z ) = C ∑ n ≥ n 0 Q n ( α − d − 2 ϑ ) = ∞ \mathcal{E}_D(\varphi_z) \geq\sum_{n\geq n_0}\rho^{n+1}C_1^2\tau^{n+z}\eta'^{2(n+z)}=C\sum_{n\geq n_0}Q^{n(\alpha-d-2\vartheta)}=\infty E D ( φ z ) ≥ n ≥ n 0 ∑ ρ n + 1 C 1 2 τ n + z η ′ 2 ( n + z ) = C n ≥ n 0 ∑ Q n ( α − d − 2 ϑ ) = ∞ for some n 0 n_0 n 0 C > 0 C>0 C > 0 φ z ∈ dom ( L η ′ ∞ ) ϑ 2 \varphi_z\in\operatorname{dom}(L^\infty_{\eta'})^{\frac{\vartheta}{2}} φ z ∈ dom ( L η ′ ∞ ) 2 ϑ
Remark 4.7 The threshold α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ k ↦ η ′ k k\mapsto\eta'^k k ↦ η ′ k L η ′ ∞ L^\infty_{\eta'} L η ′ ∞ K d K^d K d K d K^d K d ϑ = d \vartheta=d ϑ = d ( L τ ∞ ) α d (L^\infty_\tau)^{\frac{\alpha}{d}} ( L τ ∞ ) d α α < 3 d \alpha<3d α < 3 d
The quantum group E r ( 2 ) \operatorname{E}_r(2) E r ( 2 )
The concrete model and the Haar weight
Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 E r ( 2 ) \operatorname{E}_r(2) E r ( 2 ) [41 ] , in the concrete realization of [43 ] and [31 ] . Its Haar weight was obtained by Baaj [4 ] , and computed in this realization by Pal [31 ] . Let ( e i j ) i , j ∈ Z (e_{ij})_{i,j\in\mathbb{Z}} ( e ij ) i , j ∈ Z ℓ Z × Z 2 \ell^2_{\mathbb{Z}\times\mathbb{Z}} ℓ Z × Z 2 v v v n n n
v e i j = e i − 1 , j , n e i j = r i e i , j + 1 , i , j ∈ Z . (35) ve_{ij}=e_{i-1,j},\qquad ne_{ij}=r^i e_{i,j+1},\qquad i,j\in\mathbb{Z}. \tag*{(35)} v e ij = e i − 1 , j , n e ij = r i e i , j + 1 , i , j ∈ Z . (35) Then v n v ∗ = r n vnv^*=rn v n v ∗ = r n n n n C r = { 0 } ∪ { λ ∈ C : ∣ λ ∣ ∈ r Z } \mathbb{C}_r=\{0\}\cup\{\lambda\in\mathbb{C}:|\lambda|\in r^{\mathbb{Z}}\} C r = { 0 } ∪ { λ ∈ C : ∣ λ ∣ ∈ r Z } C 0 ( E r ( 2 ) ) C_0(\operatorname{E}_r(2)) C 0 ( E r ( 2 )) v k f ( n ) v^k f(n) v k f ( n ) k ∈ Z k\in\mathbb{Z} k ∈ Z f ∈ C 0 ( C r ) f\in C_0(\mathbb{C}_r) f ∈ C 0 ( C r ) Δ ( v ) = v ⊗ v \Delta(v)=v\otimes v Δ ( v ) = v ⊗ v Δ ( n ) = v ⊗ n ∔ n ⊗ v ∗ \Delta(n)=v\otimes n\mathbin{\dotplus}n\otimes v^* Δ ( n ) = v ⊗ n ∔ n ⊗ v ∗ ∔ \mathbin{\dotplus} ∔ L ∞ ( E r ( 2 ) ) L^\infty(\operatorname{E}_r(2)) L ∞ ( E r ( 2 )) C 0 ( E r ( 2 ) ) C_0(\operatorname{E}_r(2)) C 0 ( E r ( 2 )) l ∈ Z l\in\mathbb{Z} l ∈ Z τ l : = Ad ( v l ) \tau_l\mathrel{\mathop:}=\operatorname{Ad}(v^l) τ l : = Ad ( v l ) τ l ( n ) = r l n \tau_l(n)=r^l n τ l ( n ) = r l n E r ( 2 ) \operatorname{E}_r(2) E r ( 2 ) SU r ( 2 ) \operatorname{SU}_r(2) SU r ( 2 ) [43 ] .
The Fourier transform in the second variable gives a unitary operator from ℓ Z × Z 2 \ell^2_{\mathbb{Z}\times\mathbb{Z}} ℓ Z × Z 2 ℓ Z 2 ⊗ L 2 ( T ) \ell^2_{\mathbb{Z}}\otimes L^2(\mathbb{T}) ℓ Z 2 ⊗ L 2 ( T ) e i j ↦ e i ⊗ z j e_{ij}\mapsto e_i\otimes z^j e ij ↦ e i ⊗ z j T \mathbb{T} T m m m z z z T \mathbb{T} T v = v 0 ⊗ 1 v=v_0\otimes1 v = v 0 ⊗ 1 v 0 e i = e i − 1 v_0e_i=e_{i-1} v 0 e i = e i − 1 n n n diag ( r i ) i ∈ Z ⊗ 1 \operatorname{diag}(r^i)_{i\in\mathbb{Z}}\otimes1 diag ( r i ) i ∈ Z ⊗ 1 1 ⊗ M z 1\otimes M_z 1 ⊗ M z B ( ℓ Z 2 ) ⊗ ‾ L ∞ ( T ) B(\ell^2_{\mathbb{Z}})\overline{\otimes}L^\infty(\mathbb{T}) B ( ℓ Z 2 ) ⊗ L ∞ ( T ) x = ( x i j ) i , j ∈ Z x=(x_{ij})_{i,j\in\mathbb{Z}} x = ( x ij ) i , j ∈ Z L ∞ ( T ) L^\infty(\mathbb{T}) L ∞ ( T ) ℓ Z 2 ⊗ L 2 ( T ) \ell^2_{\mathbb{Z}}\otimes L^2(\mathbb{T}) ℓ Z 2 ⊗ L 2 ( T )
Proposition 5.1 Suppose that 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2
We have L ∞ ( E r ( 2 ) ) = B ( ℓ Z 2 ) ⊗ ‾ L ∞ ( T ) L^\infty(\operatorname{E}_r(2))=B(\ell^2_{\mathbb{Z}})\overline{\otimes}L^\infty(\mathbb{T}) L ∞ ( E r ( 2 )) = B ( ℓ Z 2 ) ⊗ L ∞ ( T )
φ ( x ) : = ( 1 − τ ) ∑ i ∈ Z τ i ∫ T x i i d m , x ∈ L ∞ ( E r ( 2 ) ) + (36) \varphi(x) \mathrel{\mathop:}= (1-\tau)\sum_{i\in\mathbb{Z}}\tau^i\int_{\mathbb{T}}x_{ii}\,\mathrm{d}m,\qquad x\in L^\infty(E_{\mathrm r}(2))_+ \tag*{(36)} φ ( x ) : = ( 1 − τ ) i ∈ Z ∑ τ i ∫ T x ii d m , x ∈ L ∞ ( E r ( 2 ) ) + (36) defines a normal semifinite faithful weight φ \varphi φ φ = ( 1 − τ ) h \varphi=(1-\tau)h φ = ( 1 − τ ) h h h h E r ( 2 ) E_{\mathrm r}(2) E r ( 2 ) [31 (2.1) and Theorem 2.1] . Moreover φ ∘ τ l = τ − l φ \varphi\circ\tau_l=\tau^{-l}\varphi φ ∘ τ l = τ − l φ l ∈ Z l\in\mathbb{Z} l ∈ Z
Put A : = n ∗ n A\mathrel{\mathop:}=n^*n A : = n ∗ n A = diag ( τ i ) i ∈ Z ⊗ 1 A=\operatorname{diag}(\tau^i)_{i\in\mathbb{Z}}\otimes1 A = diag ( τ i ) i ∈ Z ⊗ 1 F : τ Z → [ 0 , ∞ ] F:\tau^{\mathbb{Z}}\to[0,\infty] F : τ Z → [ 0 , ∞ ]
φ ( F ( A ) ) = ( 1 − τ ) ∑ i ∈ Z τ i F ( τ i ) . \varphi(F(A))=(1-\tau)\sum_{i\in\mathbb{Z}}\tau^iF(\tau^i). φ ( F ( A )) = ( 1 − τ ) i ∈ Z ∑ τ i F ( τ i ) . If τ = Q − d \tau=Q^{-d} τ = Q − d ∫ K d F ( ∥ x ∥ K d ) d x \int_{K_d}F(\lVert x\rVert_{K}^{d})\,\mathrm{d}x ∫ K d F (∥ x ∥ K d ) d x
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d J : L r a d ∞ ( K d ) → W ∗ ( A ) J:L^\infty_{\mathrm{rad}}(K^d)\to W^*(A) J : L rad ∞ ( K d ) → W ∗ ( A ) J ( F ( ∥ ⋅ ∥ K d ) ) = F ( A ) J(F(\lVert\cdot\rVert_K^d))=F(A) J ( F (∥ ⋅ ∥ K d )) = F ( A ) ∗ * ∗ φ ∘ J = ∫ K d ⋅ d x \varphi\circ J=\int_{K^d}\cdot\,\mathrm{d}x φ ∘ J = ∫ K d ⋅ d x f ∈ L r a d ∞ ( K d ) f\in L^\infty_{\mathrm{rad}}(K^d) f ∈ L rad ∞ ( K d ) v J ( f ) v ∗ = J ( f ( π ⋅ ) ) vJ(f)v^*=J(f(\pi\cdot)) v J ( f ) v ∗ = J ( f ( π ⋅ )) m ∈ Z m\in\mathbb{Z} m ∈ Z P m : = 1 [ 0 , τ m ] ( A ) P_m\mathrel{\mathop:}=1_{[0,\tau^m]}(A) P m : = 1 [ 0 , τ m ] ( A ) J ( 1 B m ) J(1_{B_m}) J ( 1 B m ) φ ( P m ) = τ m \varphi(P_m)=\tau^m φ ( P m ) = τ m
Proof : 1. In the picture ℓ Z 2 ⊗ L 2 ( T ) \ell^2_{\mathbb{Z}}\otimes L^2(\mathbb{T}) ℓ Z 2 ⊗ L 2 ( T ) f ( n ) f(n) f ( n ) e i ⊗ ξ e_i\otimes\xi e i ⊗ ξ e i ⊗ f ( r i z ) ξ e_i\otimes f(r^iz)\xi e i ⊗ f ( r i z ) ξ v k f ( n ) v^kf(n) v k f ( n ) L ∞ ( T ) L^\infty(\mathbb{T}) L ∞ ( T ) C 0 ( E r ( 2 ) ) ⊂ B ( ℓ Z 2 ) ⊗ ‾ L ∞ ( T ) C_0(E_{\mathrm r}(2))\subset B(\ell^2_{\mathbb{Z}})\overline{\otimes}L^\infty(\mathbb{T}) C 0 ( E r ( 2 )) ⊂ B ( ℓ Z 2 ) ⊗ L ∞ ( T ) C 0 ( E r ( 2 ) ) C_0(E_{\mathrm r}(2)) C 0 ( E r ( 2 )) ∣ n ∣ = diag ( r i ) ⊗ 1 |n|=\operatorname{diag}(r^i)\otimes1 ∣ n ∣ = diag ( r i ) ⊗ 1 e i i ⊗ 1 e_{ii}\otimes1 e ii ⊗ 1 v v v e i j ⊗ 1 e_{ij}\otimes1 e ij ⊗ 1 1 ⊗ M z 1\otimes M_z 1 ⊗ M z n n n B ( ℓ Z 2 ) ⊗ ‾ L ∞ ( T ) B(\ell^2_{\mathbb{Z}})\overline{\otimes}L^\infty(\mathbb{T}) B ( ℓ Z 2 ) ⊗ L ∞ ( T )
In the picture ℓ Z 2 ⊗ Z \ell^2_{\mathbb{Z}}\otimes\mathbb{Z} ℓ Z 2 ⊗ Z ∫ T x i i d m = ⟨ x e i 0 , e i 0 ⟩ \int_{\mathbb{T}}x_{ii}\,\mathrm{d}m=\langle xe_{i0},e_{i0}\rangle ∫ T x ii d m = ⟨ x e i 0 , e i 0 ⟩
φ ( x ) = ( 1 − τ ) ∑ i ∈ Z τ 2 i ⟨ x e i 0 , e i 0 ⟩ . \varphi(x)=(1-\tau)\sum_{i\in\mathbb{Z}}\tau^{2i}\langle xe_{i0},e_{i0}\rangle. φ ( x ) = ( 1 − τ ) i ∈ Z ∑ τ 2 i ⟨ x e i 0 , e i 0 ⟩ . which is 1 − τ 1-\tau 1 − τ [31 (2.1)] . By [31 Theorem 2.1] , this weight is left and right invariant, and the left Haar weight of a locally compact quantum group is unique up to a positive scalar [24 ] .
We have A e i j = r 2 i e i j Ae_{ij}=r^{2i}e_{ij} A e ij = r 2 i e ij A = diag ( τ i ) ⊗ 1 A=\operatorname{diag}(\tau^i)\otimes1 A = diag ( τ i ) ⊗ 1 F ( A ) = diag ( F ( τ i ) ) ⊗ 1 F(A)=\operatorname{diag}(F(\tau^i))\otimes1 F ( A ) = diag ( F ( τ i )) ⊗ 1 φ ( F ( A ) ) \varphi(F(A)) φ ( F ( A )) (36) , and the last assertion follows from (5) , since ∥ x ∥ K d = τ i \lVert x\rVert_K^d=\tau^i ∥ x ∥ K d = τ i S i S_i S i
The radial algebra L r a d ∞ ( K d ) L^\infty_{\mathrm{rad}}(K^d) L rad ∞ ( K d ) ℓ Z ∞ \ell^\infty_{\mathbb{Z}} ℓ Z ∞ J J J f ↦ diag ( f i ) ⊗ 1 f\mapsto\operatorname{diag}(f_i)\otimes1 f ↦ diag ( f i ) ⊗ 1 W ∗ ( A ) = { diag ( c i ) ⊗ 1 : c ∈ ℓ Z ∞ } W^*(A)=\{\operatorname{diag}(c_i)\otimes1:c\in\ell^\infty_{\mathbb{Z}}\} W ∗ ( A ) = { diag ( c i ) ⊗ 1 : c ∈ ℓ Z ∞ } ∗ * ∗ v ( diag ( c i ) ⊗ 1 ) v ∗ = diag ( c i + 1 ) ⊗ 1 v(\operatorname{diag}(c_i)\otimes1)v^*=\operatorname{diag}(c_{i+1})\otimes1 v ( diag ( c i ) ⊗ 1 ) v ∗ = diag ( c i + 1 ) ⊗ 1 π S i = S i + 1 \pi S_i=S_{i+1} π S i = S i + 1 v J ( f ) v ∗ = J ( f ( π ⋅ ) ) vJ(f)v^*=J(f(\pi\cdot)) v J ( f ) v ∗ = J ( f ( π ⋅ )) 1 [ 0 , τ m ] ( τ i ) = 1 1_{[0,\tau^m]}(\tau^i)=1 1 [ 0 , τ m ] ( τ i ) = 1 i ≥ m i\ge m i ≥ m P m = J ( 1 B m ) P_m=J(1_{B_m}) P m = J ( 1 B m ) φ ( P m ) = ( 1 − τ ) ∑ i ≥ m τ i = τ m \varphi(P_m)=(1-\tau)\sum_{i\ge m}\tau^i=\tau^m φ ( P m ) = ( 1 − τ ) ∑ i ≥ m τ i = τ m
Remark 5.2 The restriction of φ \varphi φ P 0 L ∞ ( E r ( 2 ) ) P 0 P_0L^\infty(E_{\mathrm r}(2))P_0 P 0 L ∞ ( E r ( 2 )) P 0 W ∗ ( A ) P 0 W^*(A)P_0 W ∗ ( A ) P 0 ( 1 − τ ) ∑ i ≥ 0 τ i δ τ i (1-\tau)\sum_{i\ge0}\tau^i\delta_{\tau^i} ( 1 − τ ) ∑ i ≥ 0 τ i δ τ i O d O_d O d P 0 P_0 P 0 B m = π m O d B_m=\pi^m\mathcal{O}^d B m = π m O d P m = τ − m ( P 0 ) P_m=\tau^{-m}(P_0) P m = τ − m ( P 0 ) τ l = Ad ( v l ) \tau_l=\operatorname{Ad}(v^l) τ l = Ad ( v l ) S U r ( 2 ) \mathrm{SU}_{\mathrm r}(2) SU r ( 2 ) E r ( 2 ) E_{\mathrm r}(2) E r ( 2 ) [43 ] , implement the dilations x ↦ π l x x\mapsto\pi^lx x ↦ π l x L ∞ ( O d ) L^\infty(\mathcal{O}^d) L ∞ ( O d ) L ∞ ( K d ) L^\infty(K^d) L ∞ ( K d ) π − N O d \pi^{-N}\mathcal{O}^d π − N O d N → ∞ N\to\infty N → ∞
Tail expectations and an exact lifting
In this subsection we put M : = L ∞ ( E r ( 2 ) ) = B ( ℓ 2 ) ⊗ ‾ L ∞ ( T ) \mathcal{M}\mathrel{\mathop:}=L^\infty(E_r(2))=B(\ell^2)\overline{\otimes}L^\infty(\mathbb{T}) M : = L ∞ ( E r ( 2 )) = B ( ℓ 2 ) ⊗ L ∞ ( T ) φ \varphi φ (36) , the projections P m = ∑ i ≥ m e i i ⊗ 1 P_m=\sum_{i\ge m}e_{ii}\otimes1 P m = ∑ i ≥ m e ii ⊗ 1 D : = ( 1 − τ ) diag ( τ i ) i ∈ Z ⊗ 1 D\mathrel{\mathop:}=(1-\tau)\operatorname{diag}(\tau^i)_{i\in\mathbb{Z}}\otimes1 D : = ( 1 − τ ) diag ( τ i ) i ∈ Z ⊗ 1 M \mathcal{M} M P m P_m P m S \mathcal{S} S ξ = ( ξ i j ) i , j ∈ Z \xi=(\xi_{ij})_{i,j\in\mathbb{Z}} ξ = ( ξ ij ) i , j ∈ Z L 2 ( T ) L^2(\mathbb{T}) L 2 ( T ) ∥ ξ ∥ S 2 = ∑ i , j ∥ ξ i j ∥ L 2 ( T ) 2 < ∞ \|\xi\|_{\mathcal{S}}^2=\sum_{i,j}\|\xi_{ij}\|_{L^2(\mathbb{T})}^2<\infty ∥ ξ ∥ S 2 = ∑ i , j ∥ ξ ij ∥ L 2 ( T ) 2 < ∞ ⟨ ξ , ζ ⟩ S = ∑ i , j ∫ T ξ i j ‾ ζ i j d m \langle\xi,\zeta\rangle_{\mathcal{S}}=\sum_{i,j}\int_{\mathbb{T}}\overline{\xi_{ij}}\zeta_{ij}\,dm ⟨ ξ , ζ ⟩ S = ∑ i , j ∫ T ξ ij ζ ij d m M \mathcal{M} M S \mathcal{S} S n φ = { x ∈ M : φ ( x ∗ x ) < ∞ } \mathfrak{n}_\varphi=\{x\in\mathcal{M}:\varphi(x^*x)<\infty\} n φ = { x ∈ M : φ ( x ∗ x ) < ∞ } x ∈ n φ x\in\mathfrak{n}_\varphi x ∈ n φ i i i x D 1 2 xD^{\frac12} x D 2 1 ( ( 1 − τ ) τ i ) 1 2 ((1-\tau)\tau^i)^{\frac12} (( 1 − τ ) τ i ) 2 1 i i i x x x x D 1 2 ∈ S xD^{\frac12}\in\mathcal{S} x D 2 1 ∈ S
∥ x D 1 2 ∥ S 2 = ( 1 − τ ) ∑ i τ i ∑ k ∫ T ∣ x k i ∣ 2 d m = φ ( x ∗ x ) . \|xD^{\frac12}\|_{\mathcal{S}}^2=(1-\tau)\sum_i\tau^i\sum_k\int_{\mathbb{T}}|x_{ki}|^2\,dm=\varphi(x^*x). ∥ x D 2 1 ∥ S 2 = ( 1 − τ ) i ∑ τ i k ∑ ∫ T ∣ x k i ∣ 2 d m = φ ( x ∗ x ) . The finite matrices with entries in L ∞ ( T ) L^\infty(\mathbb{T}) L ∞ ( T ) n φ \mathfrak{n}_\varphi n φ S \mathcal{S} S Λ ( x ) : = x D 1 2 \Lambda(x)\mathrel{\mathop:}=xD^{\frac12} Λ ( x ) : = x D 2 1 ( M , φ ) (\mathcal{M},\varphi) ( M , φ ) L 2 ( M , φ ) = S L^2(\mathcal{M},\varphi)=\mathcal{S} L 2 ( M , φ ) = S x ↦ D 1 4 x D 1 4 x\mapsto D^{\frac14}xD^{\frac14} x ↦ D 4 1 x D 4 1 M \mathcal{M} M S \mathcal{S} S m ∈ Z m\in\mathbb{Z} m ∈ Z
Φ m ( x ) : = ( 1 − P m ) x ( 1 − P m ) + τ − m φ ( P m x P m ) P m , M m : = ( 1 − P m ) M ( 1 − P m ) ⊕ C P m , (37) \Phi_m(x)\mathrel{\mathop:}=(1-P_m)x(1-P_m)+\tau^{-m}\varphi(P_mxP_m)P_m,\qquad\mathcal{M}_m\mathrel{\mathop:}=(1-P_m)\mathcal{M}(1-P_m)\oplus\mathbb{C}P_m, \tag*{(37)} Φ m ( x ) : = ( 1 − P m ) x ( 1 − P m ) + τ − m φ ( P m x P m ) P m , M m : = ( 1 − P m ) M ( 1 − P m ) ⊕ C P m , (37) where φ ( P m x P m ) \varphi(P_mxP_m) φ ( P m x P m ) x x x x ↦ ( 1 − τ ) ∑ i ≥ m τ i ∫ T x i i d m x\mapsto(1-\tau)\sum_{i\ge m}\tau^i\int_{\mathbb{T}}x_{ii}\,dm x ↦ ( 1 − τ ) ∑ i ≥ m τ i ∫ T x ii d m φ ( P m ) = τ m \varphi(P_m)=\tau^m φ ( P m ) = τ m
Lemma 5.4
For any m ∈ Z m\in\mathbb{Z} m ∈ Z Φ m \Phi_m Φ m M m \mathcal{M}_m M m φ ∘ Φ m = φ \varphi\circ\Phi_m=\varphi φ ∘ Φ m = φ M + \mathcal{M}_+ M + M m \mathcal{M}_m M m m m m Φ i Φ j = Φ min ( i , j ) \Phi_i\Phi_j=\Phi_{\min(i,j)} Φ i Φ j = Φ m i n ( i , j ) i , j ∈ Z i,j\in\mathbb{Z} i , j ∈ Z
For any m ∈ Z m\in\mathbb{Z} m ∈ Z x ∈ n φ x\in\mathfrak{n}_\varphi x ∈ n φ Φ m ( x ) ∈ n φ \Phi_m(x)\in\mathfrak{n}_\varphi Φ m ( x ) ∈ n φ Λ ( Φ m ( x ) ) = Φ m ( 2 ) Λ ( x ) \Lambda(\Phi_m(x))=\Phi_m^{(2)}\Lambda(x) Λ ( Φ m ( x )) = Φ m ( 2 ) Λ ( x ) Φ m ( 2 ) \Phi_m^{(2)} Φ m ( 2 ) S \mathcal{S} S
Φ m ( 2 ) ξ : = ( 1 − P m ) ξ ( 1 − P m ) + τ − m ⟨ ξ , P m D 1 2 ⟩ S P m D 1 2 . (38) \Phi_m^{(2)}\xi\mathrel{\mathop:}=(1-P_m)\xi(1-P_m)+\tau^{-m}\langle\xi,P_mD^{\frac12}\rangle_{\mathcal{S}}P_mD^{\frac12}. \tag*{(38)} Φ m ( 2 ) ξ : = ( 1 − P m ) ξ ( 1 − P m ) + τ − m ⟨ ξ , P m D 2 1 ⟩ S P m D 2 1 . (38) The same formula holds for the symmetric embedding. Moreover Φ m ( 2 ) → I \Phi_m^{(2)}\to I Φ m ( 2 ) → I m → + ∞ m\to+\infty m → + ∞ Φ m ( 2 ) → 0 \Phi_m^{(2)}\to0 Φ m ( 2 ) → 0 m → − ∞ m\to-\infty m → − ∞
The maps Φ m \Phi_m Φ m σ s φ = Ad ( D i s ) \sigma_s^\varphi=\operatorname{Ad}(D^{is}) σ s φ = Ad ( D i s ) s ∈ R s\in\mathbb{R} s ∈ R Ad ( 1 ⊗ R λ ) \operatorname{Ad}(1\otimes R_\lambda) Ad ( 1 ⊗ R λ ) λ ∈ T \lambda\in\mathbb{T} λ ∈ T ( R λ ξ ) ( z ) = ξ ( λ z ) (R_\lambda\xi)(z)=\xi(\lambda z) ( R λ ξ ) ( z ) = ξ ( λ z ) L 2 ( T ) L^2(\mathbb{T}) L 2 ( T )
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d m ∈ Z m\in\mathbb{Z} m ∈ Z f ∈ L r a d ∞ ( K d ) f\in L^\infty_{\mathrm{rad}}(K^d) f ∈ L rad ∞ ( K d ) Φ m J ( f ) = J ( E m f ) \Phi_mJ(f)=J(E_mf) Φ m J ( f ) = J ( E m f ) E m f E_mf E m f f f f K d ∖ B m K^d\setminus B_m K d ∖ B m m m ( f ) m_m(f) m m ( f ) B m B_m B m
Proof: 1. The compression by 1 − P m 1-P_m 1 − P m x ↦ τ − m φ ( P m x P m ) x\mapsto\tau^{-m}\varphi(P_mxP_m) x ↦ τ − m φ ( P m x P m ) M M M φ ( P m ) = τ m \varphi(P_m)=\tau^m φ ( P m ) = τ m Φ m \Phi_m Φ m M m \mathcal{M}_m M m M m \mathcal{M}_m M m P m P_m P m M m \mathcal{M}_m M m ( 1 − P m ) y P m = 0 (1-P_m)yP_m=0 ( 1 − P m ) y P m = 0 y ∈ M m y\in\mathcal{M}_m y ∈ M m x ∈ M + x\in M_+ x ∈ M + x x x i < m i<m i < m ( 1 − P m ) x ( 1 − P m ) (1-P_m)x(1-P_m) ( 1 − P m ) x ( 1 − P m ) i ≥ m i\geq m i ≥ m P m x P m P_mxP_m P m x P m φ ( x ) = φ ( ( 1 − P m ) x ( 1 − P m ) ) + φ ( P m x P m ) = φ ( Φ m ( x ) ) \varphi(x)=\varphi((1-P_m)x(1-P_m))+\varphi(P_mxP_m)=\varphi(\Phi_m(x)) φ ( x ) = φ (( 1 − P m ) x ( 1 − P m )) + φ ( P m x P m ) = φ ( Φ m ( x )) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] P m = e m m ⊗ 1 + P m + 1 P_m=e_{mm}\otimes1+P_{m+1} P m = e mm ⊗ 1 + P m + 1 y = ( 1 − P m ) y ( 1 − P m ) + c P m y=(1-P_m)y(1-P_m)+cP_m y = ( 1 − P m ) y ( 1 − P m ) + c P m M m \mathcal{M}_m M m ( 1 − P m + 1 ) y P m + 1 = 0 = P m + 1 y ( 1 − P m + 1 ) (1-P_{m+1})yP_{m+1}=0=P_{m+1}y(1-P_{m+1}) ( 1 − P m + 1 ) y P m + 1 = 0 = P m + 1 y ( 1 − P m + 1 ) P m + 1 y P m + 1 = c P m + 1 P_{m+1}yP_{m+1}=cP_{m+1} P m + 1 y P m + 1 = c P m + 1 M m ⊂ M m + 1 \mathcal{M}_m\subset\mathcal{M}_{m+1} M m ⊂ M m + 1 i ≥ j i\geq j i ≥ j Φ i Φ j = Φ j \Phi_i\Phi_j=\Phi_j Φ i Φ j = Φ j Φ j ( x ) ∈ M j ⊂ M i \Phi_j(x)\in\mathcal{M}_j\subset\mathcal{M}_i Φ j ( x ) ∈ M j ⊂ M i i ≤ j i\leq j i ≤ j ( 1 − P i ) Φ j ( x ) ( 1 − P i ) = ( 1 − P i ) x ( 1 − P i ) (1-P_i)\Phi_j(x)(1-P_i)=(1-P_i)x(1-P_i) ( 1 − P i ) Φ j ( x ) ( 1 − P i ) = ( 1 − P i ) x ( 1 − P i ) ( 1 − P i ) ( 1 − P j ) = 1 − P i (1-P_i)(1-P_j)=1-P_i ( 1 − P i ) ( 1 − P j ) = 1 − P i ( 1 − P i ) P j = 0 (1-P_i)P_j=0 ( 1 − P i ) P j = 0
φ ( P i Φ j ( x ) P i ) = φ ( ( P i − P j ) x ( P i − P j ) ) + τ − j φ ( P j x P j ) φ ( P j ) = φ ( P i x P i ) , \varphi(P_i\Phi_j(x)P_i)=\varphi((P_i-P_j)x(P_i-P_j))+\tau^{-j}\varphi(P_jxP_j)\varphi(P_j)=\varphi(P_ixP_i), φ ( P i Φ j ( x ) P i ) = φ (( P i − P j ) x ( P i − P j )) + τ − j φ ( P j x P j ) φ ( P j ) = φ ( P i x P i ) , where we used that the functional φ ( P i ⋅ P i ) \varphi(P_i\mathbin{\cdot}P_i) φ ( P i ⋅ P i ) i i i Φ i Φ j = Φ i \Phi_i\Phi_j=\Phi_i Φ i Φ j = Φ i
Let x ∈ n φ x\in\mathfrak{n}_\varphi x ∈ n φ D D D P m P_m P m ( 1 − P m ) x ( 1 − P m ) D 1 2 = ( 1 − P m ) x D 1 2 ( 1 − P m ) (1-P_m)x(1-P_m)D^{\frac12}=(1-P_m)xD^{\frac12}(1-P_m) ( 1 − P m ) x ( 1 − P m ) D 2 1 = ( 1 − P m ) x D 2 1 ( 1 − P m ) P m D 1 2 ∈ S P_mD^{\frac12}\in\mathcal{S} P m D 2 1 ∈ S
∥ P m D 1 2 ∥ S 2 = φ ( P m ) = τ m , \left\|P_mD^{\frac12}\right\|_{\mathcal{S}}^2=\varphi(P_m)=\tau^m, P m D 2 1 S 2 = φ ( P m ) = τ m , and
⟨ x D 1 2 , P m D 1 2 ⟩ S = ( 1 − τ ) ∑ i ≥ m τ i ∫ T x i i d m = φ ( P m x P m ) . \left\langle xD^{\frac12},P_mD^{\frac12}\right\rangle_{\mathcal{S}}=(1-\tau)\sum_{i\geq m}\tau^i\int_{\mathbb{T}}x_{ii}\,dm=\varphi(P_mxP_m). ⟨ x D 2 1 , P m D 2 1 ⟩ S = ( 1 − τ ) i ≥ m ∑ τ i ∫ T x ii d m = φ ( P m x P m ) . This gives Λ ( Φ m ( x ) ) = Φ m ( 2 ) Λ ( x ) \Lambda(\Phi_m(x))=\Phi_m^{(2)}\Lambda(x) Λ ( Φ m ( x )) = Φ m ( 2 ) Λ ( x ) Φ m ( x ) ∈ n φ \Phi_m(x)\in\mathfrak{n}_\varphi Φ m ( x ) ∈ n φ ⟨ D 1 4 x D 1 4 , P m D 1 2 ⟩ S = φ ( P m x P m ) \left\langle D^{\frac14}xD^{\frac14},P_mD^{\frac12}\right\rangle_{\mathcal{S}}=\varphi(P_mxP_m) ⟨ D 4 1 x D 4 1 , P m D 2 1 ⟩ S = φ ( P m x P m ) (38) is the orthogonal projection onto ( 1 − P m ) S ( 1 − P m ) (1-P_m)\mathcal{S}(1-P_m) ( 1 − P m ) S ( 1 − P m ) τ − m 2 P m D 1 2 \tau^{-\frac m2}P_mD^{\frac12} τ − 2 m P m D 2 1 ( 1 − P m ) S ( 1 − P m ) (1-P_m)\mathcal{S}(1-P_m) ( 1 − P m ) S ( 1 − P m ) Φ m ( 2 ) \Phi_m^{(2)} Φ m ( 2 ) ξ ∈ S \xi\in\mathcal{S} ξ ∈ S [ − M , M ] [-M,M] [ − M , M ] m > M m>M m > M Φ m ( 2 ) ξ = ξ \Phi_m^{(2)}\xi=\xi Φ m ( 2 ) ξ = ξ P m ξ = ξ P m = 0 P_m\xi=\xi P_m=0 P m ξ = ξ P m = 0 m ≤ − M m\leq-M m ≤ − M ( 1 − P m ) ξ = 0 (1-P_m)\xi=0 ( 1 − P m ) ξ = 0 Φ m ( 2 ) ξ = τ − m ⟨ ξ , P m D 1 2 ⟩ P m D 1 2 \Phi_m^{(2)}\xi=\tau^{-m}\langle\xi,P_mD^{\frac12}\rangle P_mD^{\frac12} Φ m ( 2 ) ξ = τ − m ⟨ ξ , P m D 2 1 ⟩ P m D 2 1 τ − m 2 ∣ ⟨ ξ , P m D 1 2 ⟩ ∣ ≤ τ − m 2 ∥ ξ ∥ S ( ( 1 − τ ) ∑ ∣ i ∣ ≤ M τ i ) 1 2 \tau^{-\frac m2}|\langle\xi,P_mD^{\frac12}\rangle|\leq\tau^{-\frac m2}\|\xi\|_{\mathcal{S}}\bigl((1-\tau)\sum_{|i|\leq M}\tau^i\bigr)^{\frac12} τ − 2 m ∣ ⟨ ξ , P m D 2 1 ⟩ ∣ ≤ τ − 2 m ∥ ξ ∥ S ( ( 1 − τ ) ∑ ∣ i ∣ ≤ M τ i ) 2 1 m → − ∞ m\to-\infty m → − ∞ τ − m 2 = τ ∣ m ∣ 2 \tau^{-\frac m2}=\tau^{\frac{|m|}{2}} τ − 2 m = τ 2 ∣ m ∣
Since D i s D^{\mathrm{i}s} D i s P m P_m P m φ ∘ Ad ( D i s ) = φ \varphi\circ\operatorname{Ad}(D^{\mathrm{i}s})=\varphi φ ∘ Ad ( D i s ) = φ (37) gives Φ m ∘ Ad ( D i s ) = Ad ( D i s ) ∘ Φ m \Phi_m\circ\operatorname{Ad}(D^{\mathrm{i}s})=\operatorname{Ad}(D^{\mathrm{i}s})\circ\Phi_m Φ m ∘ Ad ( D i s ) = Ad ( D i s ) ∘ Φ m Ad ( 1 ⊗ R λ ) \operatorname{Ad}(1\otimes R_\lambda) Ad ( 1 ⊗ R λ ) P m P_m P m φ \varphi φ φ \varphi φ Ad ( D i s ) \operatorname{Ad}(D^{\mathrm{i}s}) Ad ( D i s ) φ = ( Tr ⊗ m ) ( D ⋅ ) \varphi=(\operatorname{Tr}\otimes m)(D\mathbin{\cdot}) φ = ( Tr ⊗ m ) ( D ⋅ ) M + M_+ M + Tr ⊗ m \operatorname{Tr}\otimes m Tr ⊗ m [37 Chapter VIII] .
