Introduction
Aref’eva and Volovich observed a coincidence between radial Haar integration on a non-Archimedean unit ball and the Haar state of the quantum group S U q ( 2 ) \mathrm{SU}_{q}(2) SU q ( 2 ) [2 Section 6.1, Proposition 1 and (71)–(75), p. 10] . Let K K K Q Q Q d ≥ 1 d \ge1 d ≥ 1 μ ( O d ) = 1 \mu(\mathcal{O}^{d})=1 μ ( O d ) = 1 K d K^{d} K d τ ≔ Q − d \tau\coloneqq Q^{-d} τ : = Q − d r ≔ τ 1 2 r \coloneqq\tau^{\frac{1}{2}} r : = τ 2 1 x ↦ ∥ x ∥ K d x \mapsto\lVert x\rVert_{K}^{d} x ↦ ∥ x ∥ K d
∫ O d F ( ∥ x ∥ K d ) d x = ( 1 − τ ) ∑ n ≥ 0 τ n F ( τ n ) = h ( F ( A ) ) , A ≔ γ ∗ γ ∈ C ( S U r ( 2 ) ) . (1) \int_{\mathcal{O}^{d}} F(\lVert x\rVert_{K}^{d})\,\mathrm{d}x=(1-\tau)\sum_{n\ge0}\tau^{n}F(\tau^{n})=h(F(A)),\qquad A\coloneqq\gamma^{*}\gamma\in C(\mathrm{SU}_{r}(2)). \tag*{(1)} ∫ O d F (∥ x ∥ K d ) d x = ( 1 − τ ) n ≥ 0 ∑ τ n F ( τ n ) = h ( F ( A )) , A : = γ ∗ γ ∈ C ( SU r ( 2 )) . (1) Here h h h F F F
X τ ≔ { τ n : n ≥ 0 } ∪ { 0 } . (2) X_{\tau}\coloneqq\{\tau^{n}:n\ge0\}\cup\{0\}. \tag*{(2)} X τ : = { τ n : n ≥ 0 } ∪ { 0 } . (2) Thus r 2 = Q − d r^{2}=Q^{-d} r 2 = Q − d r = Q − 1 2 r=Q^{-\frac{1}{2}} r = Q − 2 1 1 ∗ * ∗ J J J W ∗ ( A ) \mathrm{W}^{*}(A) W ∗ ( A ) J ( F ( ∥ ⋅ ∥ K d ) ) = F ( A ) J(F(\lVert\cdot\rVert_{K}^{d}))=F(A) J ( F (∥ ⋅ ∥ K d )) = F ( A )
Identities of this kind belong to a broader picture relating harmonic analysis over local fields and q q q G L n \mathrm{GL}_{n} GL n [50 Chapter VI] , which are the specializations q = 0 q=0 q = 0 [50 Chapter VI] . Freund and Zabrodin proposed Macdonald polynomials, in which the p p p q = 0 q = 0 q = 0 t = p − 1 t = p^{-1} t = p − 1 p p p [30 Section 3] ; see also [19 Section 14] . Haran interpreted limits q → 0 q \to0 q → 0 q q q q q q p p p [38 ] , and developed a q q q p p p [39 Chapters 2–4] ; see also [48 ] and [58 ] . In these works the p p p q → 0 q \to0 q → 0 r r r
The principal conclusions are organized around four mechanisms. First, the measure correspondence extends to a comparison of closed energies, with sharp thresholds for the powers of the zonal Casimir. Second, the comparison has a boundary interpretation through the tree of balls and the random walk on the dual of S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 )
The analytic input is the regional Taibleson operator R α R^\alpha R α α > 0 \alpha> 0 α > 0 O d \mathcal{O}^d O d ∥ ξ ∥ K α \lVert\xi\rVert_K^\alpha ∥ ξ ∥ K α (8) . For d = α = 1 d = \alpha= 1 d = α = 1 Q p \mathbb{Q}_p Q p [37 ] . Its restriction R r a d α R^\alpha_{\mathrm{rad}} R rad α ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) w n ≔ ( 1 − τ ) τ n w_n \coloneqq(1-\tau)\tau^n w n : = ( 1 − τ ) τ n 0 0 0
λ n = ρ n + 1 − κ , n ≥ 0 , ρ ≔ Q α , κ ≔ 1 − τ 1 − ρ . \lambda_n = \rho^{n+1} - \kappa,\qquad n \ge0,\qquad\rho\coloneqq Q^\alpha,\qquad\kappa\coloneqq\frac{1-\tau}{1-\rho}. λ n = ρ n + 1 − κ , n ≥ 0 , ρ : = Q α , κ : = 1 − ρ 1 − τ . The eigenfunctions are Haar-type wavelets. Up to normalization they are the little 0-Laguerre functions which appear as spherical functions of G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O ) O d \mathcal{O}^d O d [48 Section 5.2] .
On the quantum side we consider, for 0 < η < 1 0 < \eta< 1 0 < η < 1
( L η f ) ( k ) = τ η k − 1 ( 1 − η k + 1 ) ( f ( k ) − f ( k + 1 ) ) + η − k ( 1 − η k ) ( f ( k ) − f ( k − 1 ) ) , k ≥ 0 , (3) (L_\eta f)(k) = \tau\eta^{k-1}(1-\eta^{k+1})(f(k)-f(k+1))+\eta^{-k}(1-\eta^k)(f(k)-f(k-1)),\qquad k \ge0, \tag*{(3)} ( L η f ) ( k ) = τ η k − 1 ( 1 − η k + 1 ) ( f ( k ) − f ( k + 1 )) + η − k ( 1 − η k ) ( f ( k ) − f ( k − 1 )) , k ≥ 0 , (3) acting on the same space ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) w w w η \eta η k ↦ p n ( η k ; τ η − 1 , 1 ; η ) k \mapsto p_n(\eta^k;\tau\eta^{-1},1;\eta) k ↦ p n ( η k ; τ η − 1 , 1 ; η ) [44 Section 14.12] , and its eigenvalues are
Λ n ( η ) = η − n ( 1 − η n ) ( 1 − τ η n ) , n ≥ 0. (4) \Lambda_n(\eta)=\eta^{-n}(1-\eta^n)(1-\tau\eta^n),\qquad n\ge0. \tag*{(4)} Λ n ( η ) = η − n ( 1 − η n ) ( 1 − τ η n ) , n ≥ 0. (4) For η = τ \eta=\tau η = τ L τ L_\tau L τ S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) τ \tau τ A A A [47 ] , [MMNN91] and Proposition 5.3 , and Λ l ( τ ) \Lambda_l(\tau) Λ l ( τ ) l l l
Theorem 1.1 (Energy bridge). Let α > 0 \alpha> 0 α > 0 η ≔ Q − α \eta\coloneqq Q^{-\alpha} η : = Q − α E R \mathcal{E}_{R} E R E η \mathcal{E}_{\eta} E η R r a d α R_{\mathrm{rad}}^{\alpha} R rad α L η L_{\eta} L η
dom E R = dom E η = { f ∈ ℓ N 0 2 ( w ) : ∑ k ≥ 0 Q ( α − d ) k ∣ f ( k + 1 ) − f ( k ) ∣ 2 < ∞ } . \operatorname{dom}\mathcal{E}_{R}=\operatorname{dom}\mathcal{E}_{\eta}=\left\{f\in\ell^2_{\mathbb{N}_0}(w):\sum_{k\ge0}Q^{(\alpha-d)k}\lvert f(k+1)-f(k)\rvert^2<\infty\right\}. dom E R = dom E η = { f ∈ ℓ N 0 2 ( w ) : k ≥ 0 ∑ Q ( α − d ) k ∣ f ( k + 1 ) − f ( k ) ∣ 2 < ∞ } . and there exist constants 0 < c 1 ≤ c 2 0<c_1\le c_2 0 < c 1 ≤ c 2 Q Q Q d d d α \alpha α
c 1 E η ( f ) ≤ E R ( f ) ≤ c 2 E η ( f ) c_1\mathcal{E}_{\eta}(f)\le\mathcal{E}_{R}(f)\le c_2\mathcal{E}_{\eta}(f) c 1 E η ( f ) ≤ E R ( f ) ≤ c 2 E η ( f ) for any f f f n ≥ 0 n\ge0 n ≥ 0
λ n = Λ n + 1 ( η ) + 1 + τ − κ − τ ρ − n − 1 . (5) \lambda_n=\Lambda_{n+1}(\eta)+1+\tau-\kappa-\tau\rho^{-n-1}. \tag*{(5)} λ n = Λ n + 1 ( η ) + 1 + τ − κ − τ ρ − n − 1 . (5) If α = d \alpha=d α = d η = τ \eta=\tau η = τ 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 ϑ > 0 \vartheta>0 ϑ > 0