We have J ( f ) = diag ( f i ) ⊗ 1 J(f)=\operatorname{diag}(f_i)\otimes1 J ( f ) = diag ( f i ) ⊗ 1 ( 1 − P m ) J ( f ) ( 1 − P m ) = diag ( f i 1 i < m ) ⊗ 1 (1-P_m)J(f)(1-P_m)=\operatorname{diag}(f_i1_{i<m})\otimes1 ( 1 − P m ) J ( f ) ( 1 − P m ) = diag ( f i 1 i < m ) ⊗ 1 τ − m φ ( P m J ( f ) P m ) = τ − m ( 1 − τ ) ∑ i ≥ m τ i f i = m m ( f ) \tau^{-m}\varphi(P_mJ(f)P_m)=\tau^{-m}(1-\tau)\sum_{i\geq m}\tau^if_i=m_m(f) τ − m φ ( P m J ( f ) P m ) = τ − m ( 1 − τ ) ∑ i ≥ m τ i f i = m m ( f ) (6) . ▪
For any t > 0 t>0 t > 0 m ∈ Z m\in\mathbb{Z} m ∈ Z
a m ( t ) ≔ e − t ρ m − e − t ρ m + 1 > 0. (39) a_m(t)\coloneqq e^{-t\rho^m}-e^{-t\rho^{m+1}}>0. \tag*{(39)} a m ( t ) : = e − t ρ m − e − t ρ m + 1 > 0. (39) Since ρ m → 0 \rho^m\to0 ρ m → 0 m → − ∞ m\to-\infty m → − ∞ ρ m → ∞ \rho^m\to\infty ρ m → ∞ m → + ∞ m\to+\infty m → + ∞ ∑ m ∈ Z a m ( t ) = 1 \sum_{m\in\mathbb{Z}}a_m(t)=1 ∑ m ∈ Z a m ( t ) = 1 ∑ m > k a m ( t ) = e − t ρ k + 1 \sum_{m>k}a_m(t)=e^{-t\rho^{k+1}} ∑ m > k a m ( t ) = e − t ρ k + 1 k ∈ Z k\in\mathbb{Z} k ∈ Z
Theorem 5.5 Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 ρ > 1 \rho>1 ρ > 1 t > 0 t>0 t > 0
T t = d e f ∑ m ∈ Z a m ( t ) Φ m , (40) \mathcal{T}_t \mathrel{\mathop{=}^{\mathrm{def}}} \sum_{m\in\mathbb{Z}} a_m(t)\Phi_m, \tag*{(40)} T t = def m ∈ Z ∑ a m ( t ) Φ m , (40) together with T 0 = Id \mathcal{T}_0=\operatorname{Id} T 0 = Id L ∞ ( E r ( 2 ) ) L^\infty(E_r(2)) L ∞ ( E r ( 2 )) φ ∘ T t = φ \varphi\circ\mathcal{T}_t=\varphi φ ∘ T t = φ σ \sigma σ φ \varphi φ T \mathbb{T} T B ( ℓ Z 2 ) ⊗ 1 B(\ell^2_{\mathbb{Z}})\otimes1 B ( ℓ Z 2 ) ⊗ 1 W ∗ ( A ) W^*(A) W ∗ ( A ) Λ ( x ) ↦ Λ ( T t ( x ) ) \Lambda(x)\mapsto\Lambda(\mathcal{T}_t(x)) Λ ( x ) ↦ Λ ( T t ( x )) L 2 ( E r ( 2 ) , φ ) L^2(E_r(2),\varphi) L 2 ( E r ( 2 ) , φ )
If d k = d e f Φ k + 1 ( 2 ) − Φ k ( 2 ) d_k\mathrel{\mathop{=}^{\mathrm{def}}}\Phi_{k+1}^{(2)}-\Phi_k^{(2)} d k = def Φ k + 1 ( 2 ) − Φ k ( 2 ) ( d k ) k ∈ Z (d_k)_{k\in\mathbb{Z}} ( d k ) k ∈ Z I I I
L = ∑ k ∈ Z ρ k + 1 d k , dom L = { ξ ∈ S : ∑ k ∈ Z ρ 2 ( k + 1 ) ∥ d k ξ ∥ S 2 < ∞ } . (41) \mathcal{L}=\sum_{k\in\mathbb{Z}}\rho^{k+1}d_k,\qquad
\operatorname{dom}\mathcal{L}=\left\{\xi\in\mathcal{S}:\sum_{k\in\mathbb{Z}}\rho^{2(k+1)}\lVert d_k\xi\rVert_{\mathcal{S}}^2<\infty\right\}. \tag*{(41)} L = k ∈ Z ∑ ρ k + 1 d k , dom L = { ξ ∈ S : k ∈ Z ∑ ρ 2 ( k + 1 ) ∥ d k ξ ∥ S 2 < ∞ } . (41) For any compact operator x ∈ K ( ℓ Z 2 ) ⊗ 1 x\in\mathcal{K}(\ell^2_{\mathbb{Z}})\otimes1 x ∈ K ( ℓ Z 2 ) ⊗ 1 ∥ T t ( x ) − x ∥ → 0 \lVert\mathcal{T}_t(x)-x\rVert\to0 ∥ T t ( x ) − x ∥ → 0 t → 0 + t\to0^+ t → 0 + τ = Q − d \tau=Q^{-d} τ = Q − d ρ = Q α \rho=Q^\alpha ρ = Q α
T t J = J T t r a d , t ≥ 0 , (42) \mathcal{T}_tJ=J\mathcal{T}^{\mathrm{rad}}_t,\qquad t\geq0, \tag*{(42)} T t J = J T t rad , t ≥ 0 , (42) where T t r a d = d e f ∑ m a m ( t ) E m \mathcal{T}^{\mathrm{rad}}_t\mathrel{\mathop{=}^{\mathrm{def}}}\sum_m a_m(t)E_m T t rad = def ∑ m a m ( t ) E m L r a d ∞ ( K d ) L^\infty_{\mathrm{rad}}(K^d) L rad ∞ ( K d ) L r a d ∞ ( K d ) ∩ L 2 ( K d ) L^\infty_{\mathrm{rad}}(K^d)\cap L^2(K^d) L rad ∞ ( K d ) ∩ L 2 ( K d ) e − t D r a d α e^{-tD^\alpha_{\mathrm{rad}}} e − t D rad α J 2 : ℓ r a d 2 ( w ) → S J_2:\ell^2_{\mathrm{rad}}(w)\to\mathcal{S} J 2 : ℓ rad 2 ( w ) → S f ↦ Λ ( J ( f ) ) f\mapsto\Lambda(J(f)) f ↦ Λ ( J ( f )) L J 2 = J 2 D r a d α \mathcal{L}J_2=J_2D^\alpha_{\mathrm{rad}} L J 2 = J 2 D rad α J 2 ( ℓ r a d 2 ( w ) ) J_2(\ell^2_{\mathrm{rad}}(w)) J 2 ( ℓ rad 2 ( w ))
Proof : Since the coefficients a m ( t ) a_m(t) a m ( t ) ∥ Φ m ∥ = 1 \lVert\Phi_m\rVert=1 ∥ Φ m ∥ = 1 (40) converges in the norm of bounded maps on M \mathcal{M} M T t \mathcal{T}_t T t x ∈ M + x\in\mathcal{M}_+ x ∈ M + ∑ m a m ( t ) Φ m ( x ) \sum_m a_m(t)\Phi_m(x) ∑ m a m ( t ) Φ m ( x ) φ \varphi φ φ ( T t ( x ) ) = ∑ m a m ( t ) φ ( Φ m ( x ) ) = φ ( x ) \varphi(\mathcal{T}_t(x))=\sum_m a_m(t)\varphi(\Phi_m(x))=\varphi(x) φ ( T t ( x )) = ∑ m a m ( t ) φ ( Φ m ( x )) = φ ( x ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] Lemma 5.4 .
By part 1 of Lemma 5.4 , the projections Φ m ( 2 ) \Phi_m^{(2)} Φ m ( 2 ) m m m I I I m → + ∞ m\to+\infty m → + ∞ 0 0 0 m → − ∞ m\to-\infty m → − ∞ d k d_k d k ∑ k d k = I \sum_k d_k=I ∑ k d k = I d k d_k d k Φ m ( 2 ) \Phi_m^{(2)} Φ m ( 2 ) m ≤ k m\leq k m ≤ k m ≥ k + 1 m\geq k+1 m ≥ k + 1 x ∈ n φ x\in\mathfrak{n}_\varphi x ∈ n φ [9 ] gives T t ( x ) ∈ n φ \mathcal{T}_t(x)\in\mathfrak{n}_\varphi T t ( x ) ∈ n φ
Λ ( T t ( x ) ) = ∑ m a m ( t ) Φ m ( 2 ) Λ ( x ) = ∑ k ∈ Z ( ∑ m > k a m ( t ) ) d k Λ ( x ) = ∑ k ∈ Z e − t ρ k + 1 d k Λ ( x ) . \Lambda(\mathcal{T}_t(x))=\sum_m a_m(t)\Phi_m^{(2)}\Lambda(x)=\sum_{k\in\mathbb{Z}}\left(\sum_{m>k}a_m(t)\right)d_k\Lambda(x)=\sum_{k\in\mathbb{Z}}e^{-t\rho^{k+1}}d_k\Lambda(x). Λ ( T t ( x )) = m ∑ a m ( t ) Φ m ( 2 ) Λ ( x ) = k ∈ Z ∑ ( m > k ∑ a m ( t ) ) d k Λ ( x ) = k ∈ Z ∑ e − t ρ k + 1 d k Λ ( x ) . Hence the L 2 L^2 L 2 T t ( 2 ) = ∑ k e − t ρ k + 1 d k \mathcal{T}_t^{(2)}=\sum_k e^{-t\rho^{k+1}}d_k T t ( 2 ) = ∑ k e − t ρ k + 1 d k (41) . By part 2 of Lemma 5.4 , the same formula holds in the symmetric embedding. In particular, ( T t ) (\mathcal{T}_t) ( T t ) φ \varphi φ [10 ] . For s , t > 0 s,t>0 s , t > 0 x ∈ n φ x\in\mathfrak{n}_\varphi x ∈ n φ Λ ( T s T t ( x ) ) = Λ ( T s + t ( x ) ) \Lambda(\mathcal{T}_s\mathcal{T}_t(x))=\Lambda(\mathcal{T}_{s+t}(x)) Λ ( T s T t ( x )) = Λ ( T s + t ( x )) T s T t ( x ) = T s + t ( x ) \mathcal{T}_s\mathcal{T}_t(x)=\mathcal{T}_{s+t}(x) T s T t ( x ) = T s + t ( x ) Λ \Lambda Λ n φ \mathfrak{n}_\varphi n φ σ \sigma σ M \mathcal{M} M
Let x ∈ M x\in\mathcal{M} x ∈ M m → + ∞ m\to+\infty m → + ∞ P m → 0 P_m\to0 P m → 0 ( 1 − P m ) x ( 1 − P m ) → x (1-P_m)x(1-P_m)\to x ( 1 − P m ) x ( 1 − P m ) → x τ φ − m ( P m x P m ) P m → 0 \tau^{-m}_\varphi(P_mxP_m)P_m\to0 τ φ − m ( P m x P m ) P m → 0 ∥ x ∥ \lVert x\rVert ∥ x ∥ Φ m ( x ) → x \Phi_m(x)\to x Φ m ( x ) → x σ \sigma σ ω \omega ω ε > 0 \varepsilon>0 ε > 0 N N N ∣ ω ( Φ m ( x ) ) − ω ( x ) ∣ ≤ ε |\omega(\Phi_m(x))-\omega(x)|\leq\varepsilon ∣ ω ( Φ m ( x )) − ω ( x ) ∣ ≤ ε m > N m>N m > N
∣ ω ( T t ( x ) ) − ω ( x ) ∣ ≤ ∑ m ≤ N a m ( t ) 2 ∥ ω ∥ ∥ x ∥ + ε = 2 ∥ ω ∥ ∥ x ∥ ( 1 − e − t ρ N + 1 ) + ε , |\omega(\mathcal{T}_t(x))-\omega(x)|\leq\sum_{m\leq N}a_m(t)2\lVert\omega\rVert\lVert x\rVert+\varepsilon=2\lVert\omega\rVert\lVert x\rVert\left(1-e^{-t\rho^{N+1}}\right)+\varepsilon, ∣ ω ( T t ( x )) − ω ( x ) ∣ ≤ m ≤ N ∑ a m ( t ) 2 ∥ ω ∥ ∥ x ∥ + ε = 2 ∥ ω ∥ ∥ x ∥ ( 1 − e − t ρ N + 1 ) + ε , which is at most 2 ε 2\varepsilon 2 ε t t t σ \sigma σ x x x ∥ P m x ∥ → 0 \|P_mx\|\to0 ∥ P m x ∥ → 0 ∥ x P m ∥ → 0 \|xP_m\|\to0 ∥ x P m ∥ → 0 ∣ τ − m φ ( P m x P m ) ∣ ≤ sup i ≥ m ∥ x i i ∥ ∞ → 0 |\tau^{-m}\varphi(P_mxP_m)|\leq\sup_{i\geq m}\|x_{ii}\|_\infty\to0 ∣ τ − m φ ( P m x P m ) ∣ ≤ sup i ≥ m ∥ x ii ∥ ∞ → 0 m → + ∞ m\to+\infty m → + ∞ ∥ Φ m ( x ) − x ∥ → 0 \|\Phi_m(x)-x\|\to0 ∥ Φ m ( x ) − x ∥ → 0 ∥ T t ( x ) − x ∥ → 0 \|T_t(x)-x\|\to0 ∥ T t ( x ) − x ∥ → 0 Φ m \Phi_m Φ m B ( ℓ Z 2 ) ⊗ 1 B(\ell^2_{\mathbb{Z}})\otimes1 B ( ℓ Z 2 ) ⊗ 1 W ∗ ( A ) W^*(A) W ∗ ( A ) T t \mathcal{T}_t T t
Suppose now that τ = Q − d \tau=Q^{-d} τ = Q − d ρ = Q α \rho=Q^\alpha ρ = Q α E m E_m E m L r a d ∞ ( K d ) L^\infty_{\mathrm{rad}}(K^d) L rad ∞ ( K d ) ( T t r a d ) (T_t^{\mathrm{rad}}) ( T t rad ) J J J k ∈ Z k\in\mathbb{Z} k ∈ Z g k g_k g k B k + 1 B_{k+1} B k + 1 B k B_k B k E m g k = g k E_mg_k=g_k E m g k = g k m ≥ k + 1 m\geq k+1 m ≥ k + 1 E m g k = 0 E_mg_k=0 E m g k = 0 m ≤ k m\leq k m ≤ k g k g_k g k B k ⊂ B m B_k\subset B_m B k ⊂ B m T t r a d g k = ∑ m > k a m ( t ) g k = e − t ρ k + 1 g k = e − t D r a d α g k T_t^{\mathrm{rad}}g_k=\sum_{m>k}a_m(t)g_k=e^{-t\rho^{k+1}}g_k=e^{-tD^\alpha_{\mathrm{rad}}}g_k T t rad g k = ∑ m > k a m ( t ) g k = e − t ρ k + 1 g k = e − t D rad α g k E m E_m E m L 2 ( K d ) L^2(K^d) L 2 ( K d ) g k g_k g k ℓ Z 2 ( w ) \ell^2_{\mathbb{Z}}(w) ℓ Z 2 ( w ) L 2 ( K d ) L^2(K^d) L 2 ( K d ) L 2 ( K d ) L^2(K^d) L 2 ( K d ) T t r a d = e − t D r a d α T_t^{\mathrm{rad}}=e^{-tD^\alpha_{\mathrm{rad}}} T t rad = e − t D rad α L r a d ∞ ( K d ) ∩ L 2 ( K d ) L^\infty_{\mathrm{rad}}(K^d)\cap L^2(K^d) L rad ∞ ( K d ) ∩ L 2 ( K d ) J 2 J_2 J 2 φ ( J ( f ) ∗ J ( f ) ) = ∑ i w i ∣ f i ∣ 2 \varphi(J(f)^*J(f))=\sum_iw_i|f_i|^2 φ ( J ( f ) ∗ J ( f )) = ∑ i w i ∣ f i ∣ 2 Φ m J = J E m \Phi_mJ=JE_m Φ m J = J E m J 2 g k J_2g_k J 2 g k d k d_k d k L J 2 g k = ρ k + 1 J 2 g k \mathcal{L}J_2g_k=\rho^{k+1}J_2g_k L J 2 g k = ρ k + 1 J 2 g k
Remark 5.6 Theorem 5.5 is the bilateral version of [Arh26, Theorem 7.2]; a first version of this construction appeared in a preliminary version of [Arh26]. The two new points are the two endpoints of the filtration: on the unit ball Φ 0 \Phi_0 Φ 0 ( M m ) m ∈ Z (\mathcal{M}_m)_{m\in\mathbb{Z}} ( M m ) m ∈ Z Φ m ( 2 ) → 0 \Phi_m^{(2)}\to0 Φ m ( 2 ) → 0 m → − ∞ m\to-\infty m → − ∞ E r ( 2 ) E_r(2) E r ( 2 ) B ( ℓ Z 2 ) ⊗ 1 B(\ell^2_{\mathbb{Z}})\otimes1 B ( ℓ Z 2 ) ⊗ 1 B ( ℓ Z 2 ) B(\ell^2_{\mathbb{Z}}) B ( ℓ Z 2 ) ( 1 − τ ) Tr ( diag ( τ i ) ⋅ ) (1-\tau)\operatorname{Tr}(\operatorname{diag}(\tau^i)\cdot) ( 1 − τ ) Tr ( diag ( τ i ) ⋅ ) Tr ( ( 1 − τ ) τ N ⋅ ) \operatorname{Tr}((1-\tau)\tau^N\cdot) Tr (( 1 − τ ) τ N ⋅ ) B ( ℓ N 0 2 ) = L ∞ ( S r 2 ) B(\ell^2_{\mathbb{N}_0})=L^\infty(S_r^2) B ( ℓ N 0 2 ) = L ∞ ( S r 2 )
From the unit ball to the whole space: the Poisson kernel
In this section 0 < τ < 1 0<\tau<1 0 < τ < 1 [8 Section 4] , is a Markov chain on { ( n , k ) ∈ N 0 2 : k ≤ n } \{(n,k)\in\mathbb{N}_0^2:k\leq n\} {( n , k ) ∈ N 0 2 : k ≤ n } k ∞ k_\infty k ∞ ν ( n , k ) ( j ) = P ( n , k ) ( k ∞ = j ) \nu_{(n,k)}(j)=\mathbb{P}_{(n,k)}(k_\infty=j) ν ( n , k ) ( j ) = P ( n , k ) ( k ∞ = j ) ( d , k ) = ( n − k , k ) (d,k)=(n-k,k) ( d , k ) = ( n − k , k ) N 0 \mathbb{N}_0 N 0
a d = 1 − τ d + 1 ( 1 − τ 2 ) τ d , b k = τ ( 1 − τ k + 1 ) 1 − τ 2 , a_d=\frac{1-\tau^{d+1}}{(1-\tau^2)\tau^d},\qquad b_k=\frac{\tau(1-\tau^{k+1})}{1-\tau^2}, a d = ( 1 − τ 2 ) τ d 1 − τ d + 1 , b k = 1 − τ 2 τ ( 1 − τ k + 1 ) , by [Arh26, Proposition 6.4]. Let ( δ k ) k ∈ N 0 (\delta_k)_{k\in\mathbb{N}_0} ( δ k ) k ∈ N 0 ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 B B B ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 ( B f ) ( k ) = b k ( f ( k ) − f ( k + 1 ) ) + b k − 1 ( f ( k ) − f ( k − 1 ) ) (Bf)(k)=b_k(f(k)-f(k+1))+b_{k-1}(f(k)-f(k-1)) ( B f ) ( k ) = b k ( f ( k ) − f ( k + 1 )) + b k − 1 ( f ( k ) − f ( k − 1 )) b − 1 = 0 b_{-1}=0 b − 1 = 0 0 ≤ B ≤ 4 sup k b k = 4 τ 1 − τ 2 0\leq B\leq4\sup_kb_k=\frac{4\tau}{1-\tau^2} 0 ≤ B ≤ 4 sup k b k = 1 − τ 2 4 τ ϱ d \varrho_d ϱ d d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 ϱ 0 = 1 \varrho_0=1 ϱ 0 = 1 ϱ ∞ = lim d ϱ d \varrho_\infty=\lim_{d}\varrho_d ϱ ∞ = lim d ϱ d ϱ ∞ ≥ 1 \varrho_\infty\geq1 ϱ ∞ ≥ 1 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) d , k , j ∈ N 0 d,k,j\in\mathbb{N}_0 d , k , j ∈ N 0
ν ( d + k , k ) ( j ) = ⟨ ϱ d ϱ ∞ ( B ) δ j , δ k ⟩ ℓ N 0 2 . (43) \nu_{(d+k,k)}(j)=\left\langle\frac{\varrho_d}{\varrho_\infty}(B)\delta_j,\delta_k\right\rangle_{\ell^2_{\mathbb{N}_0}}. \tag*{(43)} ν ( d + k , k ) ( j ) = ⟨ ϱ ∞ ϱ d ( B ) δ j , δ k ⟩ ℓ N 0 2 . (43) By [Arh26, Theorem 6.15], for any λ ∈ C \lambda\in\mathbb{C} λ ∈ C z ∈ C ∖ { 0 } z\in\mathbb{C}\setminus\{0\} z ∈ C ∖ { 0 } z + z − 1 = 2 − ( 1 − τ 2 ) λ τ z+z^{-1}=2-\frac{(1-\tau^2)\lambda}{\tau} z + z − 1 = 2 − τ ( 1 − τ 2 ) λ
ϱ ∞ ( λ ) = ( τ z , τ z − 1 ; τ ) ∞ ( τ ; τ ) ∞ 2 . (44) \varrho_\infty(\lambda)=\frac{(\tau z,\tau z^{-1};\tau)_\infty}{(\tau;\tau)_\infty^2}. \tag*{(44)} ϱ ∞ ( λ ) = ( τ ; τ ) ∞ 2 ( τ z , τ z − 1 ; τ ) ∞ . (44) As k → ∞ k \to\infty k → ∞ b k b_k b k b ∞ = d e f τ 1 − τ 2 b_\infty\overset{\mathrm{def}}{=} \frac{\tau}{1-\tau^2} b ∞ = def 1 − τ 2 τ B ∞ B_\infty B ∞ ℓ Z 2 \ell^2_{\mathbb{Z}} ℓ Z 2 ( B ∞ f ) ( k ) = b ∞ ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 ) ) (B_\infty f)(k)=b_\infty(2f(k)-f(k+1)-f(k-1)) ( B ∞ f ) ( k ) = b ∞ ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 )) ℓ Z 2 → L 2 ( T ) \ell^2_{\mathbb{Z}}\to L^2(\mathbb{T}) ℓ Z 2 → L 2 ( T ) δ k ↦ e i k θ \delta_k\mapsto e^{ik\theta} δ k ↦ e ik θ
λ ( θ ) = d e f 2 τ ( 1 − cos θ ) 1 − τ 2 , θ ∈ R , (45) \lambda(\theta) \overset{\mathrm{def}}{=} \frac{2\tau(1-\cos\theta)}{1-\tau^2}, \qquad\theta\in\mathbb{R}, \tag*{(45)} λ ( θ ) = def 1 − τ 2 2 τ ( 1 − cos θ ) , θ ∈ R , (45) and z = e i θ z=e^{i\theta} z = e i θ z + z − 1 = 2 − ( 1 − τ 2 ) λ ( θ ) τ z+z^{-1}=2-\frac{(1-\tau^2)\lambda(\theta)}{\tau} z + z − 1 = 2 − τ ( 1 − τ 2 ) λ ( θ )
Theorem 6.1 Let 0 < τ < 1 0<\tau<1 0 < τ < 1
For any d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 l ∈ Z l\in\mathbb{Z} l ∈ Z
lim k → ∞ ν ( d + k , k ) ( k + l ) = ν d Z ( l ) = d e f 1 2 π ∫ − π π ϱ d ϱ ∞ ( λ ( θ ) ) e i l θ d θ . (46) \lim_{k\to\infty}\nu_{(d+k,k)}(k+l)=\nu_d^{\mathbb{Z}}(l)\overset{\mathrm{def}}{=}\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\varrho_d}{\varrho_\infty}(\lambda(\theta))e^{il\theta}\,d\theta. \tag*{(46)} k → ∞ lim ν ( d + k , k ) ( k + l ) = ν d Z ( l ) = def 2 π 1 ∫ − π π ϱ ∞ ϱ d ( λ ( θ )) e i l θ d θ . (46) Moreover ν d Z ( l ) = ν d Z ( − l ) ≥ 0 \nu_d^{\mathbb{Z}}(l)=\nu_d^{\mathbb{Z}}(-l)\geq0 ν d Z ( l ) = ν d Z ( − l ) ≥ 0 ∑ l ∈ Z ν d Z ( l ) = 1 \sum_{l\in\mathbb{Z}}\nu_d^{\mathbb{Z}}(l)=1 ∑ l ∈ Z ν d Z ( l ) = 1
For d = 0 d=0 d = 0 k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 l ≥ 0 l\geq0 l ≥ 0
ν ( k , k ) ( k + l ) = ( τ ; τ ) k ( τ ; τ ) k + l ∑ m = 0 k τ l + 2 m ( 1 − τ k − m + 1 ) ( τ ; τ ) m ( τ ; τ ) l + m , (47) \nu_{(k,k)}(k+l)=(\tau;\tau)_k(\tau;\tau)_{k+l}\sum_{m=0}^{k}\frac{\tau^{l+2m}(1-\tau^{k-m+1})}{(\tau;\tau)_m(\tau;\tau)_{l+m}}, \tag*{(47)} ν ( k , k ) ( k + l ) = ( τ ; τ ) k ( τ ; τ ) k + l m = 0 ∑ k ( τ ; τ ) m ( τ ; τ ) l + m τ l + 2 m ( 1 − τ k − m + 1 ) , (47) and for any l ∈ Z l\in\mathbb{Z} l ∈ Z