s ≔ α ϑ . s\coloneqq\frac{\alpha}{\vartheta}. s : = ϑ α . Then dom E R ⊂ dom L Q − ϑ 2 \operatorname{dom}\mathcal{E}_{R}\subset\operatorname{dom}L^2_{Q^{-\vartheta}} dom E R ⊂ dom L Q − ϑ 2 R r a d α R_{\mathrm{rad}}^{\alpha} R rad α L Q − ϑ L_{Q^{-\vartheta}} L Q − ϑ α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ R r a d α R_{\mathrm{rad}}^{\alpha} R rad α L τ d L_{\tau}^{d} L τ d α < 3 d \alpha<3d α < 3 d
The two operators of Theorem 1.1 have no common eigenvector besides the constants: the first is diagonal in a basis of step functions, the second in a basis of polynomials in η k \eta^k η k Remark 4.3 ). The theorem therefore does not follow from a spectral identification; it is proved by reducing both forms to weighted nearest-neighbour energies. For α = d \alpha=d α = d r 2 = Q − d r^2=Q^{-d} r 2 = Q − d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) L Q − ϑ L_{Q^{-\vartheta}} L Q − ϑ α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ L τ L_{\tau} L τ r 2 = Q − d r^2=Q^{-d} r 2 = Q − d α < 3 d \alpha<3d α < 3 d η = Q − ϑ \eta=Q^{-\vartheta} η = Q − ϑ η \eta η ( τ − 2 η ) n − m (\tau^{-2}\eta)^{n-m} ( τ − 2 η ) n − m Lemma 4.5 ). The threshold d + 2 ϑ d+2\vartheta d + 2 ϑ k ↦ η k k\mapsto\eta^k k ↦ η k η k \eta^k η k α ≥ d + 2 ϑ \alpha\ge d+2\vartheta α ≥ d + 2 ϑ Proposition 4.7 ).
The energy bridge has a probabilistic interpretation for α = d \alpha=d α = d T \mathbb{T} T O d \mathcal{O}^d O d O d \mathcal{O}^d O d T \mathbb{T} T G ^ \widehat{G} G G = S U r ( 2 ) G=\mathrm{SU}_r(2) G = SU r ( 2 ) P = ( Id ⊗ ϕ ) ∘ Δ ^ P=(\operatorname{Id}\otimes\phi)\circ\widehat{\Delta} P = ( Id ⊗ ϕ ) ∘ Δ ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) ϕ \phi ϕ [40 ] , see also [41 ] , the Poisson boundary of this random walk is the Podleś sphere: the Poisson integral Θ \Theta Θ L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) a a a n = 2 s n=2s n = 2 s E ( n ) ( a ) = [ 2 s + 1 ] r Tr ( K − 1 ( P ( a ∗ a ) − a ∗ a ) s ) \mathcal{E}^{(n)}(a)=[2s+1]_r\operatorname{Tr}(K^{-1}(P(a^*a)-a^*a)_s) E ( n ) ( a ) = [ 2 s + 1 ] r Tr ( K − 1 ( P ( a ∗ a ) − a ∗ a ) s ) ∑ n E ( n ) ( a ) \sum_n\mathcal{E}^{(n)}(a) ∑ n E ( n ) ( a ) (51) .
Theorem 1.2 (Boundary energies). Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 c τ = 2 τ ( 1 + τ ) ( 1 − τ ) 2 c_{\tau}=\frac{2\tau}{(1+\tau)(1-\tau)^2} c τ = ( 1 + τ ) ( 1 − τ ) 2 2 τ
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d F ∈ L 2 ( O d ) F\in L^2(\mathcal{O}^d) F ∈ L 2 ( O d ) F F F T \mathbb{T} T 2 ( 1 − τ ) ⟨ F , M d F ⟩ 2(1-\tau)\langle F,\mathcal{M}^dF\rangle 2 ( 1 − τ ) ⟨ F , M d F ⟩ M d \mathcal{M}^d M d O d \mathcal{O}^d O d ∥ ξ ∥ K d \lVert\xi\rVert_K^d ∥ ξ ∥ K d
The restriction of P P P ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) { ( n , k ) ∈ N 0 2 : k ≤ n } \{(n,k) \in\mathbb{N}_0^2 : k \le n\} {( n , k ) ∈ N 0 2 : k ≤ n } k ∞ k_\infty k ∞ k ∞ k_\infty k ∞ ( 1 − τ ) τ k (1-\tau)\tau^k ( 1 − τ ) τ k
For any x ∈ L ∞ ( S r 2 ) x \in L^\infty(S_r^2) x ∈ L ∞ ( S r 2 ) lim n → ∞ E ( n ) ( Θ ( x ) ) = c τ ∑ l Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 \lim_{n\to\infty} \mathcal{E}^{(n)}(\Theta(x)) = c_\tau\sum_l \Lambda_l(\tau) \lVert x_l\rVert_{L^2(S_r^2)}^2 lim n → ∞ E ( n ) ( Θ ( x )) = c τ ∑ l Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] x l x_l x l l l l x x x sup n E ( n ) ( Θ ( x ) ) \sup_n \mathcal{E}^{(n)}(\Theta(x)) sup n E ( n ) ( Θ ( x )) lim n → ∞ E ( n ) ( Θ ( f ( A ) ) ) = c τ E τ ( f ) \lim_{n\to\infty} \mathcal{E}^{(n)}(\Theta(f(A))) = c_\tau\mathcal{E}_\tau(f) lim n → ∞ E ( n ) ( Θ ( f ( A ))) = c τ E τ ( f ) f : N 0 → C f:\mathbb{N}_0 \to\mathbb{C} f : N 0 → C H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) 1 1 1
These assertions are proved in Theorem 6.1 , Theorem 6.10 and Corollary 6.32 . The transition probabilities of the diagonal chain were computed by Biane [15 ] , Section 4, in his approach to Pitman’s theorem through S U q ( 2 ) \mathrm{SU}_q(2) SU q ( 2 ) k ∞ k_\infty k ∞ Proposition 6.11 ). For τ = Q − d \tau=Q^{-d} τ = Q − d 2 ( 1 − τ ) 2(1-\tau) 2 ( 1 − τ ) κ ( I − E 0 ) \kappa(I-E_0) κ ( I − E 0 ) R r a d d R_{\mathrm{rad}}^d R rad d c τ E τ c_\tau\mathcal{E}_\tau c τ E τ Theorem 1.1 for α = d \alpha=d α = d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Q d + 1 Q^d+1 Q d + 1 w k β k w_k\beta_k w k β k E τ \mathcal{E}_\tau E τ Proposition 6.4 ), so that each level contributes the same nearest-neighbour energy. The proofs rely on an algebraic property of the diagonal chain: it preserves, on each level, the polynomials of degree at most l l l τ k \tau^k τ k Lemma 6.5 ). This leads to an explicit description of the Poisson integrals of the zonal spherical functions by q q q Proposition 6.9 ), from which we deduce the second part without using the theorem of Izumi. The coefficients of this description have a generating function of basic hypergeometric type, and their limits are Gaussian binomial coefficients (Proposition 6.12 ). They give an exact formula for the energy per level of these Poisson integrals, which is bounded by a multiple of the Casimir eigenvalue uniformly in the level (Proposition 6.30 ); this gives the third part for every element of L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) ( n − k , k ) (n-k,k) ( n − k , k ) E r \mathcal{E}_r E r Theorem 6.13 ). The entire function which appears is a Jacobi theta function and the Fredholm determinant of the Green operator of the vertical network, whose eigenvalues are proportional to the squares of the quantum dimensions [ n ] r [n]_r [ n ] r Theorems 6.15 , 6.17 and 6.22). In particular, the harmonic measures seen from the vertices ( k , k ) (k,k) ( k , k ) Theorem 6.26 ), and the Poisson integral of a bounded function converges to its boundary values at the optimal rate ( n − k + 1 ) τ n − k (n-k+1)\tau^{n-k} ( n − k + 1 ) τ n − k Corollary 6.24 ).