ν 0 Z ( l ) = ( τ ; τ ) ∞ 2 ∑ m ≥ 0 τ ∣ l ∣ + 2 m ( τ ; τ ) m ( τ ; τ ) m + ∣ l ∣ , ∑ l ∈ Z ν 0 Z ( l ) e i l θ = ( τ ; τ ) ∞ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 2 , θ ∈ R . (48) \nu_0^{\mathbb{Z}}(l)=(\tau;\tau)_\infty^2\sum_{m\geq0}\frac{\tau^{|l|+2m}}{(\tau;\tau)_m(\tau;\tau)_{m+|l|}},\qquad\sum_{l\in\mathbb{Z}}\nu_0^{\mathbb{Z}}(l)e^{il\theta}=\frac{(\tau;\tau)_\infty^2}{|(\tau e^{i\theta};\tau)_\infty|^2},\qquad\theta\in\mathbb{R}. \tag*{(48)} ν 0 Z ( l ) = ( τ ; τ ) ∞ 2 m ≥ 0 ∑ ( τ ; τ ) m ( τ ; τ ) m + ∣ l ∣ τ ∣ l ∣ + 2 m , l ∈ Z ∑ ν 0 Z ( l ) e i l θ = ∣ ( τ e i θ ; τ ) ∞ ∣ 2 ( τ ; τ ) ∞ 2 , θ ∈ R . (48) Finally, there exists a constant C τ C_\tau C τ ∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ ≤ C τ τ k |\nu_{(k,k)}(k+l)-\nu_0^{\mathbb{Z}}(l)|\leq C_\tau\tau^k ∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ ≤ C τ τ k k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 l ∈ Z l\in\mathbb{Z} l ∈ Z k + l ≥ 0 k+l\geq0 k + l ≥ 0
Proof. 1. For any integer N ≥ 0 N\geq0 N ≥ 0 W N : ℓ N 0 2 → ℓ Z 2 W_N:\ell^2_{\mathbb{N}_0}\to\ell^2_{\mathbb{Z}} W N : ℓ N 0 2 → ℓ Z 2 ( W N f ) ( k ) = f ( k + N ) (W_Nf)(k)=f(k+N) ( W N f ) ( k ) = f ( k + N ) k ≥ − N k\geq-N k ≥ − N ( W N f ) ( k ) = 0 (W_Nf)(k)=0 ( W N f ) ( k ) = 0 B N = d e f W N B W N ∗ B_N\overset{\mathrm{def}}{=}W_NBW_N^* B N = def W N B W N ∗ 0 ≤ B N ≤ 4 τ 1 − τ 2 0\leq B_N\leq\frac{4\tau}{1-\tau^2} 0 ≤ B N ≤ 1 − τ 2 4 τ 0 ≤ B ∞ ≤ 4 τ 1 − τ 2 0\leq B_\infty\leq\frac{4\tau}{1-\tau^2} 0 ≤ B ∞ ≤ 1 − τ 2 4 τ u ∈ ℓ Z 2 u\in\ell^2_{\mathbb{Z}} u ∈ ℓ Z 2 N N N k ∈ Z k\in\mathbb{Z} k ∈ Z
( ( B N − B ∞ ) u ) ( k ) = ( b k + N − b ∞ ) ( u ( k ) − u ( k + 1 ) ) + ( b k + N − 1 − b ∞ ) ( u ( k ) − u ( k − 1 ) ) , ((B_N-B_\infty)u)(k)=(b_{k+N}-b_\infty)(u(k)-u(k+1))+(b_{k+N-1}-b_\infty)(u(k)-u(k-1)), (( B N − B ∞ ) u ) ( k ) = ( b k + N − b ∞ ) ( u ( k ) − u ( k + 1 )) + ( b k + N − 1 − b ∞ ) ( u ( k ) − u ( k − 1 )) , which vanishes outside a finite set independent of N N N O ( τ N ) O(\tau^N) O ( τ N ) B N u → B ∞ u B_Nu\to B_\infty u B N u → B ∞ u B N B_N B N B N → B ∞ B_N\to B_\infty B N → B ∞ p ( B N ) → p ( B ∞ ) p(B_N)\to p(B_\infty) p ( B N ) → p ( B ∞ ) p p p g g g I = [ 0 , 4 τ 1 − τ 2 ] I=[0,\frac{4\tau}{1-\tau^2}] I = [ 0 , 1 − τ 2 4 τ ] ε > 0 \varepsilon>0 ε > 0 p p p sup I ∣ g − p ∣ ≤ ε \sup_I|g-p|\leq\varepsilon sup I ∣ g − p ∣ ≤ ε ∥ g ( B N ) − p ( B N ) ∥ ≤ ε \|g(B_N)-p(B_N)\|\leq\varepsilon ∥ g ( B N ) − p ( B N ) ∥ ≤ ε ∥ g ( B ∞ ) − p ( B ∞ ) ∥ ≤ ε \|g(B_\infty)-p(B_\infty)\|\leq\varepsilon ∥ g ( B ∞ ) − p ( B ∞ ) ∥ ≤ ε g ( B N ) → g ( B ∞ ) g(B_N)\to g(B_\infty) g ( B N ) → g ( B ∞ ) B N = W N B W N ∗ ⊕ 0 B_N=W_NBW_N^*\oplus0 B N = W N B W N ∗ ⊕ 0 ℓ Z 2 = Ran W N ⊕ ( Ran W N ) ⊥ \ell^2_{\mathbb{Z}}=\operatorname{Ran}W_N\oplus(\operatorname{Ran}W_N)^\perp ℓ Z 2 = Ran W N ⊕ ( Ran W N ) ⊥ g ( B N ) = W N g ( B ) W N ∗ + g ( 0 ) ( I − W N W N ∗ ) g(B_N)=W_Ng(B)W_N^*+g(0)(I-W_NW_N^*) g ( B N ) = W N g ( B ) W N ∗ + g ( 0 ) ( I − W N W N ∗ ) N ≥ max ( 0 , − l ) N\geq\max(0,-l) N ≥ max ( 0 , − l ) ⟨ g ( B N ) δ l , δ 0 ⟩ ℓ Z 2 = ⟨ g ( B ) δ N + l , δ N ⟩ ℓ N 0 2 \langle g(B_N)\delta_l,\delta_0\rangle_{\ell^2_{\mathbb{Z}}}=\langle g(B)\delta_{N+l},\delta_N\rangle_{\ell^2_{\mathbb{N}_0}} ⟨ g ( B N ) δ l , δ 0 ⟩ ℓ Z 2 = ⟨ g ( B ) δ N + l , δ N ⟩ ℓ N 0 2 g = ϱ d ϱ ∞ g=\frac{\varrho_d}{\varrho_\infty} g = ϱ ∞ ϱ d I I I ϱ ∞ ≥ 1 \varrho_\infty\geq1 ϱ ∞ ≥ 1 (43) , we obtain
lim N → ∞ ν ( d + N , N ) ( N + l ) = ⟨ g ( B ∞ ) δ l , δ 0 ⟩ ℓ Z 2 = 1 2 π ∫ − π π g ( λ ( θ ) ) e i l θ d θ . \lim_{N\to\infty}\nu_{(d+N,N)}(N+l)=\langle g(B_\infty)\delta_l,\delta_0\rangle_{\ell^2_{\mathbb{Z}}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}g(\lambda(\theta))e^{il\theta}\,d\theta. N → ∞ lim ν ( d + N , N ) ( N + l ) = ⟨ g ( B ∞ ) δ l , δ 0 ⟩ ℓ Z 2 = 2 π 1 ∫ − π π g ( λ ( θ )) e i l θ d θ . Since λ \lambda λ ν d Z ( l ) = ν d Z ( − l ) \nu_d^{\mathbb{Z}}(l)=\nu_d^{\mathbb{Z}}(-l) ν d Z ( l ) = ν d Z ( − l ) ν d Z ( l ) ≥ 0 \nu_d^{\mathbb{Z}}(l)\geq0 ν d Z ( l ) ≥ 0 θ ↦ g ( λ ( θ ) ) \theta\mapsto g(\lambda(\theta)) θ ↦ g ( λ ( θ )) T \mathbb{T} T ∑ l ν d Z ( l ) = g ( λ ( 0 ) ) = ϱ d ( 0 ) ϱ ∞ ( 0 ) = 1 \sum_l\nu_d^{\mathbb{Z}}(l)=g(\lambda(0))=\frac{\varrho_d(0)}{\varrho_\infty(0)}=1 ∑ l ν d Z ( l ) = g ( λ ( 0 )) = ϱ ∞ ( 0 ) ϱ d ( 0 ) = 1 ϱ d ( 0 ) = 1 \varrho_d(0)=1 ϱ d ( 0 ) = 1 d ∈ N 0 ∪ { ∞ } d\in\mathbb{N}_0\cup\{\infty\} d ∈ N 0 ∪ { ∞ }
Let k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 l ≥ 0 l\geq0 l ≥ 0 [ m n ] τ = ( τ ; τ ) m ( τ ; τ ) n ( τ ; τ ) m − n \left[\begin{smallmatrix}m\\n\end{smallmatrix}\right]_\tau=\frac{(\tau;\tau)_m}{(\tau;\tau)_n(\tau;\tau)_{m-n}} [ m n ] τ = ( τ ; τ ) n ( τ ; τ ) m − n ( τ ; τ ) m 0 ≤ n ≤ m 0\leq n\leq m 0 ≤ n ≤ m τ \tau τ [12 (17.2.27)] . By [Arh26, (6.30)], we have
ν ( k , k ) ( k + l ) = ∑ n = 0 k [ k + l n ] τ [ k n ] τ ( τ ; τ ) n ( τ ; τ ) n + 1 τ 2 k + l − 2 n . \nu_{(k,k)}(k+l)=\sum_{n=0}^{k}\left[\begin{matrix}k+l\\n\end{matrix}\right]_\tau\left[\begin{matrix}k\\n\end{matrix}\right]_\tau(\tau;\tau)_n(\tau;\tau)_{n+1}\tau^{2k+l-2n}. ν ( k , k ) ( k + l ) = n = 0 ∑ k [ k + l n ] τ [ k n ] τ ( τ ; τ ) n ( τ ; τ ) n + 1 τ 2 k + l − 2 n . We put n ≔ k − m n\coloneqq k-m n : = k − m
[ k + l k − m ] τ [ k k − m ] τ ( τ ; τ ) k − m ( τ ; τ ) k − m + 1 = ( τ ; τ ) k + l ( τ ; τ ) k ( 1 − τ k − m + 1 ) ( τ ; τ ) l + m ( τ ; τ ) m , \left[\begin{matrix}k+l\\k-m\end{matrix}\right]_\tau\left[\begin{matrix}k\\k-m\end{matrix}\right]_\tau(\tau;\tau)_{k-m}(\tau;\tau)_{k-m+1}=\frac{(\tau;\tau)_{k+l}(\tau;\tau)_k(1-\tau^{k-m+1})}{(\tau;\tau)_{l+m}(\tau;\tau)_m}, [ k + l k − m ] τ [ k k − m ] τ ( τ ; τ ) k − m ( τ ; τ ) k − m + 1 = ( τ ; τ ) l + m ( τ ; τ ) m ( τ ; τ ) k + l ( τ ; τ ) k ( 1 − τ k − m + 1 ) , which gives (47) . For d = 0 d=0 d = 0 g ( λ ( θ ) ) = ( τ ; τ ) ∞ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 2 g(\lambda(\theta))=\frac{(\tau;\tau)_\infty^2}{|(\tau e^{i\theta};\tau)_\infty|^2} g ( λ ( θ )) = ∣ ( τ e i θ ; τ ) ∞ ∣ 2 ( τ ; τ ) ∞ 2 (44) with z = e i θ z=e^{i\theta} z = e i θ q q q [15 Section 1.3] , in the form [12 (17.5.4)] , we have
1 ( τ e i θ ; τ ) ∞ = ∑ a ≥ 0 τ a e i a θ ( τ ; τ ) a , \frac{1}{(\tau e^{i\theta};\tau)_\infty}=\sum_{a\geq0}\frac{\tau^a e^{ia\theta}}{(\tau;\tau)_a}, ( τ e i θ ; τ ) ∞ 1 = a ≥ 0 ∑ ( τ ; τ ) a τ a e ia θ , the series converging absolutely and uniformly in θ \theta θ e i l θ e^{il\theta} e i l θ (48) .
It remains to estimate the rate. Let l ≥ 0 l\geq0 l ≥ 0 S l ≔ ∑ m ≥ 0 τ l + 2 m ( τ ; τ ) m ( τ ; τ ) l + m S_l\coloneqq\sum_{m\geq0}\frac{\tau^{l+2m}}{(\tau;\tau)_m(\tau;\tau)_{l+m}} S l : = ∑ m ≥ 0 ( τ ; τ ) m ( τ ; τ ) l + m τ l + 2 m S l ≤ τ l ( τ ; τ ) ∞ 2 ( 1 − τ 2 ) S_l\leq\frac{\tau^l}{(\tau;\tau)_\infty^2(1-\tau^2)} S l ≤ ( τ ; τ ) ∞ 2 ( 1 − τ 2 ) τ l (47) and (48) , we have
ν ( k , k ) ( k + l ) − ν 0 Z ( l ) = ( ( τ ; τ ) k ( τ ; τ ) k + l − ( τ ; τ ) ∞ 2 ) S l \nu_{(k,k)}(k+l)-\nu_0^{\mathbb{Z}}(l)=\left((\tau;\tau)_k(\tau;\tau)_{k+l}-(\tau;\tau)_\infty^2\right)S_l ν ( k , k ) ( k + l ) − ν 0 Z ( l ) = ( ( τ ; τ ) k ( τ ; τ ) k + l − ( τ ; τ ) ∞ 2 ) S l − ( τ ; τ ) k ( τ ; τ ) k + l [ ∑ m = 0 k τ l + 2 m τ k − m + 1 ( τ ; τ ) m ( τ ; τ ) l + m + ∑ m > k τ l + 2 m ( τ ; τ ) m ( τ ; τ ) l + m ] . -(\tau;\tau)_k(\tau;\tau)_{k+l}\left[\sum_{m=0}^{k}\frac{\tau^{l+2m}\tau^{k-m+1}}{(\tau;\tau)_m(\tau;\tau)_{l+m}}+\sum_{m>k}\frac{\tau^{l+2m}}{(\tau;\tau)_m(\tau;\tau)_{l+m}}\right]. − ( τ ; τ ) k ( τ ; τ ) k + l [ m = 0 ∑ k ( τ ; τ ) m ( τ ; τ ) l + m τ l + 2 m τ k − m + 1 + m > k ∑ ( τ ; τ ) m ( τ ; τ ) l + m τ l + 2 m ] . For any j ≥ k j\geq k j ≥ k 0 ≤ ( τ ; τ ) j − ( τ ; τ ) ∞ ≤ 1 − ∏ i > j ( 1 − τ i ) ≤ ∑ i > j τ i ≤ τ k + 1 1 − τ 0\leq(\tau;\tau)_j-(\tau;\tau)_\infty\leq1-\prod_{i>j}(1-\tau^i)\leq\sum_{i>j}\tau^i\leq\frac{\tau^{k+1}}{1-\tau} 0 ≤ ( τ ; τ ) j − ( τ ; τ ) ∞ ≤ 1 − ∏ i > j ( 1 − τ i ) ≤ ∑ i > j τ i ≤ 1 − τ τ k + 1 2 τ k + 1 1 − τ S l \frac{2\tau^{k+1}}{1-\tau}S_l 1 − τ 2 τ k + 1 S l ( τ ; τ ) ∞ − 2 τ l ( τ k + 1 1 − τ + τ 2 k + 2 1 − τ 2 ) (\tau;\tau)_\infty^{-2}\tau^l\left(\frac{\tau^{k+1}}{1-\tau}+\frac{\tau^{2k+2}}{1-\tau^2}\right) ( τ ; τ ) ∞ − 2 τ l ( 1 − τ τ k + 1 + 1 − τ 2 τ 2 k + 2 ) C τ C_\tau C τ ∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ ≤ C τ τ k + l |\nu_{(k,k)}(k+l)-\nu_0^{\mathbb{Z}}(l)|\leq C_\tau\tau^{k+l} ∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ ≤ C τ τ k + l k , l ≥ 0 k,l\geq0 k , l ≥ 0 l < 0 l<0 l < 0 k ′ ≔ k + l ≥ 0 k'\coloneqq k+l\geq0 k ′ : = k + l ≥ 0 ν ( k , k ) ( j ) = ν ( j , j ) ( k ) \nu_{(k,k)}(j)=\nu_{(j,j)}(k) ν ( k , k ) ( j ) = ν ( j , j ) ( k ) ν 0 Z \nu_0^{\mathbb{Z}} ν 0 Z
∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ = ∣ ν ( k ′ , k ′ ) ( k ′ + ∣ l ∣ ) − ν 0 Z ( ∣ l ∣ ) ∣ ≤ C τ τ k ′ + ∣ l ∣ = C τ τ k . |\nu_{(k,k)}(k+l)-\nu_0^{\mathbb{Z}}(l)|=|\nu_{(k',k')}(k'+|l|)-\nu_0^{\mathbb{Z}}(|l|)|\leq C_\tau\tau^{k'+|l|}=C_\tau\tau^k. ∣ ν ( k , k ) ( k + l ) − ν 0 Z ( l ) ∣ = ∣ ν ( k ′ , k ′ ) ( k ′ + ∣ l ∣ ) − ν 0 Z ( ∣ l ∣ ) ∣ ≤ C τ τ k ′ + ∣ l ∣ = C τ τ k . This proves the last assertion.
Remark 6.2 The limit operator B ∞ B_\infty B ∞ Z \mathbb{Z} Z b ∞ b_\infty b ∞ α = d \alpha=d α = d L τ ∞ L_\tau^\infty L τ ∞ c k = 1 − τ c_k=1-\tau c k = 1 − τ L τ ∞ L_\tau^\infty L τ ∞ ℓ 2 ( w ) \ell^2(w) ℓ 2 ( w ) ℓ 2 \ell^2 ℓ 2 ⟨ B f , f ⟩ = 1 2 E τ ( f ) \langle Bf,f\rangle=\frac{1}{2}\mathcal{E}_\tau(f) ⟨ B f , f ⟩ = 2 1 E τ ( f ) ϱ d ( λ ) ϱ ∞ ( λ ) \frac{\varrho_d(\lambda)}{\varrho_\infty(\lambda)} ϱ ∞ ( λ ) ϱ d ( λ ) ζ \zeta ζ d d d ( X , Y ) (X,Y) ( X , Y ) (46) reads ν d Z ( l ) = E d [ ( e − ζ B ∞ ) 0 l ] \nu_d^{\mathbb{Z}}(l)=\mathbb{E}_d[(e^{-\zeta B_\infty})_{0l}] ν d Z ( l ) = E d [( e − ζ B ∞ ) 0 l ] Y ζ − Y 0 Y_\zeta-Y_0 Y ζ − Y 0 Y Y Y Z \mathbb{Z} Z b ∞ b_\infty b ∞ ζ \zeta ζ
The quantum plane and its covariant Casimir
The following construction identifies the quantum homogeneous space and the self-adjoint Casimir underlying the bilateral radial operator. The algebraic Casimir is classical [BCGST96, AD99]; the purpose here is its precise analytic realization, its covariant completely positive dynamics, and the distinction between the homogeneous space and the whole quantum group.
Conventions and main conclusions
Fix 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 H = ℓ 2 ( Z ) H=\ell^2(\mathbb{Z}) H = ℓ 2 ( Z ) ( e k ) k ∈ Z (e_k)_{k\in\mathbb{Z}} ( e k ) k ∈ Z V e k = e k − 1 Ve_k=e_{k-1} V e k = e k − 1 H ⊗ L 2 ( T ) H\otimes L^2(\mathbb{T}) H ⊗ L 2 ( T )
v = V ⊗ 1 , n ( e k ⊗ f ) = r k e k ⊗ z f , A = n ∗ n = diag ( τ k ) ⊗ 1. (49) v=V\otimes1,\qquad n(e_k\otimes f)=r^k e_k\otimes zf,\qquad A=n^*n=\operatorname{diag}(\tau^k)\otimes1. \tag*{(49)} v = V ⊗ 1 , n ( e k ⊗ f ) = r k e k ⊗ z f , A = n ∗ n = diag ( τ k ) ⊗ 1. (49) Here and below an unbounded generator is understood as an element affiliated with the C ∗ C^* C ∗ [42 ] . Woronowicz’s group has
v n v ∗ = r n , Δ ( v ) = v ⊗ v , Δ ( n ) = v ⊗ n + n ⊗ v ∗ ; vnv^*=rn,\qquad\Delta(v)=v\otimes v,\qquad\Delta(n)=v\otimes n+n\otimes v^*; v n v ∗ = r n , Δ ( v ) = v ⊗ v , Δ ( n ) = v ⊗ n + n ⊗ v ∗ ; Equation not extracted
(50) where the last sum is closed in the sense of the quantum-group construction [41 , 43 ] . Its reduced C ∗ C^* C ∗ A = C 0 ( E r ( 2 ) ) A=C_0(E_r(2)) A = C 0 ( E r ( 2 )) v j f ( n ) v^jf(n) v j f ( n ) j ∈ Z j\in\mathbb{Z} j ∈ Z f ∈ C 0 ( C r ) f\in C_0(C_r) f ∈ C 0 ( C r ) C r = { 0 } ∪ { r k z : k ∈ Z , z ∈ T } C_r=\{0\}\cup\{r^kz:k\in\mathbb{Z},z\in\mathbb{T}\} C r = { 0 } ∪ { r k z : k ∈ Z , z ∈ T }
φ ( x ) = ( 1 − τ ) ∑ k ∈ Z τ k ∫ T x k k ( z ) d m ( z ) , M = A ′ ′ = B ( H ) ⊗ ‾ L ∞ ( T ) . (51) \varphi(x)=(1-\tau)\sum_{k\in\mathbb{Z}}\tau^k\int_{\mathbb{T}}x_{kk}(z)\,dm(z),\qquad M=A^{\prime\prime}=B(H)\overline{\otimes}L^\infty(\mathbb{T}). \tag*{(51)} φ ( x ) = ( 1 − τ ) k ∈ Z ∑ τ k ∫ T x k k ( z ) d m ( z ) , M = A ′′ = B ( H ) ⊗ L ∞ ( T ) . (51) It is ( 1 − τ ) (1-\tau) ( 1 − τ ) [31 Theorem 2.1] . This weight is both left and right invariant. Its nontraciality must not be confused with nonunimodularity. Products such as Tr ( A x ) \operatorname{Tr}(Ax) Tr ( A x ) x ≥ 0 x\geq0 x ≥ 0 Tr ( A 1 / 2 x A 1 / 2 ) \operatorname{Tr}(A^{1/2}xA^{1/2}) Tr ( A 1/2 x A 1/2 )
On H w = ℓ 2 ( Z , w ) H_w=\ell^2(\mathbb{Z},w) H w = ℓ 2 ( Z , w ) w k = ( 1 − τ ) τ k w_k=(1-\tau)\tau^k w k = ( 1 − τ ) τ k L = L τ ∞ L=L^\infty_\tau L = L τ ∞
E 0 ( f ) = ( 1 − τ ) ∑ k ∈ Z ∣ f ( k + 1 ) − f ( k ) ∣ 2 , dom E 0 = { f ∈ H w : E 0 ( f ) < ∞ } . (52) \mathcal{E}_0(f)=(1-\tau)\sum_{k\in\mathbb{Z}}\lvert f(k+1)-f(k)\rvert^2,\qquad\operatorname{dom}\mathcal{E}_0=\{f\in H_w:\mathcal{E}_0(f)<\infty\}. \tag*{(52)} E 0 ( f ) = ( 1 − τ ) k ∈ Z ∑ ∣ f ( k + 1 ) − f ( k ) ∣ 2 , dom E 0 = { f ∈ H w : E 0 ( f ) < ∞ } . (52) Thus
( L f ) ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 ) ) . (53) (Lf)(k)=\tau^{-k}(2f(k)-f(k+1)-f(k-1)). \tag*{(53)} ( L f ) ( k ) = τ − k ( 2 f ( k ) − f ( k + 1 ) − f ( k − 1 )) . (53) We use the self-adjoint realization of Proposition 3.1, not an arbitrary extension of the difference expression on finitely supported functions. Its operator core consists of sequences vanishing near − ∞ -\infty − ∞ + ∞ +\infty + ∞ + ∞ +\infty + ∞ Δ f ( k ) → 0 \Delta f(k)\to0 Δ f ( k ) → 0
The principal conclusions are Theorems 7.1, 7.3, 7.14, 7.17 and 7.20. In particular, the answer to the intrinsic radial-Casimir problem is affirmative for the realization specified above. The answer to the stronger question about an ordinary convolution semigroup on the whole group starting at the counit is negative.
The quotient by the right torus
The quotient homomorphism π : A → C ( T ) \pi:A\to C(\mathbb{T}) π : A → C ( T ) π ( v ) = u \pi(v)=u π ( v ) = u π ( n ) = 0 \pi(n)=0 π ( n ) = 0 [41 ] . For t ∈ T t\in\mathbb{T} t ∈ T χ t ≔ ev t π \chi_t\coloneqq\operatorname{ev}_t\pi χ t : = ev t π
R t = ( Id ⊗ χ t ) Δ : v ↦ t v , n ↦ t − 1 n ; L t = ( χ t ⊗ Id ) Δ : v ↦ t v , n ↦ t n . (54) R_t=(\operatorname{Id}\otimes\chi_t)\Delta:\ v\mapsto tv,\ n\mapsto t^{-1}n;\qquad L_t=(\chi_t\otimes\operatorname{Id})\Delta:\ v\mapsto tv,\ n\mapsto tn. \tag*{(54)} R t = ( Id ⊗ χ t ) Δ : v ↦ t v , n ↦ t − 1 n ; L t = ( χ t ⊗ Id ) Δ : v ↦ t v , n ↦ t n . (54) They commute and preserve φ \varphi φ E R = ∫ T R t d m ( t ) E_R=\int_T R_t\,\mathrm{d}m(t) E R = ∫ T R t d m ( t ) E L = ∫ T L t d m ( t ) E_L=\int_T L_t\,\mathrm{d}m(t) E L = ∫ T L t d m ( t ) E = E L E R E=E_LE_R E = E L E R P j = 1 [ 0 , τ j ] ( A ) P_j=1_{[0,\tau^j]}(A) P j = 1 [ 0 , τ j ] ( A ) P j = ∑ k ≥ j e k k ⊗ 1 P_j=\sum_{k\geq j}e_{kk}\otimes1 P j = ∑ k ≥ j e k k ⊗ 1 φ ( P j ) = τ j \varphi(P_j)=\tau^j φ ( P j ) = τ j
Theorem 7.1 Define
W ( e k ⊗ f ) = e k ⊗ z − k f , Θ ( x ) = W ( x ⊗ 1 ) W ∗ . (55) W(e_k\otimes f)=e_k\otimes z^{-k}f,\qquad\Theta(x)=W(x\otimes1)W^*. \tag*{(55)} W ( e k ⊗ f ) = e k ⊗ z − k f , Θ ( x ) = W ( x ⊗ 1 ) W ∗ . (55) Then the right fixed-point algebras are
N = M R = Θ ( B ( H ) ) , P = A R = Θ ( K ( H ) + C P 0 ) . (56) N=M^R=\Theta(B(H)),\qquad P=A^R=\Theta(\mathcal{K}(H)+\mathbb{C}P_0). \tag*{(56)} N = M R = Θ ( B ( H )) , P = A R = Θ ( K ( H ) + C P 0 ) . (56) Under this identification, the weight is ψ ( x ) = ( 1 − τ ) Tr ( A x ) \psi(x)=(1-\tau)\operatorname{Tr}(Ax) ψ ( x ) = ( 1 − τ ) Tr ( A x ) E L ∣ N E_L|_N E L ∣ N δ = Δ ∣ N \delta=\Delta|_N δ = Δ ∣ N P P P C ∗ C^* C ∗ ζ = v n \zeta=vn ζ = v n
ζ = Θ ( Z ) , Z = V diag ( r k ) , ζ ∗ ζ = A , ζ ζ ∗ = τ A , δ ( ζ ) = v 2 ⊗ ζ + ζ ⊗ 1. (57) \zeta=\Theta(Z),\qquad Z=V\operatorname{diag}(r^k),\qquad\zeta^*\zeta=A,\qquad\zeta\zeta^*=\tau A,\qquad\delta(\zeta)=v^2\otimes\zeta+\zeta\otimes1. \tag*{(57)} ζ = Θ ( Z ) , Z = V diag ( r k ) , ζ ∗ ζ = A , ζ ζ ∗ = τ A , δ ( ζ ) = v 2 ⊗ ζ + ζ ⊗ 1. (57) The origin character of P P P ε 0 ( Θ ( K + c P 0 ) ) = c \varepsilon_0(\Theta(K+cP_0))=c ε 0 ( Θ ( K + c P 0 )) = c A \mathcal{A} A ( Id ⊗ ε 0 ) δ ( x ) = x (\operatorname{Id}\otimes\varepsilon_0)\delta(x)=x ( Id ⊗ ε 0 ) δ ( x ) = x x ∈ P x\in P x ∈ P
Proof : In the concrete model, R t ( e i j ⊗ z a ) = t j − i − a e i j ⊗ z a R_t(e_{ij}\otimes z^a)=t^{j-i-a}e_{ij}\otimes z^a R t ( e ij ⊗ z a ) = t j − i − a e ij ⊗ z a c i j z j − i c_{ij}z^{j-i} c ij z j − i Θ ( e i j ) = e i j ⊗ z j − i \Theta(e_{ij})=e_{ij}\otimes z^{j-i} Θ ( e ij ) = e ij ⊗ z j − i W ∗ W^* W ∗ ( c i j ) (c_{ij}) ( c ij ) (51) proves the weight identity. The left action multiplies this matrix unit by t 2 ( j − i ) t^{2(j-i)} t 2 ( j − i )
For the C ∗ C^* C ∗ J = K ( H ) ⊗ C ( T ) \mathcal{J}=\mathcal{K}(H)\otimes C(\mathbb{T}) J = K ( H ) ⊗ C ( T ) A \mathcal{A} A A \mathcal{A} A v v v n n n f ∈ C 0 ( C r ) f\in C_0(\mathbb{C}_r) f ∈ C 0 ( C r ) f ( n ) − f ( 0 ) P 0 f(n)-f(0)P_0 f ( n ) − f ( 0 ) P 0 J \mathcal{J} J A / J \mathcal{A}/\mathcal{J} A / J P 0 P_0 P 0 v P 0 vP_0 v P 0 T \mathbb{T} T A / J = C ( T ) \mathcal{A}/\mathcal{J}=C(\mathbb{T}) A / J = C ( T ) π \pi π c c c c P 0 cP_0 c P 0 J R = Θ ( K ( H ) ) \mathcal{J}^R=\Theta(\mathcal{K}(H)) J R = Θ ( K ( H )) P 0 P_0 P 0 (56) is not K ( H ) + C 1 \mathcal{K}(H)+\mathbb{C}1 K ( H ) + C 1
Coassociativity gives Δ E R = ( Id ⊗ E R ) Δ \Delta E_R=(\operatorname{Id}\otimes E_R)\Delta Δ E R = ( Id ⊗ E R ) Δ Δ ( N ) ⊂ M ⊗ ‾ N \Delta(N)\subset M\overline{\otimes}N Δ ( N ) ⊂ M ⊗ N C ∗ C^* C ∗ A \mathcal{A} A span { Δ ( x ) ( a ⊗ 1 ) : x ∈ P , a ∈ A } = A ⊗ P \operatorname{span}\{\Delta(x)(a\otimes1):x\in P,a\in\mathcal{A}\}=\mathcal{A}\otimes P span { Δ ( x ) ( a ⊗ 1 ) : x ∈ P , a ∈ A } = A ⊗ P Id ⊗ E R \operatorname{Id}\otimes E_R Id ⊗ E R A \mathcal{A} A P j P_j P j j → − ∞ j\to-\infty j → − ∞ A \mathcal{A} A [5 ] and [39 ] .
The product v n vn v n (54) ; its polar part is V ⊗ z V\otimes z V ⊗ z A 1 / 2 A^{1/2} A 1/2 Z Z Z Θ \Theta Θ P 0 P_0 P 0 ( 1 + A ) − 1 ∈ K ( H ) + C P 0 (1+A)^{-1}\in\mathcal{K}(H)+\mathbb{C}P_0 ( 1 + A ) − 1 ∈ K ( H ) + C P 0 (57) follow from (50) . The spectral projections of A A A N N N P 0 P_0 P 0 1 1 1 0 0 0 ε 0 \varepsilon_0 ε 0
Remark 7.2 The spatial twist is indispensable. Except on its diagonal, B ( H ) ⊗ 1 B(H)\otimes1 B ( H ) ⊗ 1 K ( H ) + C P 0 \mathcal{K}(H)+\mathbb{C}P_0 K ( H ) + C P 0
A closed quantum-plane Casimir
We now suppress Θ \Theta Θ B ( H ) B(H) B ( H ) ψ ( x ) = ( 1 − τ ) Tr ( A x ) \psi(x)=(1-\tau)\operatorname{Tr}(Ax) ψ ( x ) = ( 1 − τ ) Tr ( A x ) A = diag ( τ k ) A=\operatorname{diag}(\tau^k) A = diag ( τ k )
∥ x ∥ 2 , ψ 2 = ( 1 − τ ) ∑ i , j ∈ Z τ j ∣ x i j ∣ 2 . (58) \lVert x\rVert_{2,\psi}^{2}=(1-\tau)\sum_{i,j\in\mathbb{Z}}\tau^j\lvert x_{ij}\rvert^2. \tag*{(58)} ∥ x ∥ 2 , ψ 2 = ( 1 − τ ) i , j ∈ Z ∑ τ j ∣ x ij ∣ 2 . (58) The angular sector H ℓ H_\ell H ℓ x k + ℓ , k = f ( k ) x_{k+\ell,k}=f(k) x k + ℓ , k = f ( k ) H w H_w H w D : = ( Z ∗ ) − 1 = V A − 1 / 2 \mathrm{D}\mathrel{\mathop:}=(Z^*)^{-1}=VA^{-1/2} D : = ( Z ∗ ) − 1 = V A − 1/2
C f i n ( x ) = A − 1 x + x A − 1 − D ∗ x D − τ D x D ∗ . (59) \mathcal{C}_{\mathrm{fin}}(x)=A^{-1}x+xA^{-1}-\mathrm{D}^*x\mathrm{D}-\tau\mathrm{D}x\mathrm{D}^*. \tag*{(59)} C fin ( x ) = A − 1 x + x A − 1 − D ∗ x D − τ D x D ∗ . (59) Every term is a finite matrix. In particular,
C f i n ( e i j ) = ( τ − i + τ − j ) e i j − r − i − j − 2 e i + 1 , j + 1 − r 2 − i − j e i − 1 , j − 1 . (60) \mathcal{C}_{\mathrm{fin}}(e_{ij})=(\tau^{-i}+\tau^{-j})e_{ij}-r^{-i-j-2}e_{i+1,j+1}-r^{2-i-j}e_{i-1,j-1}. \tag*{(60)} C fin ( e ij ) = ( τ − i + τ − j ) e ij − r − i − j − 2 e i + 1 , j + 1 − r 2 − i − j e i − 1 , j − 1 . (60) Theorem 7.3 The positive quadratic form of (59) is closable. Its closure is the direct sum over ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z
E ℓ ( f ) = ( 1 − τ ) ( r − ℓ ∑ k ∣ f ( k + 1 ) − f ( k ) ∣ 2 + ( 1 − r − ℓ ) 2 ∑ k ∣ f ( k ) ∣ 2 ) . (61) \mathcal{E}_{\ell}(f)=(1-\tau)\left(r^{-\ell}\sum_k\lvert f(k+1)-f(k)\rvert^2+(1-r^{-\ell})^2\sum_k\lvert f(k)\rvert^2\right). \tag*{(61)} E ℓ ( f ) = ( 1 − τ ) ( r − ℓ k ∑ ∣ f ( k + 1 ) − f ( k ) ∣ 2 + ( 1 − r − ℓ ) 2 k ∑ ∣ f ( k ) ∣ 2 ) . (61) For ℓ = 0 \ell=0 ℓ = 0 ℓ ≠ 0 \ell\ne0 ℓ = 0 H w ∩ ℓ 2 ( Z ) H_w\cap\ell^2(\mathbb{Z}) H w ∩ ℓ 2 ( Z ) ℓ = 0 \ell=0 ℓ = 0 (52) . Its associated positive self-adjoint operator C \mathcal{C} C
( C ℓ f ) ( k ) = τ − k ( ( 1 + r − 2 ℓ ) f ( k ) − r − ℓ ( f ( k + 1 ) + f ( k − 1 ) ) ) . (62) (\mathcal{C}_{\ell}f)(k)=\tau^{-k}\left((1+r^{-2\ell})f(k)-r^{-\ell}(f(k+1)+f(k-1))\right). \tag*{(62)} ( C ℓ f ) ( k ) = τ − k ( ( 1 + r − 2 ℓ ) f ( k ) − r − ℓ ( f ( k + 1 ) + f ( k − 1 )) ) . (62) In particular C 0 = L τ ∞ \mathcal{C}_0=L_\tau^\infty C 0 = L τ ∞
C a l g = ( 1 − τ ) 2 λ ( P + P − ) . (63) \mathcal{C}_{\mathrm{alg}}=(1-\tau)^2\lambda(P_+P_-). \tag*{(63)} C alg = ( 1 − τ ) 2 λ ( P + P − ) . (63) Here (63) means equality of algebraic actions on ordered coordinate monomials, not the assertion that those unbounded monomials belong to the GNS space.
Proof : For a finite matrix in sector ℓ \ell ℓ (59) gives (62) . Multiplying by w k f ( k ) w_kf(k) w k f ( k ) (61) . Each form is closed by Fatou’s lemma and coordinatewise convergence. When ℓ ≠ 0 \ell\ne0 ℓ = 0 ∥ f ∥ H w + ∥ f ∥ ℓ 2 \lVert f\rVert_{H_w}+\lVert f\rVert_{\ell^2} ∥ f ∥ H w + ∥ f ∥ ℓ 2
For ℓ = 0 \ell=0 ℓ = 0 (52) . The core from Proposition 3.1 is a form core. Each member of that core, constant c c c c c c 0 0 0 J J J ( 1 − τ ) ∣ c ∣ 2 / J (1-\tau)\lvert c\rvert^2/J ( 1 − τ ) ∣ c ∣ 2 / J H w H_w H w + ∞ +\infty + ∞ ℓ = 0 \ell=0 ℓ = 0
The algebraic coordinate relations are Z Z ∗ = τ Z ∗ Z ZZ^*=\tau Z^*Z Z Z ∗ = τ Z ∗ Z (57) . They match the quantum-plane relations of [BCGST96, Proposition 3.7] under q = r q=r q = r z t h e r e = Z z_{\mathrm{there}}=Z z there = Z v t h e r e = v 2 v_{\mathrm{there}} = v^2 v there = v 2
λ ( P + P − ) ( Z ∗ j Z m ) = − r − j − m [ j ] r [ m ] r Z ∗ ( j − 1 ) Z m − 1 . (64) \lambda(P_{+}P_{-})(Z^{*j}Z^m)=-r^{-j-m}[j]_r[m]_rZ^{*(j-1)}Z^{m-1}. \tag*{(64)} λ ( P + P − ) ( Z ∗ j Z m ) = − r − j − m [ j ] r [ m ] r Z ∗ ( j − 1 ) Z m − 1 . (64) Direct multiplication on the common finite-vector domain, using D = ( Z ∗ ) − 1 D=(Z^*)^{-1} D = ( Z ∗ ) − 1
C a l g ( Z ∗ j Z m ) = − τ 1 − j − m ( 1 − τ j ) ( 1 − τ m ) Z ∗ ( j − 1 ) Z m − 1 , (65) \mathcal{C}_{\mathrm{alg}}(Z^{*j}Z^m)=-\tau^{1-j-m}(1-\tau^j)(1-\tau^m)Z^{*(j-1)}Z^{m-1}, \tag*{(65)} C alg ( Z ∗ j Z m ) = − τ 1 − j − m ( 1 − τ j ) ( 1 − τ m ) Z ∗ ( j − 1 ) Z m − 1 , (65) with value zero if j m = 0 jm=0 j m = 0 [ j ] r = r 1 − j ( 1 − τ j ) / ( 1 − τ ) [j]_r=r^{1-j}(1-\tau^j)/(1-\tau) [ j ] r = r 1 − j ( 1 − τ j ) / ( 1 − τ ) (63) . The positive realization has already been defined by the closed form, so no self-adjointness of the coordinate polynomial representation is being assumed.
Remark 7.4 There is also a coordinate-commutator formula. For any finite matrix x x x
E ( x ) = 1 2 ψ ( [ D , x ] ∗ [ D , x ] + τ [ D ∗ , x ∗ ] [ D ∗ , x ] ) . (66) \mathcal{E}(x)=\frac{1}{2}\psi\left([D,x]^*[D,x]+\tau[D^*,x^*][D^*,x]\right). \tag*{(66)} E ( x ) = 2 1 ψ ( [ D , x ] ∗ [ D , x ] + τ [ D ∗ , x ∗ ] [ D ∗ , x ] ) . (66) To verify it, put G ≔ − C f i n G\coloneqq-\mathcal{C}_{\mathrm{fin}} G : = − C fin G ( x ∗ x ) − G ( x ) ∗ x − x ∗ G ( x ) = [ D , x ] ∗ [ D , x ] + τ [ D ∗ , x ∗ ] [ D ∗ , x ] G(x^*x)-G(x)^*x-x^*G(x)=[D,x]^*[D,x]+\tau[D^*,x^*][D^*,x] G ( x ∗ x ) − G ( x ) ∗ x − x ∗ G ( x ) = [ D , x ] ∗ [ D , x ] + τ [ D ∗ , x ∗ ] [ D ∗ , x ] ψ ( G ( y ) ) = 0 \psi(G(y))=0 ψ ( G ( y )) = 0 y y y (60) . Symmetry then gives (66) . The full form is its closure; this does not assert that the two commutators are everywhere defined on bounded operators.
Remark 7.5 At ℓ = 0 \ell=0 ℓ = 0 + ∞ +\infty + ∞ form core , not an operator core. The operator core includes tails constant near + ∞ +\infty + ∞
( 1 − τ ) Δ f ( k ) = ∑ j > k w j ( L f ) ( j ) , Δ f ( k ) → 0 ( k → + ∞ ) . (67) (1-\tau)\Delta f(k)=\sum_{j>k}w_j(Lf)(j),\qquad\Delta f(k)\to0\quad(k\to+\infty). \tag*{(67)} ( 1 − τ ) Δ f ( k ) = j > k ∑ w j ( L f ) ( j ) , Δ f ( k ) → 0 ( k → + ∞ ) . (67) This is precisely the realization of Proposition 3.1 . A formal Casimir calculation alone would not determine this extension.
Spectrum and a norm-density statement
For p , ℓ ∈ Z p,\ell\in\mathbb{Z} p , ℓ ∈ Z
f p , ℓ ( k ) = r ∣ ℓ ∣ k 1 ϕ 1 ( 0 ; τ ∣ ℓ ∣ + 1 ; τ , τ p + max ( ℓ , 0 ) + k + 1 ) . (68) f_{p,\ell}(k)=r^{|\ell|k}\,{}_1\phi_1(0;\tau^{|\ell|+1};\tau,\tau^{p+\max(\ell,0)+k+1}). \tag*{(68)} f p , ℓ ( k ) = r ∣ ℓ ∣ k 1 ϕ 1 ( 0 ; τ ∣ ℓ ∣ + 1 ; τ , τ p + m a x ( ℓ , 0 ) + k + 1 ) . (68) Proposition 7.6 For each ℓ \ell ℓ f p , ℓ f_{p,\ell} f p , ℓ C ℓ \mathcal{C}_{\ell} C ℓ τ p \tau^p τ p p ∈ Z p\in\mathbb{Z} p ∈ Z C \mathcal{C} C 0 0 0
Proof. Let s ≥ 1 s\geq1 s ≥ 1 θ ≔ τ s + 1 \theta\coloneqq\tau^{s+1} θ : = τ s + 1 h ( k ) ↦ r s k h ( k ) h(k)\mapsto r^{sk}h(k) h ( k ) ↦ r s k h ( k ) ℓ 2 ( Z , ( 1 − θ ) θ k ) \ell^2(\mathbb{Z},(1-\theta)\theta^k) ℓ 2 ( Z , ( 1 − θ ) θ k ) H w H_w H w C s \mathcal{C}_s C s τ − s L θ , τ ∞ \tau^{-s}L^\infty_{\theta,\tau} τ − s L θ , τ ∞ § 3 . The forms agree first on finite supports and then by closure. Indeed the tail conductances of L θ , τ L_{\theta,\tau} L θ , τ τ s k \tau^{sk} τ s k C − s = τ s C s \mathcal{C}_{-s}=\tau^s\mathcal{C}_s C − s = τ s C s Theorem 3.6 gives (68) , completeness and simplicity. The same theorem with both parameters equal to τ \tau τ ℓ = 0 \ell=0 ℓ = 0
Lemma 7.7. Any f ∈ dom L f \in\operatorname{dom} L f ∈ dom L − ∞ -\infty − ∞ + ∞ +\infty + ∞
∥ f ∥ ∞ ≤ ∥ f ∥ H w 1 − τ + r ( 1 − τ ) ( 1 − r ) ∥ L f ∥ H w . (69) \lVert f\rVert_\infty\le\frac{\lVert f\rVert_{H_w}}{\sqrt{1-\tau}}+\frac{r}{(1-\tau)(1-r)}\lVert Lf\rVert_{H_w}. \tag*{(69)} ∥ f ∥ ∞ ≤ 1 − τ ∥ f ∥ H w + ( 1 − τ ) ( 1 − r ) r ∥ L f ∥ H w . (69) Finite spectral sums of L L L
Proof: For k ≤ 0 k \le0 k ≤ 0 ∣ f ( k ) ∣ ≤ ( 1 − τ ) − 1 / 2 τ − k / 2 ∥ f ∥ H w \lvert f(k)\rvert\le(1-\tau)^{-1/2}\tau^{-k/2}\lVert f\rVert_{H_w} ∣ f ( k )∣ ≤ ( 1 − τ ) − 1/2 τ − k /2 ∥ f ∥ H w − ∞ -\infty − ∞ (67) and Cauchy–Schwarz give ∣ Δ f ( k ) ∣ ≤ τ ( k + 1 ) / 2 ∥ L f ∥ H w / ( 1 − τ ) \lvert\Delta f(k)\rvert\le\tau^{(k+1)/2}\lVert Lf\rVert_{H_w}/(1-\tau) ∣ Δ f ( k )∣ ≤ τ ( k + 1 ) /2 ∥ L f ∥ H w / ( 1 − τ ) k ≥ 0 k \ge0 k ≥ 0 (69) by telescoping from f ( 0 ) f(0) f ( 0 )
Corollary 7.8. The eigenoperators obtained from (68) in all angular sectors linearly span a norm-dense subspace of K ( H ) + C P 0 \mathcal{K}(H)+\mathbb{C}P_0 K ( H ) + C P 0
Proof: For ℓ ≠ 0 \ell\ne0 ℓ = 0 ℓ 2 ( Z ) \ell^2(\mathbb{Z}) ℓ 2 ( Z ) (61) , hence in the operator norm of a single matrix band. This approximates every off-diagonal matrix unit. For ℓ = 0 \ell=0 ℓ = 0 P 0 P_0 P 0 − ∞ -\infty − ∞ 1 1 1 + ∞ +\infty + ∞
Actual unitary representations and the spherical transform
Algebraic equivariance is not yet equivariance for the locally compact quantum group. We now use the representations of Woronowicz, in Pal’s conventions [31 Section 3, especially (3.4)] . For p ∈ Z p \in\mathbb{Z} p ∈ Z u ( p ) u^{(p)} u ( p ) ℓ 2 ( Z ) \ell^2(\mathbb{Z}) ℓ 2 ( Z )
u a b ( p ) = v a + b f p − b + 1 , a − b ( n ) , f s , a ( z ) = ∫ T F r ( r s z u ) u − a d m ( u ) , (70) u^{(p)}_{ab}=v^{a+b}f_{p-b+1,a-b}(n),\qquad f_{s,a}(z)=\int_{\mathbb{T}}F_r(r^szu)u^{-a}\,dm(u), \tag*{(70)} u ab ( p ) = v a + b f p − b + 1 , a − b ( n ) , f s , a ( z ) = ∫ T F r ( r s z u ) u − a d m ( u ) , (70) where
F r ( z ) = ∏ h ≥ 0 1 + r h z ˉ 1 + r h z (71) F_r(z)=\prod_{h\geq0}\frac{1+r^h\bar z}{1+r^h z} \tag*{(71)} F r ( z ) = h ≥ 0 ∏ 1 + r h z 1 + r h z ˉ (71) on C r C_r C r A ∩ n φ A\cap n_\varphi A ∩ n φ 2 b + 1 2b+1 2 b + 1 R t ( u a b ( p ) ) = t 2 b u a b ( p ) R_t(u^{(p)}_{ab})=t^{2b}u^{(p)}_{ab} R t ( u ab ( p ) ) = t 2 b u ab ( p ) L t ( u a b ( p ) ) = t 2 a u a b ( p ) L_t(u^{(p)}_{ab})=t^{2a}u^{(p)}_{ab} L t ( u ab ( p ) ) = t 2 a u ab ( p )
Lemma 7.9. Put F a ( s ) ≔ ∫ T F r ( r s u ) u − a d m ( u ) F_a(s)\coloneqq\int_{\mathbb{T}}F_r(r^su)u^{-a}\,dm(u) F a ( s ) : = ∫ T F r ( r s u ) u − a d m ( u ) a , s a,s a , s
F a ( s ) = ( − 1 ) a r s a ( τ ; τ ) a 1 ϕ 1 ( 0 ; τ a + 1 ; τ , τ s ) ( a ≥ 0 ) , (72) F_a(s)=\frac{(-1)^a r^{sa}}{(\tau;\tau)_a}\,{}_1\phi_1(0;\tau^{a+1};\tau,\tau^s)\qquad(a\geq0), \tag*{(72)} F a ( s ) = ( τ ; τ ) a ( − 1 ) a r s a 1 ϕ 1 ( 0 ; τ a + 1 ; τ , τ s ) ( a ≥ 0 ) , (72) and, for b ≥ 1 b\geq1 b ≥ 1
F − b ( s ) = r s b + b ( b − 1 ) ( τ ; τ ) b 1 ϕ 1 ( 0 ; τ b + 1 ; τ , τ s + b ) . (73) F_{-b}(s)=\frac{r^{sb+b(b-1)}}{(\tau;\tau)_b}\,{}_1\phi_1(0;\tau^{b+1};\tau,\tau^{s+b}). \tag*{(73)} F − b ( s ) = ( τ ; τ ) b r s b + b ( b − 1 ) 1 ϕ 1 ( 0 ; τ b + 1 ; τ , τ s + b ) . (73) The formulas hold also when $s\leq 0.