The second step concerns quantum dynamics. A first attempt consists in lifting radial dynamics to quantum Markov semigroups on L ∞ ( S U r ( 2 ) ) L^\infty(\mathrm{SU}_r(2)) L ∞ ( SU r ( 2 )) W ∗ ( A ) W^*(A) W ∗ ( A ) L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) S r 2 = S U r ( 2 ) / T S_r^2=\mathrm{SU}_r(2)/\mathbb{T} S r 2 = SU r ( 2 ) / T L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) B ( ℓ 0 2 ) B(\ell_0^2) B ( ℓ 0 2 ) π \pi π Tr ( ( 1 − τ ) τ N ⋅ ) \operatorname{Tr}((1-\tau)\tau^N\cdot) Tr (( 1 − τ ) τ N ⋅ ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) Proposition 5.5 ). Let P n : = 1 [ 0 , τ n ] ( A ) P_n \mathrel{\mathop:}= 1_{[0,\tau^n]}(A) P n : = 1 [ 0 , τ n ] ( A ) B n = π n O d B_n=\pi^n\mathcal{O}^d B n = π n O d P n = J ( 1 B n ) P_n=J(1_{B_n}) P n = J ( 1 B n )
Φ n ( x ) = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n \Phi_n(x)=(1-P_n)x(1-P_n)+\tau^{-n}h(P_nxP_n)P_n Φ n ( x ) = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n @page 5/85 starts_mid=1 ends_mid=1
The restriction of P P P ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) { ( n , k ) ∈ N 0 2 : k ≤ n } \{(n,k) \in\mathbb{N}_0^2 : k \le n\} {( n , k ) ∈ N 0 2 : k ≤ n } k ∞ k_\infty k ∞ k ∞ k_\infty k ∞ ( 1 − τ ) τ k (1-\tau)\tau^k ( 1 − τ ) τ k
For any x ∈ L ∞ ( S r 2 ) x \in L^\infty(S_r^2) x ∈ L ∞ ( S r 2 ) lim n → ∞ E ( n ) ( Θ ( x ) ) = c τ ∑ l Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 \lim_{n\to\infty} \mathcal{E}^{(n)}(\Theta(x)) = c_\tau\sum_l \Lambda_l(\tau) \lVert x_l\rVert_{L^2(S_r^2)}^2 lim n → ∞ E ( n ) ( Θ ( x )) = c τ ∑ l Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] x l x_l x l l l l x x x sup n E ( n ) ( Θ ( x ) ) \sup_n \mathcal{E}^{(n)}(\Theta(x)) sup n E ( n ) ( Θ ( x )) lim n → ∞ E ( n ) ( Θ ( f ( A ) ) ) = c τ E τ ( f ) \lim_{n\to\infty} \mathcal{E}^{(n)}(\Theta(f(A))) = c_\tau\mathcal{E}_\tau(f) lim n → ∞ E ( n ) ( Θ ( f ( A ))) = c τ E τ ( f ) f : N 0 → C f:\mathbb{N}_0 \to\mathbb{C} f : N 0 → C H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) 1 1 1
These assertions are proved in Theorem 6.1 , Theorem 6.10 and Corollary 6.32 . The transition probabilities of the diagonal chain were computed by Biane [15 ] , Section 4, in his approach to Pitman’s theorem through S U q ( 2 ) \mathrm{SU}_q(2) SU q ( 2 ) k ∞ k_\infty k ∞ Proposition 6.11 ). For τ = Q − d \tau=Q^{-d} τ = Q − d 2 ( 1 − τ ) 2(1-\tau) 2 ( 1 − τ ) κ ( I − E 0 ) \kappa(I-E_0) κ ( I − E 0 ) R r a d d R_{\mathrm{rad}}^d R rad d c τ E τ c_\tau\mathcal{E}_\tau c τ E τ Theorem 1.1 for α = d \alpha=d α = d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Q d + 1 Q^d+1 Q d + 1 w k β k w_k\beta_k w k β k E τ \mathcal{E}_\tau E τ Proposition 6.4 ), so that each level contributes the same nearest-neighbour energy. The proofs rely on an algebraic property of the diagonal chain: it preserves, on each level, the polynomials of degree at most l l l τ k \tau^k τ k Lemma 6.5 ). This leads to an explicit description of the Poisson integrals of the zonal spherical functions by q q q Proposition 6.9 ), from which we deduce the second part without using the theorem of Izumi. The coefficients of this description have a generating function of basic hypergeometric type, and their limits are Gaussian binomial coefficients (Proposition 6.12 ). They give an exact formula for the energy per level of these Poisson integrals, which is bounded by a multiple of the Casimir eigenvalue uniformly in the level (Proposition 6.30 ); this gives the third part for every element of L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) ( n − k , k ) (n-k,k) ( n − k , k ) E r \mathcal{E}_r E r Theorem 6.13 ). The entire function which appears is a Jacobi theta function and the Fredholm determinant of the Green operator of the vertical network, whose eigenvalues are proportional to the squares of the quantum dimensions [ n ] r [n]_r [ n ] r Theorems 6.15 , 6.17 and 6.22). In particular, the harmonic measures seen from the vertices ( k , k ) (k,k) ( k , k ) Theorem 6.26 ), and the Poisson integral of a bounded function converges to its boundary values at the optimal rate ( n − k + 1 ) τ n − k (n-k+1)\tau^{n-k} ( n − k + 1 ) τ n − k Corollary 6.24 ).