Proof. For s > 0 s>0 s > 0 (71) by Euler’s identity ( x ; τ ) ∞ = ∑ n ≥ 0 ( − 1 ) n τ n ( n − 1 ) / 2 ( τ ; τ ) n x n (x;\tau)_\infty=\sum_{n\geq0}\frac{(-1)^n\tau^{n(n-1)/2}}{(\tau;\tau)_n}x^n ( x ; τ ) ∞ = ∑ n ≥ 0 ( τ ; τ ) n ( − 1 ) n τ n ( n − 1 ) /2 x n [12 (17.5.1)] and its denominator by the q q q 1 ( x ; τ ) ∞ = ∑ n ≥ 0 x n ( τ ; τ ) n \frac{1}{(x;\tau)_\infty}=\sum_{n\geq0}\frac{x^n}{(\tau;\tau)_n} ( x ; τ ) ∞ 1 = ∑ n ≥ 0 ( τ ; τ ) n x n ∣ x ∣ < 1 |x|<1 ∣ x ∣ < 1 [12 (17.5.4)] , see also [15 Section 1.3] and [22 (1.6) and (1.7)] . Both expansions are absolutely convergent on the unit circle. Collecting the coefficient of u a u^a u a (72) and (73) . For completeness, the continuation to integral s ≤ 0 s\leq0 s ≤ 0
Ξ ( w , z ) = ∑ j ≥ 0 ( − 1 ) j τ j ( j − 1 ) / 2 ( w τ j ; τ ) ∞ ( τ ; τ ) j z j = ( w ; τ ) ∞ 1 ϕ 1 ( 0 ; w ; τ , z ) . (74) \Xi(w,z)=\sum_{j\geq0}\frac{(-1)^j\tau^{j(j-1)/2}(w\tau^j;\tau)_\infty}{(\tau;\tau)_j}z^j=(w;\tau)_\infty{}_1\phi_1(0;w;\tau,z). \tag*{(74)} Ξ ( w , z ) = j ≥ 0 ∑ ( τ ; τ ) j ( − 1 ) j τ j ( j − 1 ) /2 ( w τ j ; τ ) ∞ z j = ( w ; τ ) ∞ 1 ϕ 1 ( 0 ; w ; τ , z ) . (74) It is symmetric in w , z w,z w , z [12 (17.5.1)] gives an absolutely locally uniformly convergent double series symmetric in its two indices. This is also [22 Proposition 2.1] . For n ≥ 1 n\geq1 n ≥ 1 n n n w = τ 1 − n w=\tau^{1-n} w = τ 1 − n
Ξ ( τ 1 − n , z ) = ( − 1 ) n τ n ( n − 1 ) / 2 z n Ξ ( τ n + 1 , τ n z ) . (75) \Xi(\tau^{1-n},z)=(-1)^n\tau^{n(n-1)/2}z^n\Xi(\tau^{n+1},\tau^n z). \tag*{(75)} Ξ ( τ 1 − n , z ) = ( − 1 ) n τ n ( n − 1 ) /2 z n Ξ ( τ n + 1 , τ n z ) . (75) Both formulas of the lemma are equivalently F a ( s ) = ( − 1 ) a r s a Ξ ( τ a + 1 , τ s ) / ( τ ; τ ) ∞ F_a(s)=(-1)^a r^{sa}\Xi(\tau^{a+1},\tau^s)/(\tau;\tau)_\infty F a ( s ) = ( − 1 ) a r s a Ξ ( τ a + 1 , τ s ) / ( τ ; τ ) ∞ s ≤ 0 s\leq0 s ≤ 0
F r ( r s u ) = u s − 1 F r ( r 2 − s u ) , F a ( s ) = F a − s + 1 ( 2 − s ) . (76) F_r(r^s u)=u^{s-1}F_r(r^{2-s}u),\qquad F_a(s)=F_{a-s+1}(2-s). \tag*{(76)} F r ( r s u ) = u s − 1 F r ( r 2 − s u ) , F a ( s ) = F a − s + 1 ( 2 − s ) . (76) The product identity holds off the exceptional points and extends there by continuity. To check the same reflection for the expression in Ξ \Xi Ξ n : = 1 − s n\mathrel{\mathop:}=1-s n : = 1 − s (75) . The resulting factor is ( − 1 ) a + n r ( n + 1 ) ( a + n ) (-1)^{a+n}r^{(n+1)(a+n)} ( − 1 ) a + n r ( n + 1 ) ( a + n ) Ξ ( τ a + n + 1 , τ n + 1 ) \Xi(\tau^{a+n+1},\tau^{n+1}) Ξ ( τ a + n + 1 , τ n + 1 ) F a − s + 1 ( 2 − s ) F_{a-s+1}(2-s) F a − s + 1 ( 2 − s ) 2 − s > 0 2-s>0 2 − s > 0 (75) with n = b n=b n = b (73) . Thus all lattice radii are covered.
Theorem 7.10 The right-fixed coefficients u a 0 ( p ) u_{a0}^{(p)} u a 0 ( p ) P \mathcal{P} P L \mathcal{L} L τ p \tau^p τ p ℓ = − a \ell=-a ℓ = − a F a ( p + 1 + k ) F_a(p+1+k) F a ( p + 1 + k )
u 00 ( p ) = F p ( Λ ) , F p ( τ k ) = 1 ϕ 1 ( 0 ; τ ; τ , τ p + k + 1 ) , F p ( 0 ) = 1 , (77) u_{00}^{(p)}=F_p(\Lambda),\qquad F_p(\tau^k)={}_1\phi_1(0;\tau;\tau,\tau^{p+k+1}),\qquad F_p(0)=1, \tag*{(77)} u 00 ( p ) = F p ( Λ ) , F p ( τ k ) = 1 ϕ 1 ( 0 ; τ ; τ , τ p + k + 1 ) , F p ( 0 ) = 1 , (77) where F p F_p F p ∣ F p ∣ ≤ 1 |F_p|\leq1 ∣ F p ∣ ≤ 1 X = { 0 } ∪ τ Z X=\{0\}\cup\tau^{\mathbb{Z}} X = { 0 } ∪ τ Z u a 0 ( p ) u_{a0}^{(p)} u a 0 ( p ) a , p ∈ Z a,p\in\mathbb{Z} a , p ∈ Z P \mathcal{P} P
δ ( u a 0 ( p ) ) = ∑ b ∈ Z u a b ( p ) ⊗ u b 0 ( p ) . (78) \delta(u_{a0}^{(p)})=\sum_{b\in\mathbb{Z}}u_{ab}^{(p)}\otimes u_{b0}^{(p)}. \tag*{(78)} δ ( u a 0 ( p ) ) = b ∈ Z ∑ u ab ( p ) ⊗ u b 0 ( p ) . (78) The GNS spectrum of L \mathcal{L} L
Proof. In (70) , take b = 0 b=0 b = 0 f p + 1 , a ( r k z ) = z a F a ( p + 1 + k ) f_{p+1,a}(r^k z)=z^aF_a(p+1+k) f p + 1 , a ( r k z ) = z a F a ( p + 1 + k ) u a 0 ( p ) u_{a0}^{(p)} u a 0 ( p ) − a -a − a Θ − 1 \Theta^{-1} Θ − 1 a ≥ 0 a\geq0 a ≥ 0 (72) makes it ( − 1 ) a r ( p + 1 ) a ( τ ; τ ) a − 1 f p , − a (-1)^a r^{(p+1)a}(\tau;\tau)_a^{-1}f_{p,-a} ( − 1 ) a r ( p + 1 ) a ( τ ; τ ) a − 1 f p , − a a = − b < 0 a=-b<0 a = − b < 0 (73) makes it r b ( p + b ) ( τ ; τ ) b − 1 f p , b r^{b(p+b)}(\tau;\tau)_b^{-1}f_{p,b} r b ( p + b ) ( τ ; τ ) b − 1 f p , b (77) is the case a = 0 a=0 a = 0 0 0 0 (78) is the corepresentation identity, with the sum understood in the usual weak/strict coefficient sense. As a normalization check, Pal’s orthogonality gives
∥ u a 0 ( p ) ∥ 2 , φ 2 = ( 1 − τ ) r 2 ( a − p ) , ∥ F p ∥ H w 2 = ( 1 − τ ) τ − p . (79) \lVert u^{(p)}_{a0}\rVert_{2,\varphi}^{2}=(1-\tau)r^{2(a-p)},\qquad\lVert F_p\rVert_{H_w}^{2}=(1-\tau)\tau^{-p}. \tag*{(79)} ∥ u a 0 ( p ) ∥ 2 , φ 2 = ( 1 − τ ) r 2 ( a − p ) , ∥ F p ∥ H w 2 = ( 1 − τ ) τ − p . (79) Remark 7.11 The Fourier-coefficient calculation is the step which connects the analytic closure to the integrated quantum-group action. An equality of formal q q q (78) for the analytic eigenoperators.
The radial heat process and its entrance law
We write T t = e − t L T_t=e^{-tL} T t = e − t L X = { 0 } ∪ τ Z X=\{0\}\cup\tau^{\mathbb{Z}} X = { 0 } ∪ τ Z 0 0 0 k → + ∞ k\to+\infty k → + ∞ k → − ∞ k\to-\infty k → − ∞ C 0 ( X ) C_0(X) C 0 ( X ) − ∞ -\infty − ∞ + ∞ +\infty + ∞
Proposition 7.12 The operator L L L ℓ Z ∞ \ell^\infty_{\mathbb{Z}} ℓ Z ∞ C 0 ( X ) C_0(X) C 0 ( X ) μ t \mu_t μ t X X X μ 0 = δ 0 \mu_0=\delta_0 μ 0 = δ 0
μ t ( f ) = ( T t f ) ( 0 ) , ∫ X F p d μ t = e − t τ p , p ∈ Z . (80) \mu_t(f)=(T_tf)(0),\qquad\int_X F_p\,d\mu_t=e^{-t\tau^p},\qquad p\in\mathbb{Z}. \tag*{(80)} μ t ( f ) = ( T t f ) ( 0 ) , ∫ X F p d μ t = e − t τ p , p ∈ Z . (80) For every t > 0 t>0 t > 0 μ t ( { 0 } ) = 0 \mu_t(\{0\})=0 μ t ({ 0 }) = 0 P t ( k , ⋅ ) P_t(k,\mathord\cdot) P t ( k , ⋅ ) ℓ Z 1 \ell^1_{\mathbb{Z}} ℓ Z 1 μ t \mu_t μ t k → + ∞ k\to+\infty k → + ∞
μ τ j t ( k + j ) = μ t ( k ) ( j , k ∈ Z , t > 0 ) . (81) \mu_{\tau^j t}(k+j)=\mu_t(k)\qquad(j,k\in\mathbb{Z},\ t>0). \tag*{(81)} μ τ j t ( k + j ) = μ t ( k ) ( j , k ∈ Z , t > 0 ) . (81) Proof: The form (52) is a Dirichlet form [14 Section 1.4] . The associated minimal continuous-time chain [30 Chapter 2] has rates τ − k \tau^{-k} τ − k Z \mathbb{Z} Z 0 0 0 2 2 2 Theorem 7.3 , so this minimal process realizes exactly L L L + ∞ +\infty + ∞
For a bounded constant-tail core function g g g L g Lg Lg ∥ T t g − g ∥ ∞ ≤ t ∥ L g ∥ ∞ \lVert T_tg-g\rVert_\infty\leq t\lVert Lg\rVert_\infty ∥ T t g − g ∥ ∞ ≤ t ∥ Lg ∥ ∞ t > 0 t>0 t > 0 T t g ∈ dom L T_tg\in\operatorname{dom}L T t g ∈ dom L T t g ∈ C 0 ( X ) T_tg\in C_0(X) T t g ∈ C 0 ( X ) C 0 ( X ) C_0(X) C 0 ( X ) 0 0 0
We check that it has mass one. For the coordinate function a ( k ) = τ k a(k)=\tau^k a ( k ) = τ k − L -L − L ( − L a ) ( k ) = c ∗ = τ + τ − 1 − 2 (-La)(k)=c_*=\tau+\tau^{-1}-2 ( − L a ) ( k ) = c ∗ = τ + τ − 1 − 2 E k [ a ( X t ) ] ≤ τ k + c ∗ t \mathbb{E}_k[a(X_t)]\leq\tau^k+c_*t E k [ a ( X t )] ≤ τ k + c ∗ t k ≥ 0 k\geq0 k ≥ 0 X t < − M X_t<-M X t < − M ( 1 + c ∗ t ) τ M (1+c_*t)\tau^M ( 1 + c ∗ t ) τ M 1 k ≥ − M ∈ C 0 ( X ) 1_{k\geq-M}\in C_0(X) 1 k ≥ − M ∈ C 0 ( X ) 1 − ( 1 + c ∗ t ) τ M 1-(1+c_*t)\tau^M 1 − ( 1 + c ∗ t ) τ M M → ∞ M\to\infty M → ∞
The eigenfunction identity in Theorem 7.10 , first in H w H_w H w C 0 ( X ) C_0(X) C 0 ( X ) (80) . As p → − ∞ p\to-\infty p → − ∞ F p ( τ k ) = F 0 ( τ p + k ) → 0 F_p(\tau^k)=F_0(\tau^{p+k})\to0 F p ( τ k ) = F 0 ( τ p + k ) → 0 k k k F p ( 0 ) = 1 F_p(0)=1 F p ( 0 ) = 1 ∣ F p ∣ ≤ 1 \lvert F_p\rvert\leq1 ∣ F p ∣ ≤ 1 (80) gives μ t ( { 0 } ) = 0 \mu_t(\{0\})=0 μ t ({ 0 }) = 0 t > 0 t>0 t > 0 C 0 ( X ) C_0(X) C 0 ( X ) Z \mathbb{Z} Z ℓ 1 \ell^1 ℓ 1
Finally, the shift Q f ( k ) = f ( k + 1 ) Qf(k)=f(k+1) Q f ( k ) = f ( k + 1 ) Q L Q − 1 = τ − 1 L QLQ^{-1}=\tau^{-1}L Q L Q − 1 = τ − 1 L Q T t = T t / τ Q QT_t=T_{t/\tau}Q Q T t = T t / τ Q (81) .
Remark 7.13 For t > 0 t>0 t > 0
μ t ( k ) = τ k ∑ p ∈ Z τ p e − t τ p 1 ϕ 1 ( 0 ; τ ; τ , τ p + k + 1 ) . (82) \mu_t(k)=\tau^k\sum_{p\in\mathbb{Z}}\tau^p e^{-t\tau^p}{}_1\phi_1(0;\tau;\tau,\tau^{p+k+1}). \tag*{(82)} μ t ( k ) = τ k p ∈ Z ∑ τ p e − t τ p 1 ϕ 1 ( 0 ; τ ; τ , τ p + k + 1 ) . (82) Indeed, expand 1 { k } \mathbf{1}_{\{k\}} 1 { k } (79) . The resulting series for T t 1 { k } T_t\mathbf{1}_{\{k\}} T t 1 { k } ∑ p τ p e − t τ p < ∞ \sum_p\tau^p e^{-t\tau^p}<\infty ∑ p τ p e − t τ p < ∞ ∣ F p ∣ ≤ 1 |F_p|\leq1 ∣ F p ∣ ≤ 1 (82) . Positivity is justified by the process above, not by the signs in this spectral sum.
Covariant completely positive heat dynamics
Let η T = h T π \eta_T=h_T\pi η T = h T π A \mathcal{A} A M M M t > 0 t>0 t > 0 M M M
ω t ( x ) = μ t ( E x ) = ∑ k ∈ Z μ t ( k ) ∫ T x k k ( z ) d m ( z ) , ω 0 = η T . (83) \omega_t(x)=\mu_t(Ex)=\sum_{k\in\mathbb{Z}}\mu_t(k)\int_{\mathbb{T}}x_{kk}(z)\,dm(z),\qquad\omega_0=\eta_T. \tag*{(83)} ω t ( x ) = μ t ( E x ) = k ∈ Z ∑ μ t ( k ) ∫ T x k k ( z ) d m ( z ) , ω 0 = η T . (83) The expression at t = 0 t=0 t = 0 A \mathcal{A} A
Theorem 7.14 The maps
S t = ( Id ⊗ ω t ) Δ ∣ N ( t > 0 ) , S 0 = Id N , (84) S_t=(\operatorname{Id}\otimes\omega_t)\Delta|_N\quad(t>0),\qquad S_0=\operatorname{Id}_N, \tag*{(84)} S t = ( Id ⊗ ω t ) Δ ∣ N ( t > 0 ) , S 0 = Id N , (84) form a normal unital completely positive semigroup on N N N φ ∣ N \varphi|_N φ ∣ N e − t C e^{-t\mathcal{C}} e − t C
δ S t = ( Id ⊗ S t ) δ , S t ( F ( A ) ) = ( T t F ) ( A ) . (85) \delta S_t=(\operatorname{Id}\otimes S_t)\delta,\qquad S_t(F(A))=(T_tF)(A). \tag*{(85)} δ S t = ( Id ⊗ S t ) δ , S t ( F ( A )) = ( T t F ) ( A ) . (85) The semigroup preserves P \mathcal{P} P σ \sigma σ N N N
S t ( u a 0 ( p ) ) = e − t τ p u a 0 ( p ) . (86) S_t(u^{(p)}_{a_0})=e^{-t\tau^p}u^{(p)}_{a_0}. \tag*{(86)} S t ( u a 0 ( p ) ) = e − t τ p u a 0 ( p ) . (86) The states in (83) satisfy ω s ∗ ω t = ω s + t \omega_s*\omega_t=\omega_{s+t} ω s ∗ ω t = ω s + t A \mathcal{A} A η T \eta_T η T
( Id ⊗ ω t ) Δ = S t E R , ( Id ⊗ ω 0 ) Δ = E R , (87) (\operatorname{Id}\otimes\omega_t)\Delta=S_tE_R,\qquad(\operatorname{Id}\otimes\omega_0)\Delta=E_R, \tag*{(87)} ( Id ⊗ ω t ) Δ = S t E R , ( Id ⊗ ω 0 ) Δ = E R , (87) not the identity on A \mathcal{A} A
Proof. Each ω t \omega_t ω t t > 0 t>0 t > 0 R ω t = ( Id ⊗ ω t ) Δ R_{\omega_t}=(\operatorname{Id}\otimes\omega_t)\Delta R ω t = ( Id ⊗ ω t ) Δ N N N R ω t E R = E R R ω t = R ω t R_{\omega_t}E_R=E_RR_{\omega_t}=R_{\omega_t} R ω t E R = E R R ω t = R ω t [35 Theorem 2.1] , with attention to the opposite convention for left/right names. In particular its GNS implementation is a contraction.
Using the torus weights and (80) , ω t ( u b 0 ( p ) ) = δ b 0 e − t τ p \omega_t(u^{(p)}_{b_0})=\delta_{b0}e^{-t\tau^p} ω t ( u b 0 ( p ) ) = δ b 0 e − t τ p (78) by this normal functional. This proves (86) . The complete orthogonal eigenbasis of Theorem 7.10 implies that the GNS implementation is e − t L e^{-t\mathcal{L}} e − t L S t S_t S t P \mathcal{P} P σ \sigma σ N N N
Coassociativity gives (85) for t > 0 t>0 t > 0 (84) , and for t = 0 t=0 t = 0 P \mathcal{P} P S t ( P j ) → 1 S_t(P_j)\to1 S t ( P j ) → 1 j → − ∞ j\to-\infty j → − ∞ ∥ ( 1 − S t ( P j ) ) P 0 ∥ ≤ ( 1 + c ∗ t ) τ − j → 0 \|(1-S_t(P_j))P_0\|\leq(1+c_*t)\tau^{-j}\to0 ∥ ( 1 − S t ( P j )) P 0 ∥ ≤ ( 1 + c ∗ t ) τ − j → 0 [26 Chapter 2] on K ( H ) + C P 0 K(H)+\mathbb{C}P_0 K ( H ) + C P 0 σ s ω ( x ) = A i s x A − i s \sigma^{\omega}_s(x)=A^{is}xA^{-is} σ s ω ( x ) = A i s x A − i s ℓ \ell ℓ τ i s ℓ \tau^{is\ell} τ i s ℓ S t S_t S t
The GNS implementation is self-adjoint. Put D ψ : = ( 1 − τ ) A D_\psi\mathrel{\mathop:}= (1-\tau)A D ψ : = ( 1 − τ ) A x ↦ D ψ 1 / 4 x D ψ 1 / 4 x\mapsto D_\psi^{1/4}xD_\psi^{1/4} x ↦ D ψ 1/4 x D ψ 1/4 x ↦ x D ψ 1 / 2 x\mapsto xD_\psi^{1/2} x ↦ x D ψ 1/2 ℓ \ell ℓ τ ℓ / 4 \tau^{\ell/4} τ ℓ /4 S t S_t S t
For clarity, point-σ \sigma σ N N N P \mathcal{P} P x ∈ B ( H ) x\in B(H) x ∈ B ( H ) Q Q Q S t ( x − Q x Q ) S_t(x-QxQ) S t ( x − Q x Q ) S t ( 1 − Q ) S_t(1-Q) S t ( 1 − Q ) S t ( Q ) → Q S_t(Q)\to Q S t ( Q ) → Q Q x Q ∈ P QxQ\in\mathcal{P} Q x Q ∈ P S t ( Q x Q ) → Q x Q S_t(QxQ)\to QxQ S t ( Q x Q ) → Q x Q σ \sigma σ
On A A A E E E C 0 ( X ) C_0(X) C 0 ( X ) ω t → δ 0 E = η T \omega_t\to\delta_0E=\eta_T ω t → δ 0 E = η T (87) . Its composition law follows from E R ∣ N = Id N E R|_N=\operatorname{Id}_N E R ∣ N = Id N S t S_t S t A A A ω s ∗ ω t = ω s + t \omega_s*\omega_t=\omega_{s+t} ω s ∗ ω t = ω s + t s = 0 s=0 s = 0 η T ( p k ) = 0 \eta_T(p_k)=0 η T ( p k ) = 0 k k k ∑ k p k = 1 \sum_k p_k=1 ∑ k p k = 1 η T \eta_T η T M M M
Proposition 7.15 The semigroup of Theorem 7.14 is the unique covariant completely positive contraction semigroup on P \mathcal{P} P T t T_t T t N N N
Proof. For a covariant map S S S ν : = ϵ 0 S \nu\mathrel{\mathop:}=\epsilon_0S ν : = ϵ 0 S Id ⊗ ϵ 0 \operatorname{Id}\otimes\epsilon_0 Id ⊗ ϵ 0 S = ( Id ⊗ ν ) δ S=(\operatorname{Id}\otimes\nu)\delta S = ( Id ⊗ ν ) δ ϵ 0 \epsilon_0 ϵ 0 ν = ν E L \nu=\nu E_L ν = ν E L ν \nu ν S S S T t T_t T t μ t E L \mu_tE_L μ t E L (84) . These slices can be read first in a nondegenerate multiplier realization, or verified on the coefficient span, which is norm dense by Theorem 7.10. Normal extensions coincide on the σ \sigma σ P \mathcal{P} P
Remark 7.16 The complete positivity in Theorem 7.14 is established by actual convolution states, not by a merely formal Lindblad expression. Conversely, after the theorem has been proved, (59) is an explicit infinitesimal expression of Lindblad type [27 ] with the two unbounded jump operators D D D r D ∗ rD^* r D ∗
No Feller lifting to the whole quantum group
For a probability vector μ \mu μ Z \mathbb{Z} Z
F ( μ ) : = ∑ k ∈ Z μ ( k − 1 ) μ ( k ) . (88) \mathcal{F}(\mu)\mathrel{\mathop:}=\sum_{k\in\mathbb{Z}}\sqrt{\mu(k-1)\mu(k)}. \tag*{(88)} F ( μ ) : = k ∈ Z ∑ μ ( k − 1 ) μ ( k ) . (88) Cauchy–Schwarz gives F ( μ ) ≤ 1 \mathcal{F}(\mu) \le1 F ( μ ) ≤ 1 μ \sqrt{\mu} μ ℓ 2 ( Z ) \ell^2(\mathbb{Z}) ℓ 2 ( Z )
Theorem 7.17. Let t > 0 t > 0 t > 0 R : A → A R: \mathcal{A} \to\mathcal{A} R : A → A { F ( A ) : F ∈ C 0 ( X ) } \{F(A): F \in C_0(X)\} { F ( A ) : F ∈ C 0 ( X )} T t T_t T t
∥ R ( v P 0 ) − v P 0 ∥ ≥ 1 − F ( μ t ) > 0. (89) \lVert R(vP_0)-vP_0\rVert\ge1-\mathcal{F}(\mu_t)>0. \tag*{(89)} ∥ R ( v P 0 ) − v P 0 ∥ ≥ 1 − F ( μ t ) > 0. (89) Consequently, for any family ( R t ) t > 0 (R_t)_{t>0} ( R t ) t > 0
∥ R τ j ( v P 0 ) − v P 0 ∥ ≥ δ τ ≔ 1 − F ( μ 1 ) > 0 ( j ≥ 0 ) . (90) \lVert R_{\tau_j}(vP_0)-vP_0\rVert\ge\delta_\tau\coloneqq1-\mathcal{F}(\mu_1)>0 \qquad(j \ge0). \tag*{(90)} ∥ R τ j ( v P 0 ) − v P 0 ∥ ≥ δ τ : = 1 − F ( μ 1 ) > 0 ( j ≥ 0 ) . (90) There is no such family point-norm converging to the identity on A \mathcal{A} A
Proof. Let R ( x ) = V 1 ∗ Π ( x ) V 1 R(x)=V_1^*\Pi(x)V_1 R ( x ) = V 1 ∗ Π ( x ) V 1 [32 Theorem 4.1] with ∥ V 1 ∥ ≤ 1 \lVert V_1\rVert\le1 ∥ V 1 ∥ ≤ 1 ξ i ≔ V 1 ( e i ⊗ 1 ) \xi_i \coloneqq V_1(e_i\otimes1) ξ i : = V 1 ( e i ⊗ 1 ) p k ≔ e k k ⊗ 1 p_k \coloneqq e_{kk}\otimes1 p k : = e k k ⊗ 1
∥ Π ( p k ) ξ i ∥ 2 = P t ( i , k ) , ∑ k P t ( i , k ) = 1. (91) \lVert\Pi(p_k)\xi_i\rVert^2=P_t(i,k), \qquad\sum_k P_t(i,k)=1. \tag*{(91)} ∥ Π ( p k ) ξ i ∥ 2 = P t ( i , k ) , k ∑ P t ( i , k ) = 1. (91) Therefore ∥ ξ i ∥ = 1 \lVert\xi_i\rVert=1 ∥ ξ i ∥ = 1 ξ i = ∑ k Π ( p k ) ξ i \xi_i=\sum_k\Pi(p_k)\xi_i ξ i = ∑ k Π ( p k ) ξ i Π \Pi Π p k p_k p k v P 0 p k = p k − 1 v P 0 p k vP_0p_k=p_{k-1}vP_0p_k v P 0 p k = p k − 1 v P 0 p k ∥ v P 0 ∥ = 1 \lVert vP_0\rVert=1 ∥ v P 0 ∥ = 1
∣ ⟨ R ( v P 0 ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ ∣ ≤ ∑ k ≥ 0 P t ( i , k − 1 ) P t ( j , k ) ≤ ∑ k ∈ Z P t ( i , k − 1 ) P t ( j , k ) . (92) \left|\langle R(vP_0)(e_j\otimes1),e_i\otimes1\rangle\right|
\le\sum_{k\ge0}\sqrt{P_t(i,k-1)P_t(j,k)}
\le\sum_{k\in\mathbb{Z}}\sqrt{P_t(i,k-1)P_t(j,k)}. \tag*{(92)} ∣ ⟨ R ( v P 0 ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ ∣ ≤ k ≥ 0 ∑ P t ( i , k − 1 ) P t ( j , k ) ≤ k ∈ Z ∑ P t ( i , k − 1 ) P t ( j , k ) . (92) The sums follow by approximating the two Stinespring vectors by finite sums of their orthogonal components, not by assuming v P 0 vP_0 v P 0 i = j − 1 i=j-1 i = j − 1 j → + ∞ j\to+\infty j → + ∞ Proposition 7.12 the two probability vectors converge in ℓ 1 \ell^1 ℓ 1 μ t \mu_t μ t ∥ α − β ∥ 2 2 ≤ ∥ α − β ∥ 1 \lVert\sqrt{\alpha}-\sqrt{\beta}\rVert_2^2\le\lVert\alpha-\beta\rVert_1 ∥ α − β ∥ 2 2 ≤ ∥ α − β ∥ 1 F ( μ t ) \mathcal{F}(\mu_t) F ( μ t ) v P 0 vP_0 v P 0 (89) . Scaling (81) leaves the affinity unchanged and proves (90) .