The second step concerns quantum dynamics. A first attempt consists in lifting radial dynamics to quantum Markov semigroups on L ∞ ( S U r ( 2 ) ) L^\infty(\mathrm{SU}_r(2)) L ∞ ( SU r ( 2 )) W ∗ ( A ) W^*(A) W ∗ ( A ) L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) S r 2 = S U r ( 2 ) / T S_r^2=\mathrm{SU}_r(2)/\mathbb{T} S r 2 = SU r ( 2 ) / T L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) B ( ℓ 0 2 ) B(\ell_0^2) B ( ℓ 0 2 ) π \pi π Tr ( ( 1 − τ ) τ N ⋅ ) \operatorname{Tr}((1-\tau)\tau^N\cdot) Tr (( 1 − τ ) τ N ⋅ ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) Proposition 5.5 ). Let P n : = 1 [ 0 , τ n ] ( A ) P_n \mathrel{\mathop:}= 1_{[0,\tau^n]}(A) P n : = 1 [ 0 , τ n ] ( A ) B n = π n O d B_n=\pi^n\mathcal{O}^d B n = π n O d P n = J ( 1 B n ) P_n=J(1_{B_n}) P n = J ( 1 B n )
Φ n ( x ) = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n \Phi_n(x)=(1-P_n)x(1-P_n)+\tau^{-n}h(P_nxP_n)P_n Φ n ( x ) = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n are Haar-preserving conditional expectations, and probability mixtures of them produce quantum Markov semigroups. Exact liftings of the Taibleson dynamics are obtained in this way; constructions from increasing conditional expectations also occur in [76 Theorems 1.1 and 1.7] . Such liftings do not use the coproduct, and on C ( S U r ( 2 ) ) C(SU_r(2)) C ( S U r ( 2 ))
Theorem 1.3 (Quantum dynamics on the Podleś sphere) Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2
There exists a Haar-preserving, GNS-symmetric and KMS-symmetric quantum Markov semigroup ( T t ) t ≥ 0 (\mathcal{T}_t)_{t\geq0} ( T t ) t ≥ 0 L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) h h h C ( S r 2 ) C(S_r^2) C ( S r 2 ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0 T t J = J e − t R r a d α \mathcal{T}_tJ=Je^{-tR_{\mathrm{rad}}^\alpha} T t J = J e − t R rad α t ≥ 0 t\geq0 t ≥ 0
There exists a Haar-preserving semigroup ( S t ) t ≥ 0 (\mathcal{S}_t)_{t\geq0} ( S t ) t ≥ 0 C ( S r 2 ) C(S_r^2) C ( S r 2 ) S t = ( Id ⊗ ω t ) ∘ Δ \mathcal{S}_t=(\operatorname{Id}\otimes\omega_t)\circ\Delta S t = ( Id ⊗ ω t ) ∘ Δ ω t \omega_t ω t S t ( u j 0 ( l ) ) = e − t Λ l ( τ ) u j 0 ( l ) \mathcal{S}_t(u_{j_0}^{(l)})=e^{-t\Lambda_l(\tau)}u_{j_0}^{(l)} S t ( u j 0 ( l ) ) = e − t Λ l ( τ ) u j 0 ( l ) u j 0 ( l ) u_{j_0}^{(l)} u j 0 ( l ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) C ∗ ( A ) C^*(A) C ∗ ( A ) e − t L τ e^{-tL_\tau} e − t L τ
Every strongly continuous semigroup ( S t ) t ≥ 0 (S_t)_{t\geq0} ( S t ) t ≥ 0 C ( S r 2 ) C(S_r^2) C ( S r 2 ) Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ \Delta\circ S_t=(\operatorname{Id}\otimes S_t)\circ\Delta Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ C ∗ ( A ) C^*(A) C ∗ ( A ) τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0 R r a d α R_{\mathrm{rad}}^\alpha R rad α α ≤ d \alpha\leq d α ≤ d α ≤ d \alpha\leq d α ≤ d ( S t ) (\mathcal{S}_t) ( S t ) λ ↦ λ α / d \lambda\mapsto\lambda^{\alpha/d} λ ↦ λ α / d L τ α / d L_\tau^{\alpha/d} L τ α / d
These assertions are proved in Theorems 7.2 , 7.7 and 7.10 and in Corollary 7.8. In part 1, the semigroup is a probability mixture of the expectations Φ n \Phi_n Φ n L p L^p L p H ∞ H^\infty H ∞ Theorem 7.4 ). In part 2, positivity is not visible on the Casimir eigenvalues. It is obtained from the radial semigroup itself: the law at time t t t 0 ∈ X τ 0\in X_\tau 0 ∈ X τ μ t \mu_t μ t ω t = μ t ∘ E \omega_t=\mu_t\circ E ω t = μ t ∘ E E E E S U r ( 2 ) SU_r(2) S U r ( 2 ) α > d \alpha>d α > d Proposition 7.9 ), whereas, by Theorem 1.1 , the Taibleson energy of order α \alpha α Λ l ( τ ) α / d \Lambda_l(\tau)^{\alpha/d} Λ l ( τ ) α / d d < α < 3 d d<\alpha<3d d < α < 3 d L τ α / d L_\tau^{\alpha/d} L τ α / d
The third step explains why S U r ( 2 ) SU_r(2) S U r ( 2 ) b b b C ( S U r ( 2 ) ) C(SU_r(2)) C ( S U r ( 2 )) α \alpha α [54 ] and [21 ] . It does not belong to C ( S r 2 ) C(S_r^2) C ( S r 2 )
Theorem 1.4 (Frontier between S r 2 S_r^2 S r 2 S U r ( 2 ) SU_r(2) S U r ( 2 ) Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d ( S t ) t ≥ 0 (S_t)_{t\geq0} ( S t ) t ≥ 0 C ( S U r ( 2 ) ) C(SU_r(2)) C ( S U r ( 2 )) C ∗ ( A ) C^*(A) C ∗ ( A ) J e − t R r a d α J − 1 Je^{-tR_{\mathrm{rad}}^\alpha}J^{-1} J e − t R rad α J − 1 ∥ S ρ − j ( b ) − b ∥ ≥ δ > 0 \|S_{\rho-j}(b)-b\|\geq\delta>0 ∥ S ρ − j ( b ) − b ∥ ≥ δ > 0 j ≥ 1 j\geq1 j ≥ 1 δ \delta δ j j j
There exists δ ′ > 0 \delta' > 0 δ ′ > 0 S : C ( S U r ( 2 ) ) → C ( S U r ( 2 ) ) S: C(\mathrm{SU}_r(2)) \to C(\mathrm{SU}_r(2)) S : C ( SU r ( 2 )) → C ( SU r ( 2 )) C ∗ ( A ) C^*(A) C ∗ ( A ) e − t L τ e^{-tL_\tau} e − t L τ 0 < t ≤ Λ 1 ( τ ) − 1 0 < t \le\Lambda_1(\tau)^{-1} 0 < t ≤ Λ 1 ( τ ) − 1 ∥ S ( b ) − b ∥ ≥ δ ′ \lVert S(b)-b\rVert\ge\delta' ∥ S ( b ) − b ∥ ≥ δ ′
There exists t 0 > 0 t_0 > 0 t 0 > 0 0 < t ≤ t 0 0 < t \le t_0 0 < t ≤ t 0 S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) u j k ( l ) ↦ e − t Λ l u j k ( l ) u_{jk}^{(l)} \mapsto e^{-t\Lambda_l}u_{jk}^{(l)} u j k ( l ) ↦ e − t Λ l u j k ( l ) l ∈ 1 2 N 0 l \in\frac{1}{2}\mathbb{N}_0 l ∈ 2 1 N 0 C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) S r 2 S_r^2 S r 2 P t \mathcal{P}_t P t Theorem 1.3 .