Ordinary weakly continuous convolution semigroups starting at the counit yield point-norm continuous convolution semigroups on C 0 C_0 C 0 [35 Theorem 3.2] . Applying the preceding obstruction excludes them. This does not exclude the idempotent-started family in Theorem 7.14 , whose value at zero on A \mathcal{A} A E R E_R E R
A sharp covariance threshold for Taibleson energies
This section concerns the precise class relevant to the manuscript: normal unital, Haar-weight-preserving covariant semigroups on N N N P P P
Lemma 7.18. There is a constant C τ ≥ 1 C_\tau\ge1 C τ ≥ 1
0 ≤ 1 − F p ( x ) ≤ C τ max ( 1 , τ p ) ( 1 − F 0 ( x ) ) ( x ∈ X , p ∈ Z ) . (93) 0\le1-F_p(x)\le C_\tau\max(1,\tau^p)(1-F_0(x)) \qquad(x\in X,p\in\mathbb{Z}). \tag*{(93)} 0 ≤ 1 − F p ( x ) ≤ C τ max ( 1 , τ p ) ( 1 − F 0 ( x )) ( x ∈ X , p ∈ Z ) . (93) Proof: The functions are real and ∣ F p ∣ ≤ 1 |F_p| \le1 ∣ F p ∣ ≤ 1 F 0 ( τ k ) < 1 F_0(\tau^k) < 1 F 0 ( τ k ) < 1 k k k u ↦ F r ( r k + 1 u ) u \mapsto F_r(r^{k+1}u) u ↦ F r ( r k + 1 u ) s s s ( − r s u − 1 ; τ ) ∞ / ( − r s u ; τ ) ∞ (-r^s u^{-1};\tau)_\infty/(-r^s u;\tau)_\infty ( − r s u − 1 ; τ ) ∞ / ( − r s u ; τ ) ∞ s s s (76) reduces nonconstancy to this fact. An average of a nonconstant continuous unit-modulus function cannot have modulus one.
At + ∞ +\infty + ∞ 1 − F 0 ( τ k ) = τ k + 1 ( 1 − τ ) 2 + O ( τ 2 k ) 1-F_0(\tau^k)=\frac{\tau^{k+1}}{(1-\tau)^2}+O(\tau^{2k}) 1 − F 0 ( τ k ) = ( 1 − τ ) 2 τ k + 1 + O ( τ 2 k ) − ∞ -\infty − ∞ F 0 ( τ k ) → 0 F_0(\tau^k)\to0 F 0 ( τ k ) → 0 c , C > 0 c,C>0 c , C > 0 c min ( 1 , τ k ) ≤ 1 − F 0 ( τ k ) ≤ C min ( 1 , τ k ) c\min(1,\tau^k)\le1-F_0(\tau^k)\le C\min(1,\tau^k) c min ( 1 , τ k ) ≤ 1 − F 0 ( τ k ) ≤ C min ( 1 , τ k ) k k k F p ( τ k ) = F 0 ( τ k + p ) F_p(\tau^k)=F_0(\tau^{k+p}) F p ( τ k ) = F 0 ( τ k + p ) min ( 1 , τ k + p ) ≤ max ( 1 , τ p ) min ( 1 , τ k ) \min(1,\tau^{k+p})\le\max(1,\tau^p)\min(1,\tau^k) min ( 1 , τ k + p ) ≤ max ( 1 , τ p ) min ( 1 , τ k )
Proposition 7.19 For every semigroup in the class specified above there are finite numbers Ψ ( p ) ≥ 0 \Psi(p)\ge0 Ψ ( p ) ≥ 0 p ∈ Z p\in\mathbb{Z} p ∈ Z
S t ( u a 0 ( p ) ) = e − t Ψ ( p ) u a 0 ( p ) , Ψ ( p ) ≤ C τ max ( 1 , τ p ) Ψ ( 0 ) . (94) S_t(u_{a_0}^{(p)})=e^{-t\Psi(p)}u_{a_0}^{(p)},\qquad\Psi(p)\le C_\tau\max(1,\tau^p)\Psi(0). \tag*{(94)} S t ( u a 0 ( p ) ) = e − t Ψ ( p ) u a 0 ( p ) , Ψ ( p ) ≤ C τ max ( 1 , τ p ) Ψ ( 0 ) . (94) Proof: Put ν t = ε 0 S t \nu_t=\varepsilon_0S_t ν t = ε 0 S t Proposition 7.15 , covariance implies S t = ( I d ⊗ ν t ) δ S_t=({\rm Id}\otimes\nu_t)\delta S t = ( Id ⊗ ν t ) δ ν t = ν t E L \nu_t=\nu_tE_L ν t = ν t E L S t ( u a 0 ( p ) ) = ν t ( F p ( A ) ) u a 0 ( p ) S_t(u_{a_0}^{(p)})=\nu_t(F_p(A))u_{a_0}^{(p)} S t ( u a 0 ( p ) ) = ν t ( F p ( A )) u a 0 ( p ) t t t e − t Ψ ( p ) e^{-t\Psi(p)} e − t Ψ ( p ) Ψ ( p ) ≥ 0 \Psi(p)\ge0 Ψ ( p ) ≥ 0 ν t \nu_t ν t m t m_t m t X X X (93) and adding the missing mass (the coefficient on the right is at least one) gives
1 − e − t Ψ ( p ) ≤ C τ max ( 1 , τ p ) ( 1 − e − t Ψ ( 0 ) ) . 1-e^{-t\Psi(p)}\le C_\tau\max(1,\tau^p)(1-e^{-t\Psi(0)}). 1 − e − t Ψ ( p ) ≤ C τ max ( 1 , τ p ) ( 1 − e − t Ψ ( 0 ) ) . Divide by t t t t → 0 + t\to0^+ t → 0 +
Theorem 7.20 Let K K K Q Q Q d ≥ 1 d\ge1 d ≥ 1 τ = Q − d \tau=Q^{-d} τ = Q − d D r a d α D^\alpha_{\mathrm{rad}} D rad α 0 < α ≤ d 0<\alpha\le d 0 < α ≤ d 0 < α ≤ d 0<\alpha\le d 0 < α ≤ d Theorem 7.14 with generator L α / d \mathcal{L}^{\alpha/d} L α / d s = α / d s=\alpha/d s = α / d
( 1 + τ ) − 2 s E L s ( f ) ≤ E D α ( f ) ≤ ( 1 − τ ) − 2 s E L s ( f ) . (95) (1+\tau)^{-2s}\mathcal{E}_{L^s}(f)\le\mathcal{E}_{D^\alpha}(f)\le(1-\tau)^{-2s}\mathcal{E}_{L^s}(f). \tag*{(95)} ( 1 + τ ) − 2 s E L s ( f ) ≤ E D α ( f ) ≤ ( 1 − τ ) − 2 s E L s ( f ) . (95) For s > 1 s>1 s > 1 L s L^s L s C 0 ( X ) C_0(X) C 0 ( X )
Proof: For s = α d ≤ 1 s=\frac{\alpha}{d}\le1 s = d α ≤ 1 λ ↦ λ s \lambda\mapsto\lambda^s λ ↦ λ s [34 Chapters 5 and 12] preserves normal complete positivity, Haar invariance, covariance, the two symmetries and point-norm continuity on P \mathcal{P} P L s L^s L s Theorem 4.6 , with ϑ = d \vartheta=d ϑ = d α ≤ d < 3 d \alpha\le d<3d α ≤ d < 3 d Theorem 4.2 by the operator-monotonicity of x ↦ x s x\mapsto x^s x ↦ x s 0 < s ≤ 1 0<s\le1 0 < s ≤ 1 D r a d α = ( D r a d d ) s D^\alpha_{\mathrm{rad}}=(D^d_{\mathrm{rad}})^s D rad α = ( D rad d ) s x ↦ x s x\mapsto x^s x ↦ x s [7 Theorem V.1.9] . Its proof through the representation λ s = sin π s π ∫ 0 ∞ t s − 1 λ ( λ + t ) − 1 d t \lambda^s=\frac{\sin\pi s}{\pi}\int_0^\infty t^{s-1}\lambda(\lambda+t)^{-1}\,dt λ s = π s i n π s ∫ 0 ∞ t s − 1 λ ( λ + t ) − 1 d t [18 Theorem VI.2.1] . Conversely, the finite exponents in (94) imply that every F p F_p F p α ≥ 3 d \alpha\ge3d α ≥ 3 d Theorem 4.6 (3). Suppose d < α < 3 d d<\alpha<3d d < α < 3 d s ≔ α d > 1 s \coloneqq\frac{\alpha}{d}>1 s : = d α > 1 F 0 F_0 F 0 F p ( k ) F_p(k) F p ( k ) F p ( τ k ) F_p(\tau^k) F p ( τ k ) D r a d α D^\alpha_{\mathrm{rad}} D rad α F p ( k ) = F 0 ( k + p ) F_p(k)=F_0(k+p) F p ( k ) = F 0 ( k + p )
E D α ( F p ) ∥ F p ∥ H w 2 = τ p s E D α ( F 0 ) ∥ F 0 ∥ H w 2 . \frac{\mathcal{E}_{D^\alpha}(F_p)}{\lVert F_p\rVert_{H_w}^2}=\tau^{ps}\frac{\mathcal{E}_{D^\alpha}(F_0)}{\lVert F_0\rVert_{H_w}^2}. ∥ F p ∥ H w 2 E D α ( F p ) = τ p s ∥ F 0 ∥ H w 2 E D α ( F 0 ) . Equivalence of forms forces Ψ ( p ) ≥ c τ p s \Psi(p)\ge c\tau^{ps} Ψ ( p ) ≥ c τ p s p p p Ψ ( p ) ≤ C τ p \Psi(p)\le C\tau^p Ψ ( p ) ≤ C τ p p ≤ 0 p\le0 p ≤ 0 p → − ∞ p\to-\infty p → − ∞ s > 1 s>1 s > 1
For the last assertion, a conservative Feller Markov realization of e − t L s e^{-tL_s} e − t L s e − t τ p s e^{-t\tau^{ps}} e − t τ p s F p F_p F p τ p s ≤ C τ max ( 1 , τ p ) \tau^{ps}\le C_\tau\max(1,\tau^p) τ p s ≤ C τ max ( 1 , τ p )
Contraction of quantum Casimir dynamics
Framework and scope
Fix 0 < r < 1 0<r<1 0 < r < 1 τ ≔ r 2 \tau\coloneqq r^2 τ : = r 2 Theorem 7.1 :
H + = ℓ N 0 2 , H = ℓ Z 2 , P c = K ( H + ) + C 1 , P e = K ( H ) + C P 0 , (96) H_+=\ell^2_{\mathbb{N}_0},\quad H=\ell^2_{\mathbb{Z}},\quad P_c=\mathcal{K}(H_+)+\mathbb{C}1,\quad P_e=\mathcal{K}(H)+\mathbb{C}P_0, \tag*{(96)} H + = ℓ N 0 2 , H = ℓ Z 2 , P c = K ( H + ) + C 1 , P e = K ( H ) + C P 0 , (96) where P a = ∑ k ≥ a e k k P_a=\sum_{k\ge a}e_{kk} P a = ∑ k ≥ a e k k P 0 P_0 P 0 B ( H + ) B(H_+) B ( H + ) B ( H ) B(H) B ( H )
h c ( x ) = ( 1 − τ ) ∑ k ≥ 0 τ k x k k , ψ ( x ) = ( 1 − τ ) ∑ k ∈ Z τ k x k k , w k = ( 1 − τ ) τ k . (97) h_c(x)=(1-\tau)\sum_{k\ge0}\tau^k x_{kk},\quad\psi(x)=(1-\tau)\sum_{k\in\mathbb{Z}}\tau^k x_{kk},\quad w_k=(1-\tau)\tau^k. \tag*{(97)} h c ( x ) = ( 1 − τ ) k ≥ 0 ∑ τ k x k k , ψ ( x ) = ( 1 − τ ) k ∈ Z ∑ τ k x k k , w k = ( 1 − τ ) τ k . (97) Expressions involving the weight are interpreted in the extended positive sense. Write H c = L 2 ( B ( H + ) , h c ) \mathcal{H}_c=L^2(B(H_+),h_c) H c = L 2 ( B ( H + ) , h c ) H e = L 2 ( B ( H ) , ψ ) \mathcal{H}_e=L^2(B(H),\psi) H e = L 2 ( B ( H ) , ψ ) ∥ x ∥ 2 , ψ 2 = ∑ i , k w k ∣ x i k ∣ 2 \lVert x\rVert_{2,\psi}^2=\sum_{i,k}w_k|x_{ik}|^2 ∥ x ∥ 2 , ψ 2 = ∑ i , k w k ∣ x ik ∣ 2
The compact Casimir C c \mathcal{C}_c C c
C c u j 0 ( l ) = Λ l u j 0 ( l ) , Λ l = τ − l ( 1 − τ l ) ( 1 − τ l + 1 ) , l ≥ ∣ j ∣ , l , j ∈ Z . (98) \mathcal{C}_c u^{(l)}_{j0}=\Lambda_l u^{(l)}_{j0},\quad\Lambda_l=\tau^{-l}(1-\tau^l)(1-\tau^{l+1}),\quad l\ge|j|,\quad l,j\in\mathbb{Z}. \tag*{(98)} C c u j 0 ( l ) = Λ l u j 0 ( l ) , Λ l = τ − l ( 1 − τ l ) ( 1 − τ l + 1 ) , l ≥ ∣ j ∣ , l , j ∈ Z . (98) The compact quantum Markov semigroup with this GNS implementation is denoted by S t c S^c_t S t c ℓ \ell ℓ matrix-band index , rather than an a priori representation label.
On the plane, a band x k + ℓ , k = f ( k ) x_{k+\ell,k}=f(k) x k + ℓ , k = f ( k ) H w = ℓ 2 ( Z , w ) H_w=\ell^2(\mathbb{Z},w) H w = ℓ 2 ( Z , w ) C e \mathcal{C}_e C e Theorem 7.3 , whose band forms and expressions are
E e , ℓ ( f ) = ( 1 − τ ) ( r − ℓ ∑ k ∣ f ( k + 1 ) − f ( k ) ∣ 2 + ( 1 − r − ℓ ) 2 ∑ k ∣ f ( k ) ∣ 2 ) , (99) \mathcal{E}_{e,\ell}(f)=(1-\tau)\left(r^{-\ell}\sum_k|f(k+1)-f(k)|^2+(1-r^{-\ell})^2\sum_k|f(k)|^2\right), \tag*{(99)} E e , ℓ ( f ) = ( 1 − τ ) ( r − ℓ k ∑ ∣ f ( k + 1 ) − f ( k ) ∣ 2 + ( 1 − r − ℓ ) 2 k ∑ ∣ f ( k ) ∣ 2 ) , (99) ( C e , ℓ f ) ( k ) = τ − k ( ( 1 + τ − k ) f ( k ) − r − ℓ f ( k + 1 ) − r − ℓ f ( k − 1 ) ) . (100) (\mathcal{C}_{e,\ell}f)(k)=\tau^{-k}\left((1+\tau^{-k})f(k)-r^{-\ell}f(k+1)-r^{-\ell}f(k-1)\right). \tag*{(100)} ( C e , ℓ f ) ( k ) = τ − k ( ( 1 + τ − k ) f ( k ) − r − ℓ f ( k + 1 ) − r − ℓ f ( k − 1 ) ) . (100) At ℓ = 0 \ell= 0 ℓ = 0 (99) is omitted from both the value and the domain condition. At ℓ ≠ 0 \ell\ne0 ℓ = 0 H w ∩ ℓ 2 ( Z ) \mathcal{H}_w \cap\ell^2(\mathbb{Z}) H w ∩ ℓ 2 ( Z ) + ∞ +\infty + ∞ Proposition 3.1 . In particular, C e , 0 = L ∞ \mathcal{C}_{e,0} = L^\infty C e , 0 = L ∞
We also use Proposition 7.6 , Proposition 7.12 and Theorem 7.14 . The band eigenfunctions are
f p , ℓ ( k ) = r ∣ ℓ ∣ k 1 ϕ 1 ( 0 ; τ ∣ ℓ ∣ + 1 ; τ , τ p + max ( ℓ , 0 ) + k + 1 ) , p ∈ Z , (101) f_{p,\ell}(k) = r^{|\ell|k}{}_1\phi_1(0;\tau^{|\ell|+1};\tau,\tau^{p+\max(\ell,0)+k+1}), \qquad p \in\mathbb{Z}, \tag*{(101)} f p , ℓ ( k ) = r ∣ ℓ ∣ k 1 ϕ 1 ( 0 ; τ ∣ ℓ ∣ + 1 ; τ , τ p + m a x ( ℓ , 0 ) + k + 1 ) , p ∈ Z , (101) with simple eigenvalues τ p \tau^p τ p X p , ℓ X_{p,\ell} X p , ℓ P e \mathcal{P}_e P e S t e S^e_t S t e e − t C e e^{-t\mathcal{C}_e} e − t C e P e \mathcal{P}_e P e [31 , 22 ] .
The proofs in § 9.2 –8.6 establish the analytic and probabilistic contraction without invoking convergence of quantum-group coproducts. § 9.7 then adds compatibility with those coproducts, using the published C ∗ C^* C ∗
Expanding corners and the invariant weights
For N ≥ 0 N \ge0 N ≥ 0 J N : H + → H J_N : H_+ \to H J N : H + → H J N e k = e k − N J_N e_k = e_{k-N} J N e k = e k − N
Q N ≔ P − N = J N J N ∗ , ι N ( x ) ≔ J N x J N ∗ , ρ N ( x ) ≔ J N ∗ x J N . (102) Q_N \coloneqq P_{-N} = J_NJ_N^*, \qquad\iota_N(x) \coloneqq J_NxJ_N^*, \qquad\rho_N(x) \coloneqq J_N^*xJ_N. \tag*{(102)} Q N : = P − N = J N J N ∗ , ι N ( x ) : = J N x J N ∗ , ρ N ( x ) : = J N ∗ x J N . (102) Proposition 8.1 The map ι N \iota_N ι N ∗ * ∗ P c \mathcal{P}_c P c Q N P e Q N Q_N\mathcal{P}_eQ_N Q N P e Q N Q N Q_N Q N P e \mathcal{P}_e P e Q N → 1 Q_N \to1 Q N → 1 M ( P e ) M(\mathcal{P}_e) M ( P e )
ψ ( ι N ( x ) ) = τ − N h c ( x ) ( x ≥ 0 ) . (103) \psi(\iota_N(x)) = \tau^{-N}h_c(x) \qquad(x \ge0). \tag*{(103)} ψ ( ι N ( x )) = τ − N h c ( x ) ( x ≥ 0 ) . (103) Consequently
J N Λ c ( x ) = τ N / 2 Λ e ( ι N ( x ) ) (104) J_N\Lambda_c(x) = \tau^{N/2}\Lambda_e(\iota_N(x)) \tag*{(104)} J N Λ c ( x ) = τ N /2 Λ e ( ι N ( x )) (104) defines an isometry from H c \mathcal{H}_c H c H e \mathcal{H}_e H e R N R_N R N ξ ↦ Q N ξ Q N \xi\mapsto Q_N\xi Q_N ξ ↦ Q N ξ Q N R N → I R_N \to I R N → I
Proof: The projection Q N − P 0 Q_N-P_0 Q N − P 0 Q N P e Q N = K ( Q N H ) + C Q N = ι N ( P c ) Q_N\mathcal{P}_eQ_N = \mathcal{K}(Q_NH)+\mathbb{C}Q_N = \iota_N(\mathcal{P}_c) Q N P e Q N = K ( Q N H ) + C Q N = ι N ( P c ) P 0 P_0 P 0 N N N Q N Q_N Q N Q N P 0 = P 0 Q N = P 0 Q_NP_0=P_0Q_N=P_0 Q N P 0 = P 0 Q N = P 0 Q N Q_N Q N (103) follows by setting k = j − N k=j-N k = j − N (97) . It implies the isometry assertion. In weighted matrix coordinates, J N J_N J N − N -N − N τ N / 2 \tau^{N/2} τ N /2 R N R_N R N
The same formula holds for the symmetric embeddings. Indeed, if D c = ( 1 − τ ) diag ( τ k ) k ≥ 0 D_c=(1-\tau)\operatorname{diag}(\tau^k)_{k\ge0} D c = ( 1 − τ ) diag ( τ k ) k ≥ 0 D e = ( 1 − τ ) diag ( τ k ) k ∈ Z D_e=(1-\tau)\operatorname{diag}(\tau^k)_{k\in\mathbb{Z}} D e = ( 1 − τ ) diag ( τ k ) k ∈ Z
τ N / 2 D e 1 / 4 ι N ( x ) D e 1 / 4 = J N ( D c 1 / 4 x D c 1 / 4 ) J N ∗ . \tau^{N/2}D_e^{1/4}\iota_N(x)D_e^{1/4}=J_N(D_c^{1/4}xD_c^{1/4})J_N^*. τ N /2 D e 1/4 ι N ( x ) D e 1/4 = J N ( D c 1/4 x D c 1/4 ) J N ∗ . Thus the isometries are literal corner embeddings in the Hilbert–Schmidt standard forms. They preserve positive cones. A C ∗ C^* C ∗ (103) cannot be discarded.
The compact Casimir in every angular band
Put k ℓ ≔ max ( 0 , − ℓ ) k_\ell\coloneqq\max(0,-\ell) k ℓ : = max ( 0 , − ℓ ) ℓ \ell ℓ x k + ℓ , k = f ( k ) x_{k+\ell,k}=f(k) x k + ℓ , k = f ( k ) k ≥ k ℓ k\ge k_\ell k ≥ k ℓ
c ℓ , k = ( 1 − τ k ) ( 1 − τ k + ℓ ) , v ℓ , k = 1 + τ − ℓ − ( 1 + τ ) τ − k − τ − ℓ ( c ℓ , k + 1 + c ℓ , k ) . (105) c_{\ell,k}=\sqrt{(1-\tau^k)(1-\tau^{k+\ell})}, \qquad
v_{\ell,k}=1+\tau^{-\ell}-(1+\tau)\tau^{-k}-\tau^{-\ell}(c_{\ell,k+1}+c_{\ell,k}). \tag*{(105)} c ℓ , k = ( 1 − τ k ) ( 1 − τ k + ℓ ) , v ℓ , k = 1 + τ − ℓ − ( 1 + τ ) τ − k − τ − ℓ ( c ℓ , k + 1 + c ℓ , k ) . (105) In particular c ℓ , k ℓ = 0 c_{\ell,k_\ell}=0 c ℓ , k ℓ = 0
Theorem 8.2 (Angular Jacobi expression) The compact Casimir (98) has the band expression
( C c , ℓ f ) ( k ) = ( τ − k ( 1 + τ − ℓ ) − ( 1 + τ ) ) f ( k ) − τ − ℓ − k ( c ℓ , k + 1 f ( k + 1 ) + c ℓ , k f ( k − 1 ) ) . (106) (\mathcal{C}_{c,\ell}f)(k)=\left(\tau^{-k}(1+\tau^{-\ell})-(1+\tau)\right)f(k)-\tau^{-\ell-k}\left(c_{\ell,k+1}f(k+1)+c_{\ell,k}f(k-1)\right). \tag*{(106)} ( C c , ℓ f ) ( k ) = ( τ − k ( 1 + τ − ℓ ) − ( 1 + τ ) ) f ( k ) − τ − ℓ − k ( c ℓ , k + 1 f ( k + 1 ) + c ℓ , k f ( k − 1 ) ) . (106) The term at k − 1 k-1 k − 1
E c , ℓ ( f ) = ( 1 − τ ) ( τ − ℓ ∑ k ≥ k ℓ c ℓ , k + 1 ∣ Δ f ( k ) ∣ 2 + ∑ k ≥ k ℓ v ℓ , k ∣ f ( k ) ∣ 2 ) , (107) \mathcal{E}_{c,\ell}(f)=(1-\tau)\left(\tau^{-\ell}\sum_{k\ge k_\ell}c_{\ell,k+1}\lvert\Delta f(k)\rvert^2+\sum_{k\ge k_\ell}v_{\ell,k}\lvert f(k)\rvert^2\right), \tag*{(107)} E c , ℓ ( f ) = ( 1 − τ ) ( τ − ℓ k ≥ k ℓ ∑ c ℓ , k + 1 ∣ Δ f ( k ) ∣ 2 + k ≥ k ℓ ∑ v ℓ , k ∣ f ( k ) ∣ 2 ) , (107) on the maximal finite-energy domain in ℓ 2 ( { k ≥ k ℓ } , w ) \ell^2(\{k\ge k_\ell\},w) ℓ 2 ({ k ≥ k ℓ } , w )
v ℓ , k ≥ ( 1 − τ − ℓ ) 2 , (108) v_{\ell,k}\ge(1-\tau^{-\ell})^2, \tag*{(108)} v ℓ , k ≥ ( 1 − τ − ℓ ) 2 , (108) with v 0 , k = 0 v_{0,k}=0 v 0 , k = 0 ℓ 2 \ell^2 ℓ 2
Proof: We first identify the representation-theoretic operator and then its closure. Let s ≥ 0 s\ge0 s ≥ 0 b ∗ γ b^*\gamma b ∗ γ r k 1 − τ k + 1 r^k\sqrt{1-\tau^{k+1}} r k 1 − τ k + 1 ( b ∗ γ ) s (b^*\gamma)^s ( b ∗ γ ) s k k k
q s ( k ) = r s k ( τ k + 1 ; τ ) s , k ≥ 0. (109) q_s(k)=r^{sk}\sqrt{(\tau^{k+1};\tau)_s},\qquad k\ge0. \tag*{(109)} q s ( k ) = r s k ( τ k + 1 ; τ ) s , k ≥ 0. (109) Here ( a ; τ ) s = ∏ j = 0 s − 1 ( 1 − a τ j ) (a;\tau)_s=\prod_{j=0}^{s-1}(1-a\tau^j) ( a ; τ ) s = ∏ j = 0 s − 1 ( 1 − a τ j ) s = 0 s=0 s = 0
The m + 1 m+1 m + 1 ( b ∗ γ ) s A j c (b^*\gamma)^sA_j^c ( b ∗ γ ) s A j c 0 ≤ j ≤ m 0\le j\le m 0 ≤ j ≤ m − 2 s -2s − 2 s 2 ( s + m ) 2(s+m) 2 ( s + m ) s s s s , s + 1 , … , s + m s,s+1,\ldots,s+m s , s + 1 , … , s + m s + m s+m s + m q s q_s q s m m m w k q s ( k ) 2 w_kq_s(k)^2 w k q s ( k ) 2
ν s ( k ) = τ ( s + 1 ) k ( τ s + 1 ; τ ) k ( τ ; τ ) k . (110) \nu_s(k)=\tau^{(s+1)k}\frac{(\tau^{s+1};\tau)_k}{(\tau;\tau)_k}. \tag*{(110)} ν s ( k ) = τ ( s + 1 ) k ( τ ; τ ) k ( τ s + 1 ; τ ) k . (110) Polynomials in τ k \tau^k τ k q s ( k ) > 0 q_s(k)>0 q s ( k ) > 0
For completeness the needed polynomial difference calculation is explicit. On polynomials let
( L s h ) ( k ) = B s ( k ) ( h ( k ) − h ( k + 1 ) ) + D s ( k ) ( h ( k ) − h ( k − 1 ) ) , B s ( k ) = τ − s − k ( 1 − τ s + k + 1 ) , D s ( k ) = τ − k ( 1 − τ k ) . (111) (L_s h)(k)=B_s(k)(h(k)-h(k+1))+D_s(k)(h(k)-h(k-1)),
\qquad
B_s(k)=\tau^{-s-k}(1-\tau^{s+k+1}),\quad D_s(k)=\tau^{-k}(1-\tau^k). \tag*{(111)} ( L s h ) ( k ) = B s ( k ) ( h ( k ) − h ( k + 1 )) + D s ( k ) ( h ( k ) − h ( k − 1 )) , B s ( k ) = τ − s − k ( 1 − τ s + k + 1 ) , D s ( k ) = τ − k ( 1 − τ k ) . (111) The ratio ν s ( k + 1 ) / ν s ( k ) \nu_s(k+1)/\nu_s(k) ν s ( k + 1 ) / ν s ( k ) h ( k ) = τ m k h(k)=\tau^{mk} h ( k ) = τ mk
λ s , m = τ − m ( 1 − τ m ) ( 1 − τ 2 s + m + 1 ) . \lambda_{s,m}=\tau^{-m}(1-\tau^m)(1-\tau^{2s+m+1}). λ s , m = τ − m ( 1 − τ m ) ( 1 − τ 2 s + m + 1 ) . Consequently its orthogonal polynomial of degree m m m
Λ s + m − Λ s = τ − s λ s , m , (112) \Lambda_{s+m}-\Lambda_s=\tau^{-s}\lambda_{s,m}, \tag*{(112)} Λ s + m − Λ s = τ − s λ s , m , (112) the compact Casimir on the polynomial band is M g s ( Λ s + τ − s L s ) M g s − 1 M_{g_s}(\Lambda_s+\tau^{-s}L_s)M_{g_s}^{-1} M g s ( Λ s + τ − s L s ) M g s − 1
g s ( k + 1 ) g s ( k ) = r s 1 − τ k + s + 1 1 − τ k + 1 \frac{g_s(k+1)}{g_s(k)}=r^s\sqrt{\frac{1-\tau^{k+s+1}}{1-\tau^{k+1}}} g s ( k ) g s ( k + 1 ) = r s 1 − τ k + 1 1 − τ k + s + 1 gives (106) for ℓ = s \ell=s ℓ = s − s -s − s k k k k − s k-s k − s r s r^s r s
Multiplying (106) by w k f ( k ) w_kf(k) w k f ( k ) (107) for finite supports. The elementary inequality ( 1 − u ) ( 1 − v ) ≤ 1 − ( u + v ) / 2 \sqrt{(1-u)(1-v)}\leq1-(u+v)/2 ( 1 − u ) ( 1 − v ) ≤ 1 − ( u + v ) /2
v ℓ , k ≥ ( 1 − r − ℓ ) 2 + ( 1 + τ ) τ k ( r − ℓ + r ℓ 2 − 1 ) . v_{\ell,k}\geq(1-r^{-\ell})^2+(1+\tau)\tau^k\left(\frac{r^{-\ell}+r^\ell}{2}-1\right). v ℓ , k ≥ ( 1 − r − ℓ ) 2 + ( 1 + τ ) τ k ( 2 r − ℓ + r ℓ − 1 ) . proving (108) . All coefficients in the form are bounded for fixed ℓ \ell ℓ ℓ ≠ 0 \ell\ne0 ℓ = 0 ℓ 2 \ell^2 ℓ 2 ℓ = 0 \ell=0 ℓ = 0 ( 1 − τ ) ∑ k ≥ 0 ( 1 − τ k + 1 ) ∣ Δ f ( k ) ∣ 2 (1-\tau)\sum_{k\geq0}(1-\tau^{k+1})|\Delta f(k)|^2 ( 1 − τ ) ∑ k ≥ 0 ( 1 − τ k + 1 ) ∣Δ f ( k ) ∣ 2 J J J ( 1 − τ ) ∣ c ∣ 2 / J (1-\tau)|c|^2/J ( 1 − τ ) ∣ c ∣ 2 / J
Finally each polynomial eigenvector g s p m g_sp_m g s p m s > 0 s>0 s > 0 s = 0 s=0 s = 0 O ( τ k ) O(\tau^k) O ( τ k ) (98) . This proves equality of the closed realizations, not merely of their formal expressions.
Corollary 8.3 (Oscillator expression) On H + H_+ H + d c e k ≔ τ − k − 1 e k − 1 d_{ce_k}\coloneqq\sqrt{\tau^{-k}-1}\,e_{k-1} d c e k : = τ − k − 1 e k − 1 d c e 0 = 0 d_{ce_0}=0 d c e 0 = 0 h 0 = A c − 1 − 1 + τ 2 h_0=A_c^{-1}-\frac{1+\tau}{2} h 0 = A c − 1 − 2 1 + τ
C c ( x ) = h 0 x + x h 0 − d c ∗ x d c − τ d c x d c ∗ . (113) \mathcal{C}_c(x)=h_0x+xh_0-d_c^*xd_c-\tau d_cxd_c^*. \tag*{(113)} C c ( x ) = h 0 x + x h 0 − d c ∗ x d c − τ d c x d c ∗ . (113) Moreover, τ N / 2 J N d c J N ∗ \tau^{N/2}J_Nd_cJ_N^* τ N /2 J N d c J N ∗ H H H D = V A − 1 / 2 D=VA^{-1/2} D = V A − 1/2
Proof: One has d c ∗ d c = A c − 1 − 1 d_c^*d_c=A_c^{-1}-1 d c ∗ d c = A c − 1 − 1 τ d c d c ∗ = A c − 1 − τ \tau d_cd_c^*=A_c^{-1}-\tau τ d c d c ∗ = A c − 1 − τ (106) . After translation the coefficient of the lowering shift is τ − k − τ N \sqrt{\tau^{-k}-\tau^N} τ − k − τ N k ≥ − N k\geq-N k ≥ − N − N -N − N τ − k / 2 \tau^{-k/2} τ − k /2 k k k
Bounded band defects, resolvents and Mosco convergence
Define the nonnegative self-adjoint operator on H e \mathcal{H}_e H e
A N = τ N J N C c J N ∗ ⊕ 0 R N ⊥ H e . (114) A_N=\tau^N J_N\mathcal{C}_cJ_N^*\oplus0_{R_N^\perp\mathcal{H}_e}. \tag*{(114)} A N = τ N J N C c J N ∗ ⊕ 0 R N ⊥ H e . (114) Its band restriction is denoted by A N , ℓ A_{N,\ell} A N , ℓ H w H_w H w U f ( k ) = r f ( k + 1 ) Uf(k)=rf(k+1) U f ( k ) = r f ( k + 1 ) N = 0 N=0 N = 0 A 0 , ℓ A_{0,\ell} A 0 , ℓ
Theorem 8.4 (A bounded defect in each band). For every fixed ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z B ℓ B_\ell B ℓ H w H_w H w
A N , ℓ = C e , ℓ + τ N U N B ℓ U − N , dom A N , ℓ = dom C e , ℓ . (115) A_{N,\ell}=\mathcal{C}_{e,\ell}+\tau^NU^NB_\ell U^{-N},\qquad\operatorname{dom}A_{N,\ell}=\operatorname{dom}\mathcal{C}_{e,\ell}. \tag*{(115)} A N , ℓ = C e , ℓ + τ N U N B ℓ U − N , dom A N , ℓ = dom C e , ℓ . (115) One may take B ℓ = A 0 , ℓ − C e , ℓ B_\ell=A_{0,\ell}-\mathcal{C}_{e,\ell} B ℓ = A 0 , ℓ − C e , ℓ ∥ B ℓ ∥ ≤ M ℓ \lVert B_\ell\rVert\leq M_\ell ∥ B ℓ ∥ ≤ M ℓ
M ℓ = max { 1 + τ , ( 1 + τ − ℓ ) τ 1 − k ℓ } + 2 r 1 − ℓ max { τ − k ℓ , 1 + τ ℓ } . (116) M_\ell=\max\{1+\tau,(1+\tau^{-\ell})\tau^{1-k_\ell}\}+2r^{1-\ell}\max\{\tau^{-k_\ell},1+\tau^\ell\}. \tag*{(116)} M ℓ = max { 1 + τ , ( 1 + τ − ℓ ) τ 1 − k ℓ } + 2 r 1 − ℓ max { τ − k ℓ , 1 + τ ℓ } . (116) In particular, for λ > 0 \lambda>0 λ > 0 t ≥ 0 t\geq0 t ≥ 0
∥ ( A N , ℓ + λ ) − 1 − ( C e , ℓ + λ ) − 1 ∥ ≤ λ − 2 M ℓ τ N , (117) \lVert(A_{N,\ell}+\lambda)^{-1}-(\mathcal{C}_{e,\ell}+\lambda)^{-1}\rVert\leq\lambda^{-2}M_\ell\tau^N, \tag*{(117)} ∥( A N , ℓ + λ ) − 1 − ( C e , ℓ + λ ) − 1 ∥ ≤ λ − 2 M ℓ τ N , (117) ∥ e − t A N , ℓ − e − t C e , ℓ ∥ ≤ t M ℓ τ N . (118) \lVert e^{-tA_{N,\ell}}-e^{-t\mathcal{C}_{e,\ell}}\rVert\leq tM_\ell\tau^N. \tag*{(118)} ∥ e − t A N , ℓ − e − t C e , ℓ ∥ ≤ t M ℓ τ N . (118) Proof: Pass from weighted sequences to the counting orthonormal basis by f ( k ) ↦ w k f ( k ) f(k)\mapsto\sqrt{w_k}f(k) f ( k ) ↦ w k f ( k ) A 0 , ℓ − C e , ℓ A_{0,\ell}-\mathcal{C}_{e,\ell} A 0 , ℓ − C e , ℓ − ( 1 + τ ) -(1+\tau) − ( 1 + τ ) k ≥ k ℓ k\geq k_\ell k ≥ k ℓ − ( 1 + τ − ℓ ) τ − k -(1+\tau^{-\ell})\tau^{-k} − ( 1 + τ − ℓ ) τ − k k < k ℓ k<k_\ell k < k ℓ (116) . The entry joining k k k k + 1 k+1 k + 1
b ℓ ( k ) = { r − ℓ − 1 τ − k ( 1 − c ℓ , k + 1 ) , k ≥ k ℓ , r − ℓ − 1 τ − k , k < k ℓ . b_\ell(k)=
\begin{cases}
r^{-\ell-1}\tau^{-k}(1-c_{\ell,k+1}), & k\geq k_\ell,\\
r^{-\ell-1}\tau^{-k}, & k<k_\ell.
\end{cases} b ℓ ( k ) = { r − ℓ − 1 τ − k ( 1 − c ℓ , k + 1 ) , r − ℓ − 1 τ − k , k ≥ k ℓ , k < k ℓ . Using 1 − ( 1 − u ) ( 1 − v ) ≤ u + v 1-\sqrt{(1-u)(1-v)}\leq u+v 1 − ( 1 − u ) ( 1 − v ) ≤ u + v r 1 − ℓ ( 1 + τ ℓ ) r^{1-\ell}(1+\tau^\ell) r 1 − ℓ ( 1 + τ ℓ ) r 1 − ℓ τ − k ℓ r^{1-\ell}\tau^{-k_\ell} r 1 − ℓ τ − k ℓ (116) . These calculations include the edge crossing the artificial left endpoint.
Finite supports are a form core for both operators: for A 0 , ℓ A_{0,\ell} A 0 , ℓ C e , ℓ \mathcal{C}_{e,\ell} C e , ℓ (99) . On this common form core the two energies differ by ⟨ B ℓ f , f ⟩ \langle B_\ell f,f\rangle ⟨ B ℓ f , f ⟩ ( 1 + ∥ B ℓ ∥ ) ∥ f ∥ 2 (1+\lVert B_\ell\rVert)\lVert f\rVert^2 ( 1 + ∥ B ℓ ∥) ∥ f ∥ 2 [18 Theorem VI.2.1] now gives A 0 , ℓ = C e , ℓ + B ℓ A_{0,\ell}=\mathcal{C}_{e,\ell}+B_\ell A 0 , ℓ = C e , ℓ + B ℓ U − 1 C e , ℓ U = τ C e , ℓ U^{-1}\mathcal{C}_{e,\ell}U=\tau\mathcal{C}_{e,\ell} U − 1 C e , ℓ U = τ C e , ℓ (115) . The resolvent identity proves (117) . The bounded-perturbation Duhamel formula [13 Corollary III.1.7] , with both heat semigroups contractive, proves (118) .
In particular the test bands ℓ = 1 \ell=1 ℓ = 1 ℓ = − 1 \ell=-1 ℓ = − 1 (100) , without a change of Casimir normalization. The bounds are not uniform over all angular momenta. That distinction is necessary, as Proposition 8.7 below shows.
Theorem 8.5 (Full Hilbert-space and form contraction). The operators A N A_N A N C e \mathcal{C}_e C e
q N ( ξ ) = τ N E c ( J N ∗ ξ ) (119) q_N(\xi)=\tau^N\mathcal{E}_c(J_N^*\xi) \tag*{(119)} q N ( ξ ) = τ N E c ( J N ∗ ξ ) (119) (with zero energy on R N ⊥ H e R_N^\perp\mathcal{H}_e R N ⊥ H e E e \mathcal{E}_e E e [29 ] . The same is true for the lower semicontinuous extended-valued forms q ^ N \widehat{q}_N q N q ^ N ( ξ ) = q N ( ξ ) \widehat{q}_N(\xi)=q_N(\xi) q N ( ξ ) = q N ( ξ ) ξ ∈ R N H e \xi\in R_N\mathcal{H}_e ξ ∈ R N H e q ^ N ( ξ ) = + ∞ \widehat{q}_N(\xi)=+\infty q N ( ξ ) = + ∞
e − t A N ξ → e − t C e ξ (120) e^{-tA_N}\xi\to e^{-t\mathcal{C}_e}\xi \tag*{(120)} e − t A N ξ → e − t C e ξ (120) uniformly for t t t [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ξ ∈ H e \xi\in\mathcal{H}_e ξ ∈ H e
Proof: The resolvent and heat claims follow immediately from Theorem 8.4 on any finite collection of bands. Orthogonal band truncations and the uniform bounds λ − 1 \lambda^{-1} λ − 1 1 1 1
Here is also a direct proof of Mosco convergence, which verifies the boundary realization independently. In band ℓ \ell ℓ a N = k ℓ − N a_N=k_\ell-N a N = k ℓ − N k ≥ a N k\geq a_N k ≥ a N
c ℓ , k ( N ) = ( 1 − τ N + k ) ( 1 − τ N + k + ℓ ) , v ℓ , k ( N ) = 1 + τ − ℓ − ( 1 + τ ) τ N + k − r − ℓ ( c ℓ , k + 1 ( N ) + c ℓ , k ( N ) ) . c_{\ell,k}^{(N)}=\sqrt{(1-\tau^{N+k})(1-\tau^{N+k+\ell})},\qquad v_{\ell,k}^{(N)}=1+\tau^{-\ell}-(1+\tau)\tau^{N+k}-r^{-\ell}(c_{\ell,k+1}^{(N)}+c_{\ell,k}^{(N)}). c ℓ , k ( N ) = ( 1 − τ N + k ) ( 1 − τ N + k + ℓ ) , v ℓ , k ( N ) = 1 + τ − ℓ − ( 1 + τ ) τ N + k − r − ℓ ( c ℓ , k + 1 ( N ) + c ℓ , k ( N ) ) . Substituting (104) into (107) gives
q N , ℓ ( f ) = ( 1 − τ ) − ℓ ( r − ℓ ∑ k ≥ a N c ℓ , k + 1 ( N ) ∣ Δ f ( k ) ∣ 2 + ∑ k ≥ a N v ℓ , k ( N ) ∣ f ( k ) ∣ 2 ) . (121) q_{N,\ell}(f)=(1-\tau)^{-\ell}\left(r^{-\ell}\sum_{k\geq a_N}c_{\ell,k+1}^{(N)}|\Delta f(k)|^2+\sum_{k\geq a_N}v_{\ell,k}^{(N)}|f(k)|^2\right). \tag*{(121)} q N , ℓ ( f ) = ( 1 − τ ) − ℓ ( r − ℓ k ≥ a N ∑ c ℓ , k + 1 ( N ) ∣Δ f ( k ) ∣ 2 + k ≥ a N ∑ v ℓ , k ( N ) ∣ f ( k ) ∣ 2 ) . (121) The coordinates below a N a_N a N ℓ , k \ell,k ℓ , k r − ℓ r^{-\ell} r − ℓ ( 1 − r − ℓ ) 2 (1-r^{-\ell})^2 ( 1 − r − ℓ ) 2 ξ N ⇀ ξ \xi_N\rightharpoonup\xi ξ N ⇀ ξ H e \mathcal{H}_e H e (121) , pass to the limit and then take the supremum over finite collections. This gives
E e ( ξ ) ≤ lim inf N q N ( ξ N ) . \mathcal{E}_e(\xi)\leq\liminf_N q_N(\xi_N). E e ( ξ ) ≤ N lim inf q N ( ξ N ) . For the upper bound, finite matrices are a form core for E e \mathcal{E}_e E e F F F R N F = F R_NF=F R N F = F q N ( F ) → E e ( F ) q_N(F)\to\mathcal{E}_e(F) q N ( F ) → E e ( F ) ξ \xi ξ F m F_m F m N N N ∣ q N ( F m ) − E e ( F m ) ∣ |q_N(F_m)-\mathcal{E}_e(F_m)| ∣ q N ( F m ) − E e ( F m ) ∣ R N F m = F m R_NF_m=F_m R N F m = F m F m + 1 F_{m+1} F m + 1 ξ \xi ξ + ∞ +\infty + ∞ [29 Theorem 2.4.1 and Corollary 2.6.1] , see also [25 ] . The argument above proves the two Mosco conditions directly, for both choices of the transported extended-valued form.
Remark 8.6 (Zero extension is not Markovian). The matrix-level statement refers to the genuine completely Dirichlet forms on the varying corners , or to the preceding quadratic-form
convergence after matrix amplification. It does not say that q N q_N q N S τ N t C \mathcal{S}^{\mathcal{C}}_{\tau_N t} S τ N t C Q N x Q N Q_NxQ_N Q N x Q N i ≥ − N i \ge-N i ≥ − N j < − N j < -N j < − N ( e i + e j ) ( e i + e j ) ∗ (e_i+e_j)(e_i+e_j)^* ( e i + e j ) ( e i + e j ) ∗ i , j i,j i , j
( P τ N t C ( i + N , i + N ) 1 1 1 ) \begin{pmatrix}
\mathcal{P}^{\mathcal{C}}_{\tau_N t}(i+N,i+N) & 1 \\
1 & 1
\end{pmatrix} ( P τ N t C ( i + N , i + N ) 1 1 1 ) For t > 0 t>0 t > 0 § 9.5 .
Proposition 8.7 (Failure of global norm convergence) For every N ≥ 0 N \ge0 N ≥ 0 t > 0 t>0 t > 0 λ > 0 \lambda>0 λ > 0
∥ e − t A N − e − t C e ∥ = 1 , (122) \left\|e^{-tA_N}-e^{-t\mathcal{C}_e}\right\|=1, \tag*{(122)} e − t A N − e − t C e = 1 , (122) ∥ ( A N + λ ) − 1 − ( C e + λ ) − 1 ∥ = λ − 1 . (123) \left\|(A_N+\lambda)^{-1}-(\mathcal{C}_e+\lambda)^{-1}\right\|=\lambda^{-1}. \tag*{(123)} ( A N + λ ) − 1 − ( C e + λ ) − 1 = λ − 1 . (123) The same equalities hold with the first operators replaced, respectively, by the compressed operators J N e − t τ N C c J N ∗ J_Ne^{-t\tau_N\mathcal{C}_c}J_N^* J N e − t τ N C c J N ∗ J N ( λ + τ N C c ) − 1 J N ∗ J_N(\lambda+\tau_N\mathcal{C}_c)^{-1}J_N^* J N ( λ + τ N C c ) − 1 J N ∗ (117) –(118) .
Proof. Fix j = − N − 1 j=-N-1 j = − N − 1 i → + ∞ i\to+\infty i → + ∞ ξ i = w j − 1 / 2 e i j \xi_i=w_j^{-1/2}e_{ij} ξ i = w j − 1/2 e ij R N ⊥ H e R_N^\perp\mathcal{H}_e R N ⊥ H e ℓ = i − j \ell=i-j ℓ = i − j (99) , in form order,
C e , ℓ ≥ ( 1 − r − ℓ ) 2 M τ − k . \mathcal{C}_{e,\ell}\ge(1-r^{-\ell})^2M_{\tau^{-k}}. C e , ℓ ≥ ( 1 − r − ℓ ) 2 M τ − k . The order-reversing property of the resolvent, equivalently its variational characterization, gives
0 ≤ ⟨ ( λ + C e ) − 1 ξ i , ξ i ⟩ ≤ 1 λ + ( 1 − r − ℓ ) 2 τ − j → 0. 0\le\left\langle(\lambda+\mathcal{C}_e)^{-1}\xi_i,\xi_i\right\rangle\le\frac{1}{\lambda+(1-r^{-\ell})^2\tau^{-j}}\to0. 0 ≤ ⟨ ( λ + C e ) − 1 ξ i , ξ i ⟩ ≤ λ + ( 1 − r − ℓ ) 2 τ − j 1 → 0. Since the resolvent is bounded by λ − 1 \lambda^{-1} λ − 1 ξ i \xi_i ξ i e − 2 t u ≤ C t , λ ( λ + u ) − 1 e^{-2tu}\le C_{t,\lambda}(\lambda+u)^{-1} e − 2 t u ≤ C t , λ ( λ + u ) − 1 u ≥ 0 u\ge0 u ≥ 0 ∥ e − t C e ξ i ∥ → 0 \|e^{-t\mathcal{C}_e}\xi_i\|\to0 ∥ e − t C e ξ i ∥ → 0 e − t A N e^{-tA_N} e − t A N ( λ + A N ) − 1 (\lambda+A_N)^{-1} ( λ + A N ) − 1 λ − 1 \lambda^{-1} λ − 1 (122) –(123) . The upper bounds follow because the differences are differences of positive operators in [ 0 , I ] [0,I] [ 0 , I ] [ 0 , λ − 1 I ] [0,\lambda^{-1}I] [ 0 , λ − 1 I ]
For the compressed operators, the compact spectrum (98) , with finite multiplicities 2 ℓ + 1 2\ell+1 2 ℓ + 1 Λ ℓ → ∞ \Lambda_\ell\to\infty Λ ℓ → ∞ C e \mathcal{C}_e C e τ p \tau^p τ p τ p → 0 \tau^p\to0 τ p → 0 p → + ∞ p\to+\infty p → + ∞ 1 1 1 λ − 1 \lambda^{-1} λ − 1
Contraction of spherical eigenoperators and completely positive maps
We now pass from Hilbert convergence to the norm topology of the quantum homogeneous spaces. This passage is not a consequence of strong resolvent convergence alone.
For m , s ≥ 0 m,s\ge0 m , s ≥ 0
p m ( x ; τ s , τ s ; τ ) = ∑ a = 0 m ( τ − m ; τ ) a ( τ m + 2 s + 1 ; τ ) a ( τ s + 1 ; τ ) a ( τ ; τ ) a ( τ x ) a . (124) p_m(x;\tau^s,\tau^s;\tau)=\sum_{a=0}^{m}\frac{(\tau^{-m};\tau)_a(\tau^{m+2s+1};\tau)_a}{(\tau^{s+1};\tau)_a(\tau;\tau)_a}(\tau x)^a. \tag*{(124)} p m ( x ; τ s , τ s ; τ ) = a = 0 ∑ m ( τ s + 1 ; τ ) a ( τ ; τ ) a ( τ − m ; τ ) a ( τ m + 2 s + 1 ; τ ) a ( τ x ) a . (124) It is the degree-m m m (111) , normalized to be one at zero. One can verify this directly by equating coefficients in that equation; orthogonality and uniqueness follow from its detailed balance and triangularity.
Theorem 8.8 (Norm convergence of nonradial spherical coefficients) Fix p ∈ Z p \in\mathbb{Z} p ∈ Z s ≥ 0 s \ge0 s ≥ 0 N ≥ max ( 0 , p + s ) N \ge\max(0,p+s) N ≥ max ( 0 , p + s )
F p , s ( N ) ( k ) = { r s k ( τ − N + k + 1 ; τ ) s p N − p − s ( τ N + k ; τ s , τ s ; τ ) , k ≥ − N , 0 , k < − N . (125) F_{p,s}^{(N)}(k)=
\begin{cases}
r^{sk}\sqrt{(\tau^{-N+k+1};\tau)_{s}p_{N-p-s}(\tau^{N+k};\tau^{s},\tau^{s};\tau)}, & k\ge-N,\\
0, & k<-N.
\end{cases} \tag*{(125)} F p , s ( N ) ( k ) = { r s k ( τ − N + k + 1 ; τ ) s p N − p − s ( τ N + k ; τ s , τ s ; τ ) , 0 , k ≥ − N , k < − N . (125) This is the band coefficient of an operator X p , s ( N ) ∈ ι N ( P c ) X_{p,s}^{(N)} \in\iota_N(\mathcal{P}_c) X p , s ( N ) ∈ ι N ( P c ) N − p N-p N − p
A N , s F p , s ( N ) = λ N , p F p , s ( N ) , λ N , p = τ p ( 1 − τ N − p ) ( 1 − τ N − p + 1 ) . (126) A_{N,s}F_{p,s}^{(N)}=\lambda_{N,p}F_{p,s}^{(N)}, \qquad\lambda_{N,p}=\tau^p(1-\tau^{N-p})(1-\tau^{N-p+1}). \tag*{(126)} A N , s F p , s ( N ) = λ N , p F p , s ( N ) , λ N , p = τ p ( 1 − τ N − p ) ( 1 − τ N − p + 1 ) . (126) Then F p , s ( N ) → f p , s F_{p,s}^{(N)} \to f_{p,s} F p , s ( N ) → f p , s H w H_w H w Z \mathbb{Z} Z X p , s ( N ) → X p , s X_{p,s}^{(N)} \to X_{p,s} X p , s ( N ) → X p , s
Proof: The formula in Theorem 8.2 identifies the compact eigenfunction as g s p m g_{s p m} g s p m (125) ; then (126) follows from (98) .
Set m = N − p − s m=N-p-s m = N − p − s (124) use
( τ − m ; τ ) a = ( − 1 ) a τ − m a + a ( a − 1 ) / 2 ∏ b = 0 a − 1 ( 1 − τ m − b ) . (\tau^{-m};\tau)_a=(-1)^a\tau^{-ma+a(a-1)/2}\prod_{b=0}^{a-1}(1-\tau^{m-b}). ( τ − m ; τ ) a = ( − 1 ) a τ − ma + a ( a − 1 ) /2 b = 0 ∏ a − 1 ( 1 − τ m − b ) . For fixed k k k a a a N N N
τ a ( p + s + k + 1 ) + a ( a − 1 ) / 2 ( τ s + 1 ; τ ) ∞ ( τ ; τ ) ∞ . (127) \frac{\tau^{a(p+s+k+1)+a(a-1)/2}}{(\tau^{s+1};\tau)_\infty(\tau;\tau)_\infty}. \tag*{(127)} ( τ s + 1 ; τ ) ∞ ( τ ; τ ) ∞ τ a ( p + s + k + 1 ) + a ( a − 1 ) /2 . (127) The majorant is summable. Each fixed coefficient converges to its 1 ϕ 1 {}_1\phi_1 1 ϕ 1 (125) converges to one. Dominated convergence proves pointwise convergence to (101) .
To upgrade it to Hilbert convergence, choose an interval enclosing τ p \tau^p τ p τ Z ∪ { 0 } \tau^{\mathbb{Z}} \cup\{0\} τ Z ∪ { 0 } N N N A N , s A_{N,s} A N , s λ N , p \lambda_{N,p} λ N , p Λ l \Lambda_l Λ l (126) . The norm-resolvent convergence of Theorem 8.4 gives norm convergence of the corresponding projections to the projection onto C f p , s \mathbb{C}f_{p,s} C f p , s f p , s f_{p,s} f p , s g N → f p , s / ∥ f p , s ∥ g_N \to f_{p,s}/\lVert f_{p,s}\rVert g N → f p , s / ∥ f p , s ∥ F p , s ( N ) F_{p,s}^{(N)} F p , s ( N ) g N g_N g N f p , s f_{p,s} f p , s ∥ f p , s ∥ \lVert f_{p,s}\rVert ∥ f p , s ∥ H w H_w H w H w H_w H w
For k → − ∞ k\to-\infty k → − ∞ ∣ F p , s ( N ) ( k ) ∣ ≤ w k − 1 / 2 ∥ F p , s ( N ) ∥ H w \lvert F_{p,s}^{(N)}(k)\rvert\le w_k^{-1/2}\lVert F_{p,s}^{(N)}\rVert_{H_w} ∣ F p , s ( N ) ( k )∣ ≤ w k − 1/2 ∥ F p , s ( N ) ∥ H w k → + ∞ k\to+\infty k → + ∞ (127) gives ∣ F p , s ( N ) ( k ) ∣ ≤ C p , s , r r s k \lvert F_{p,s}^{(N)}(k)\rvert\le C_{p,s,r}r^{sk} ∣ F p , s ( N ) ( k )∣ ≤ C p , s , r r s k s > 0 s>0 s > 0 s = 0 s=0 s = 0 ∣ F p , 0 ( N ) ( k ) − 1 ∣ ≤ C p , r τ k \lvert F_{p,0}^{(N)}(k)-1\rvert\le C_{p,r}\tau^k ∣ F p , 0 ( N ) ( k ) − 1 ∣ ≤ C p , r τ k k k k N N N
For s > 0 s>0 s > 0 s s s ( − s ) (-s) ( − s ) s s s f p , s ( k − s ) = r − s 2 f p , − s ( k ) f_{p,s}(k-s)=r^{-s^2}f_{p,-s}(k) f p , s ( k − s ) = r − s 2 f p , − s ( k ) r s 2 ( X p , s ( N ) ) ∗ r^{s^2}(X_{p,s}^{(N)})^* r s 2 ( X p , s ( N ) ) ∗ X p , − s X_{p,-s} X p , − s
Theorem 8.9 (Point-norm quantum Markov contraction). For x ∈ P e x \in\mathcal{P}_e x ∈ P e t ≥ 0 t \ge0 t ≥ 0
S ~ t ( N ) ( x ) = ι N ( S τ N t c ( ρ N ( x ) ) ) . (128) \widetilde{S}_t^{(N)}(x)=\iota_N\left(S_{\tau_Nt}^{c}(\rho_N(x))\right). \tag*{(128)} S t ( N ) ( x ) = ι N ( S τ N t c ( ρ N ( x )) ) . (128) These are completely positive contractions; their value at t = 0 t=0 t = 0 x ↦ Q N x Q N x \mapsto Q_NxQ_N x ↦ Q N x Q N x ∈ P e x \in\mathcal{P}_e x ∈ P e T < ∞ T<\infty T < ∞
sup 0 ≤ t ≤ T ∥ S ~ t ( N ) ( x ) − S t e ( x ) ∥ → 0. (129) \sup_{0\le t\le T}\left\|\widetilde{S}_t^{(N)}(x)-S_t^e(x)\right\|\to0. \tag*{(129)} 0 ≤ t ≤ T sup S t ( N ) ( x ) − S t e ( x ) → 0. (129) The same assertion holds after every fixed finite matrix amplification. The normal extensions converge point-σ \sigma σ B ( H ) B(H) B ( H ) ψ \psi ψ
Proof: Compression, S t c S_t^c S t c ι N \iota_N ι N ρ N ι N = id \rho_N\iota_N=\operatorname{id} ρ N ι N = id (128) , including its compression value at zero. Formula (103) proves invariant-weight preservation on the corner. The GNS implementation of (128) is exactly J N e − t τ N C c J N ∗ J_Ne^{-t\tau_NC_c}J_N^* J N e − t τ N C c J N ∗ τ N / 2 \tau^{N/2} τ N /2 τ − N / 2 \tau^{-N/2} τ − N /2
For a limiting eigenoperator X = X p , ℓ X=X_{p,\ell} X = X p , ℓ X N X_N X N Theorem 8.8 . Both the compact and plane semigroups act on these operators with the respective eigenvalues λ N , p \lambda_{N,p} λ N , p τ p \tau^p τ p
sup t ≤ T ∥ S ~ t ( N ) X − S t e X ∥ ≤ 2 ∥ X N − X ∥ + T ∣ λ N , p − τ p ∣ ∥ X ∥ → 0. \sup_{t\le T}\left\|\widetilde{S}_t^{(N)}X-S_t^eX\right\|\le2\left\|X_N-X\right\|+T\left|\lambda_{N,p}-\tau^p\right|\left\|X\right\|\to0. t ≤ T sup S t ( N ) X − S t e X ≤ 2 ∥ X N − X ∥ + T ∣ λ N , p − τ p ∣ ∥ X ∥ → 0. The norm density of their span in P e \mathcal{P}_e P e (129) . Finite matrices over P e \mathcal{P}_e P e
For the normal statement, let F m F_m F m 1 1 1 S ~ t ( N ) ( x − F m x F m ) \widetilde{S}_t^{(N)}(x-F_mxF_m) S t ( N ) ( x − F m x F m ) 2 ∥ x ∥ 2\|x\| 2∥ x ∥ S ~ t ( N ) ( 1 − F m ) \widetilde{S}_t^{(N)}(1-F_m) S t ( N ) ( 1 − F m ) S ~ t ( N ) ( 1 ) = Q N \widetilde{S}_t^{(N)}(1)=Q_N S t ( N ) ( 1 ) = Q N m m m (129) applies to F m F_m F m F m x F m F_mxF_m F m x F m S t e ( F m ) ↑ 1 S_t^e(F_m)\mathrel{\uparrow}1 S t e ( F m ) ↑ 1 t t t m m m [ 0 , T ] [0,T] [ 0 , T ] t t t m m m N → ∞ N\to\infty N → ∞ σ \sigma σ
Theorem 8.9 concerns the full noncommutative quotient, not merely radial functions. In particular it applies to P 0 P_0 P 0 P e \mathcal{P}_e P e j < − N j<-N j < − N S ~ t ( N ) ( e j j ) = 0 \widetilde{S}_t^{(N)}(e_{jj})=0 S t ( N ) ( e j j ) = 0 ( S t e ( e j j ) ) j j ≥ e − 2 τ t − j (S_t^e(e_{jj}))_{jj}\ge e^{-2\tau t-j} ( S t e ( e j j ) ) j j ≥ e − 2 τ t − j j → − ∞ j\to-\infty j → − ∞ ∥ S ~ t ( N ) − S t e ∥ B ( P e ) ≥ 1 \left\|\widetilde{S}_t^{(N)}-S_t^e\right\|_{B(\mathcal{P}_e)}\ge1 S t ( N ) − S t e B ( P e ) ≥ 1
A quantitative contraction of entrance laws Let μ t c \mu_t^c μ t c μ t e \mu_t^e μ t e t > 0 t>0 t > 0 Z \mathbb{Z} Z
μ N , t ( k ) = { μ t c ( N + k ) , k ≥ − N , 0 , k < − N . (130) \mu_{N,t}(k)=
\begin{cases}
\mu_{t}^{c}(N+k), & k\ge-N,\\
0, & k<-N.
\end{cases} \tag*{(130)} μ N , t ( k ) = { μ t c ( N + k ) , 0 , k ≥ − N , k < − N . (130) Theorem 8.10 (Total-variation estimate). For every N ≥ 0 N \ge0 N ≥ 0 t > 0 t > 0 t > 0
∥ μ N , t − μ t e ∥ ℓ 1 ≤ 2 ( 1 − e − ( 1 + τ ) τ N t ) ≤ 2 ( 1 + τ ) t τ N . (131) \lVert\mu_{N,t}-\mu_t^e\rVert_{\ell^1} \le2\left(1-e^{-(1+\tau)\tau^N t}\right) \le2(1+\tau)t\tau^N. \tag*{(131)} ∥ μ N , t − μ t e ∥ ℓ 1 ≤ 2 ( 1 − e − ( 1 + τ ) τ N t ) ≤ 2 ( 1 + τ ) t τ N . (131) Thus convergence is uniform on 0 < t ≤ T 0 < t \le T 0 < t ≤ T T T T t = 0 t = 0 t = 0 Z \mathbb{Z} Z
Proof: The plane chain has rates q ( k , k ± 1 ) = τ − k q(k,k\pm1)=\tau^{-k} q ( k , k ± 1 ) = τ − k k ≥ − N k \ge-N k ≥ − N
q N ( k , k + 1 ) = τ − k − τ N + 1 , q N ( k , k − 1 ) = τ − k − τ N . (132) q_N(k,k+1)=\tau^{-k}-\tau^{N+1},\qquad q_N(k,k-1)=\tau^{-k}-\tau^N. \tag*{(132)} q N ( k , k + 1 ) = τ − k − τ N + 1 , q N ( k , k − 1 ) = τ − k − τ N . (132) The second one is zero at k = − N k=-N k = − N Z \mathbb{Z} Z − N -N − N
Both minimal chains are conservative. For the plane this follows from recurrence of the symmetric embedded walk and the holding times at a fixed site. For the compact embedded walk, its scale ratios telescope:
∏ k = 1 m q c ( k , k − 1 ) q c ( k , k + 1 ) = ∏ k = 1 m 1 − τ k 1 − τ k + 1 = 1 − τ 1 − τ m + 1 . \prod_{k=1}^{m}\frac{q_c(k,k-1)}{q_c(k,k+1)}
=\prod_{k=1}^{m}\frac{1-\tau^k}{1-\tau^{k+1}}
=\frac{1-\tau}{1-\tau^{m+1}}. k = 1 ∏ m q c ( k , k + 1 ) q c ( k , k − 1 ) = k = 1 ∏ m 1 − τ k + 1 1 − τ k = 1 − τ m + 1 1 − τ . The associated scale sum diverges, so the walk hits zero almost surely and returns there indefinitely. Its independent positive holding times there have divergent sum. This proves nonexplosion. The form-core statements ensure that the minimal chains are the specified self-adjoint realizations, not additional boundary extensions.
All q N q_N q N q q q
sup k ∈ Z ∑ ϵ ∈ { − 1 , 1 } ( q ( k , k + ϵ ) − q N ( k , k + ϵ ) ) ≤ c N : = ( 1 + τ ) τ N . (133) \sup_{k\in\mathbb{Z}}\sum_{\epsilon\in\{-1,1\}}\left(q(k,k+\epsilon)-q_N(k,k+\epsilon)\right)\le c_N\mathrel{\vcentcolon=}(1+\tau)\tau^N. \tag*{(133)} k ∈ Z sup ϵ ∈ { − 1 , 1 } ∑ ( q ( k , k + ϵ ) − q N ( k , k + ϵ ) ) ≤ c N : = ( 1 + τ ) τ N . (133) Above − N -N − N c N c_N c N 2 τ − k ≤ 2 τ N + 1 ≤ c N 2\tau^{-k}\le2\tau^{N+1}\le c_N 2 τ − k ≤ 2 τ N + 1 ≤ c N [28 ] for the coupling method. While they agree, use common jumps with rates q N q_N q N c N c_N c N t t t 1 − e − c N t 1-e^{-c_Nt} 1 − e − c N t ℓ 1 \ell^1 ℓ 1
Now let the common starting site tend to + ∞ +\infty + ∞ ℓ 1 \ell^1 ℓ 1 Proposition 7.12 . Translation of the compact law is exactly (130) . Passing to the limit proves (131) .
The same coupling also bounds the operator norm of the difference between the plane radial semigroup and the completed, absorbing-below-− N -N − N 2 ( 1 − e − c N t ) 2(1-e^{-c_Nt}) 2 ( 1 − e − c N t ) Proposition 8.7 about full angular Hilbert operators.
In the full plane-motion representation M e = B ( H ) ⊗ ‾ L ∞ ( T ) M_e=\mathrm{B}(H)\mathbin{\overline{\otimes}}L^\infty(\mathbb{T}) M e = B ( H ) ⊗ L ∞ ( T ) t > 0 t>0 t > 0
ω ^ N , t ( a ) = ∑ k μ N , t ( k ) ∫ T a k k ( z ) d m ( z ) , ω t e ( a ) = ∑ k μ t e ( k ) ∫ T a k k ( z ) d m ( z ) . (134) \widehat{\omega}_{N,t}(a)=\sum_k\mu_{N,t}(k)\int_{\mathbb{T}}a_{kk}(z)\,\mathrm{d}m(z),\qquad
\omega_t^e(a)=\sum_k\mu_t^e(k)\int_{\mathbb{T}}a_{kk}(z)\,\mathrm{d}m(z). \tag*{(134)} ω N , t ( a ) = k ∑ μ N , t ( k ) ∫ T a k k ( z ) d m ( z ) , ω t e ( a ) = k ∑ μ t e ( k ) ∫ T a k k ( z ) d m ( z ) . (134) Corollary 8.11 These are normal states and
∥ ω ^ N , t − ω t e ∥ = ∥ μ N , t − μ t e ∥ 1 ≤ 2 ( 1 − e − c N t ) . (135) \lVert\widehat{\omega}_{N,t}-\omega_t^e\rVert=\lVert\mu_{N,t}-\mu_t^e\rVert_1\leq2(1-e^{-cNt}). \tag*{(135)} ∥ ω N , t − ω t e ∥ = ∥ μ N , t − μ t e ∥ 1 ≤ 2 ( 1 − e − c N t ) . (135) The equality of norms also holds on C 0 ( E r ( 2 ) ) C_0(E_r(2)) C 0 ( E r ( 2 )) C ∗ C^* C ∗ M e M_e M e
Proof : The upper bound follows directly from the diagonal formula. Finite diagonal sign tests lie in C 0 ( E r ( 2 ) ) C_0(E_r(2)) C 0 ( E r ( 2 )) ℓ 1 \ell^1 ℓ 1 M e M_e M e
The transported states ω ^ N , t \widehat{\omega}_{N,t} ω N , t fixed plane coproduct. Their semigroup law belongs to the transported compact structure. The limiting convolution semigroup starts at the torus idempotent; its action on the quotient starts at the identity.
Compatibility with the quantum-group contraction
This section uses a published theorem, rather than deriving coproduct convergence from the preceding Hilbert convergence. Write A c = C ( S U r ( 2 ) ) A_c=C(SU_r(2)) A c = C ( S U r ( 2 )) A e = C 0 ( E r ( 2 ) ) A_e=C_0(E_r(2)) A e = C 0 ( E r ( 2 ))
Δ c ( b ) = b ⊗ b − r γ ∗ ⊗ γ , Δ c ( γ ) = γ ⊗ b + b ∗ ⊗ γ . \Delta_c(b)=b\otimes b-r\gamma^*\otimes\gamma,\qquad\Delta_c(\gamma)=\gamma\otimes b+b^*\otimes\gamma. Δ c ( b ) = b ⊗ b − r γ ∗ ⊗ γ , Δ c ( γ ) = γ ⊗ b + b ∗ ⊗ γ . In the Euclidean convention
Δ e ( v ) = v ⊗ v , Δ e ( n ) = v ⊗ n + n ⊗ v ∗ . \Delta_e(v)=v\otimes v,\qquad\Delta_e(n)=v\otimes n+n\otimes v^*. Δ e ( v ) = v ⊗ v , Δ e ( n ) = v ⊗ n + n ⊗ v ∗ . Their faithful concrete representations differ in the circular orientation needed for the contraction. Put
κ 0 ( b ) : = v 1 − A P 0 , κ 0 ( γ ) : = n ∗ P 0 , κ N : = Ad ( v N ) κ 0 . (136) \kappa_0(b)\mathrel{\mathop:}=v\sqrt{1-AP_0},\qquad\kappa_0(\gamma)\mathrel{\mathop:}=n^*P_0,\qquad\kappa_N\mathrel{\mathop:}=\operatorname{Ad}(v^N)\kappa_0. \tag*{(136)} κ 0 ( b ) : = v 1 − A P 0 , κ 0 ( γ ) : = n ∗ P 0 , κ N : = Ad ( v N ) κ 0 . (136) Here the first expression is functional calculus only on P 0 H P_0H P 0 H κ 0 \kappa_0 κ 0 z ↦ z ‾ z\mapsto\overline{z} z ↦ z ∗ * ∗ P 0 A e P 0 P_0A_eP_0 P 0 A e P 0 κ N \kappa_N κ N Q N A e Q N Q_NA_eQ_N Q N A e Q N
κ N θ c ( x ) = Θ e ( ι N ( x ) ) . (137) \kappa_N\theta_c(x)=\Theta_e(\iota_N(x)). \tag*{(137)} κ N θ c ( x ) = Θ e ( ι N ( x )) . (137) In particular it identifies the right fixed-point corners, without silently replacing them by left quotients. This is why the adjoint on n n n (136) matters.
Proposition 8.12 (Published C ∗ C^* C ∗ With the above conventions, define completely positive contractions
Γ N ( a ) : = ( κ N ⊗ κ N ) Δ c ( κ N − 1 ( Q N a Q N ) ) , a ∈ A e . (138) \Gamma_N(a)\mathrel{\mathop:}=(\kappa_N\otimes\kappa_N)\Delta_c\left(\kappa_N^{-1}(Q_NaQ_N)\right),\qquad a\in A_e. \tag*{(138)} Γ N ( a ) : = ( κ N ⊗ κ N ) Δ c ( κ N − 1 ( Q N a Q N ) ) , a ∈ A e . (138) Then ∥ Γ N ( a ) − Δ e ( a ) ∥ → 0 \lVert\Gamma_N(a)-\Delta_e(a)\rVert\to0 ∥ Γ N ( a ) − Δ e ( a )∥ → 0
Source and convention check. This is the real positive parameter instance of Woronowicz’s C ∗ C^* C ∗ [43 ] , explicitly stated as [RR21, Theorem 6.6, equation (6.7)]. In that source the braiding is trivial for q = r > 0 q=r>0 q = r > 0 γ ↦ n ∗ P 0 \gamma\mapsto n^*P_0 γ ↦ n ∗ P 0 Ad ( v N ) \operatorname{Ad}(v^N) Ad ( v N ) a a a
There is also a useful direct check on the norm topology of this input. Inside the initial corner write α = κ 0 ( b ) \alpha=\kappa_0(b) α = κ 0 ( b ) γ 0 = κ 0 ( γ ) \gamma_0=\kappa_0(\gamma) γ 0 = κ 0 ( γ ) t N = ( τ A ; τ ) N P 0 t_N=(\tau_A;\tau)_NP_0 t N = ( τ A ; τ ) N P 0 α N v − N = t N 1 2 \alpha^Nv^{-N}=t_N^{\frac12} α N v − N = t N 2 1 a 1 : = α ⊗ α a_1\mathrel{\mathop:}=\alpha\otimes\alpha a 1 : = α ⊗ α b 1 : = − r γ 0 ∗ ⊗ γ 0 b_1\mathrel{\mathop:}=-r\gamma_0^*\otimes\gamma_0 b 1 : = − r γ 0 ∗ ⊗ γ 0 a 1 b 1 = τ b 1 a 1 a_1b_1=\tau b_1a_1 a 1 b 1 = τ b 1 a 1 q q q [17 Proposition IV.2.2] , applied to the pair ( b 1 , a 1 ) (b_1,a_1) ( b 1 , a 1 ) a 1 b 1 = τ b 1 a 1 a_1b_1=\tau b_1a_1 a 1 b 1 = τ b 1 a 1
U N = Δ c ( α N ) ( v − N ⊗ v − N ) = ∑ j = 0 N [ N j ] τ b 1 j ( t N − j 1 2 ⊗ t N − j 1 2 ) ( v − j ⊗ v − j ) U_N=\Delta_c(\alpha^N)(v^{-N}\otimes v^{-N})=\sum_{j=0}^{N}\begin{bmatrix}N\\j\end{bmatrix}_{\tau}b_1^j(t_{N-j}^{\frac12}\otimes t_{N-j}^{\frac12})(v^{-j}\otimes v^{-j}) U N = Δ c ( α N ) ( v − N ⊗ v − N ) = j = 0 ∑ N [ N j ] τ b 1 j ( t N − j 2 1 ⊗ t N − j 2 1 ) ( v − j ⊗ v − j ) converges in norm: the j j j r j / ( τ ; τ ) ∞ r^j/(\tau;\tau)_\infty r j / ( τ ; τ ) ∞ a a a
Γ N ( a ) = U N ∗ Δ c ( t N − 1 2 a t N − 1 2 ) U N . \Gamma_N(a)=U_N^*\Delta_c(t_N^{-\frac12}at_N^{-\frac12})U_N. Γ N ( a ) = U N ∗ Δ c ( t N − 2 1 a t N − 2 1 ) U N . Thus the norm limit exists on the initial corner. The identity Γ N ( Ad ( v m ) a ) = Ad ( v m ⊗ v m ) Γ N − m ( a ) \Gamma_N(\operatorname{Ad}(v^m)a)=\operatorname{Ad}(v^m\otimes v^m)\Gamma_{N-m}(a) Γ N ( Ad ( v m ) a ) = Ad ( v m ⊗ v m ) Γ N − m ( a ) N ≥ m N\ge m N ≥ m Δ e \Delta_e Δ e ( a ; τ ) 0 (a;\tau)_0 ( a ; τ ) 0
Theorem 8.13 (Coactions and covariance). Let δ N = Γ N ∣ P e \delta_N=\Gamma_N|_{P_e} δ N = Γ N ∣ P e (137) . Then δ N ( x ) → δ e ( x ) \delta_N(x)\to\delta_e(x) δ N ( x ) → δ e ( x ) x ∈ P e x\in P_e x ∈ P e δ e ( P e ) ⊂ A e ⊗ P e \delta_e(P_e)\subset A_e\otimes P_e δ e ( P e ) ⊂ A e ⊗ P e
δ N S ~ t ( N ) = ( Id ⊗ S ~ t ( N ) ) δ N . (139) \delta_N\widetilde S_t^{(N)}=(\operatorname{Id}\otimes\widetilde S_t^{(N)})\delta_N. \tag*{(139)} δ N S t ( N ) = ( Id ⊗ S t ( N ) ) δ N . (139) Both sides converge in norm to δ e S t e = ( Id ⊗ S t e ) δ e \delta_eS_t^e=(\operatorname{Id}\otimes S_t^e)\delta_e δ e S t e = ( Id ⊗ S t e ) δ e t t t x ∈ P e x\in P_e x ∈ P e
sup 0 ≤ t ≤ T ∥ S ~ t ( N ) x − S t e x ∥ ≤ ∥ Γ N x − Δ e x ∥ + 2 ( 1 − e − c N T ) ∥ x ∥ . (140) \sup_{0\le t\le T}\left\|\widetilde S_t^{(N)}x-S_t^ex\right\|\le\left\|\Gamma_Nx-\Delta_ex\right\|+2\left(1-e^{-cNT}\right)\|x\|. \tag*{(140)} 0 ≤ t ≤ T sup S t ( N ) x − S t e x ≤ ∥ Γ N x − Δ e x ∥ + 2 ( 1 − e − c N T ) ∥ x ∥. (140) Proof: The dictionary (137) follows by applying κ N \kappa_N κ N θ c ( e i j ) = e i j ⊗ z j − i \theta_c(e_{ij})=e_{ij}\otimes z^{j-i} θ c ( e ij ) = e ij ⊗ z j − i e i − N , j − N ⊗ z j − i = Θ e ( e i − N , j − N ) e_{i-N,j-N}\otimes z^{j-i}=\Theta_e(e_{i-N,j-N}) e i − N , j − N ⊗ z j − i = Θ e ( e i − N , j − N ) δ N ( x ) \delta_N(x) δ N ( x ) A e ⊗ P e A_e\otimes P_e A e ⊗ P e
For x x x κ N \kappa_N κ N (139) . For a general x x x Q N Q_N Q N A e A_e A e { S t e ( x ) : 0 ≤ t ≤ T } \{S_t^e(x):0\le t\le T\} { S t e ( x ) : 0 ≤ t ≤ T } δ N − δ e \delta_N-\delta_e δ N − δ e δ e ( x ) \delta_e(x) δ e ( x ) A e ⊗ P e A_e \otimes P_e A e ⊗ P e Theorem 8.9 uniformly on [ 0 , T ] [0,T] [ 0 , T ] δ N ( x ) → δ e ( x ) \delta_N(x) \to\delta_e(x) δ N ( x ) → δ e ( x )
The normal state in (134) restricts to the compact entrance state on κ N ( A c ) \kappa_N(A_c) κ N ( A c ) μ τ N t c ( k ) \mu_{\tau_N t}^{c}(k) μ τ N t c ( k )
S ~ t ( N ) x = ( Id ⊗ ω ^ N , t ) Γ N ( x ) , S t e x = ( Id ⊗ ω t e ) Δ e ( x ) . \widetilde{S}^{(N)}_t x = (\operatorname{Id} \otimes\widehat{\omega}_{N,t})\Gamma_N(x), \qquad S^e_t x = (\operatorname{Id} \otimes\omega^e_t)\Delta_e(x). S t ( N ) x = ( Id ⊗ ω N , t ) Γ N ( x ) , S t e x = ( Id ⊗ ω t e ) Δ e ( x ) . Subtracting and using Corollary 8.11 proves (140) for t > 0 t>0 t > 0 A e A_e A e
Remark 8.14 The deformation parameter r r r N → ∞ N \to\infty N → ∞ S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) E r ( 2 ) \mathrm{E}_r(2) E r ( 2 ) Γ N \Gamma_N Γ N S ~ t ( N ) \widetilde{S}^{(N)}_t S t ( N )
Conclusions, dependencies and limits of the result
The central new analytic calculation is (106) , with the positive potential (108) . It yields the bounded band defect (115) and the full Mosco limit, selecting exactly the self-adjoint realization of Theorem 7.3 . The nonradial spherical coefficients converge in both GNS and algebra norm. Consequently the contraction extends to completely positive dynamics on the entire quotient algebra. The elementary rate comparison (133) gives quantitative convergence of the entrance laws and of the associated normal states. Adding the published quantum-group contraction makes the dynamics compatible with the limiting coaction.
The convergence is stronger than a radial special-function limit, but weaker than global Hilbert operator-norm convergence, which is explicitly false for the comparison conventions of the question. Uniform norm estimates are available after restricting the angular momentum to any fixed finite set. The zero-extension used to formulate a self-adjoint operator on a common Hilbert space must not be confused with a completely positive extension to the whole von Neumann algebra.
This section does not establish a uniform completely bounded norm rate, a classification of covariant generators, a higher-rank contraction, or an independent reconstruction of Woronowicz’s quantum-group contraction. The existence and integrated covariance of the limiting plane semigroup were established in § 8 ; the present arguments prove their compatibility with the compact limit. The C ∗ C^* C ∗
AI statement. The main mathematical ideas and the overall research direction originated with the author. The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions and feedback during the development and preparation of this manuscript.
Competing interests The author declares that he has no competing interests.
Data availability No data sets were generated during this study.
Bibliography
Cédric Arhancet
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ORCID: 0000-0002-5179-6972