Here Λ l = r − 2 l − 1 − r 2 + r 2 l + 2 \Lambda_l=r^{-2l}-1-r^2+r^{2l+2} Λ l = r − 2 l − 1 − r 2 + r 2 l + 2 l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_0 l ∈ 2 1 N 0 Λ l ( τ ) \Lambda_l(\tau) Λ l ( τ ) l l l b b b Lemma 8.1 ). Both the Taibleson and the Casimir dynamics have entrance laws with a persistent atom along times tending to zero; for the Casimir dynamics we prove this by a Paley–Zygmund argument based on the first two zonal spherical functions (Proposition 8.5). Theorem 1.4 is proved in Corollary 8.4 and Theorem 8.6 . Part 3 shows in particular that the heat semigroup of the Casimir operator is not a quantum Markov semigroup on C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) S U q ( 2 ) \mathrm{SU}_q(2) SU q ( 2 ) [65 ] and [21 Section 11] .
The last step relates the fixed-parameter picture with the limits q → 0 q\to0 q → 0 [38 ] , [39 ] and [48 ] . We denote by M α M^\alpha M α O d \mathcal{O}^d O d ∥ ξ ∥ K \lVert\xi\rVert_K ∥ ξ ∥ K M r a d α = R r a d α + κ ( I − E 0 ) M^\alpha_{\mathrm{rad}}=R^\alpha_{\mathrm{rad}}+\kappa(I-E_0) M rad α = R rad α + κ ( I − E 0 ) E 0 E_0 E 0
Theorem 1.5 (Crystal limit) Let α > 0 \alpha>0 α > 0 0 < η ≤ Q − α 0<\eta\le Q^{-\alpha} 0 < η ≤ Q − α s ( η ) ≔ α log Q log 1 η ∈ ( 0 , 1 ] s(\eta)\coloneqq\frac{\alpha\log Q}{\log\frac{1}{\eta}}\in(0,1] s ( η ) : = l o g η 1 α l o g Q ∈ ( 0 , 1 ] ( e − t L η s ( η ) ) t ≥ 0 (e^{-tL^{s(\eta)}_\eta})_{t\ge0} ( e − t L η s ( η ) ) t ≥ 0 L η s ( η ) L^{s(\eta)}_\eta L η s ( η ) M r a d α M^\alpha_{\mathrm{rad}} M rad α η → 0 + \eta\to0^+ η → 0 + t > 0 t>0 t > 0
The convergence of the eigenfunctions is a variant of the limits q → 0 q\to0 q → 0 q q q [38 ] and Koornwinder–Onn [48 Theorem 2.1] , in which the limit functions are the little 0 0 0 G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O ) O d \mathcal{O}^d O d [48 Section 5.2] . The new point is the convergence of Markov generators, which requires the subordination exponent s ( η ) → 0 s(\eta)\to0 s ( η ) → 0 η = τ \eta=\tau η = τ Theorem 1.3 , and for η = Q − α \eta=Q^{-\alpha} η = Q − α Theorem 1.1 .
Finally, the quantum group S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) r 2 = Q − d r^2=Q^{-d} r 2 = Q − d C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 U q ( n ) U_q(n) U q ( n ) U q ( n − 1 ) × T U_q(n-1)\times\mathbb{T} U q ( n − 1 ) × T [57 ] , its zonal spherical functions are little q 2 q^2 q 2 L η L_\eta L η η = q 2 \eta=q^2 η = q 2 τ = q 2 ( n − 1 ) \tau=q^{2(n-1)} τ = q 2 ( n − 1 )
Theorem 1.6 (Quantum projective spaces) Let n ≥ 2 n\ge2 n ≥ 2 0 < q < 1 0<q<1 0 < q < 1 q 2 ( n − 1 ) = Q − d q^{2(n-1)}=Q^{-d} q 2 ( n − 1 ) = Q − d q 2 = Q − α q^2=Q^{-\alpha} q 2 = Q − α α = d n − 1 \alpha=\frac{d}{n-1} α = n − 1 d O d \mathcal{O}^d O d C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 L Q − α L_{Q^{-\alpha}} L Q − α C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 ) α \alpha α n = d + 1 n=d+1 n = d + 1 1 1 1 O d \mathcal{O}^d O d C P q d \mathbb{C}P_q^d C P q d q = Q − 1 2 q=Q^{-\frac{1}{2}} q = Q − 2 1 α ′ > 0 \alpha'>0 α ′ > 0 L Q − α α ′ α L_{Q^{-\alpha}}^{\frac{\alpha'}{\alpha}} L Q − α α α ′ α ′ \alpha' α ′ α ′ < d + 2 α \alpha'<d+2\alpha α ′ < d + 2 α
there exists a covariant strongly continuous semigroup of unital completely positive maps on C ( C P q n − 1 ) C(\mathbb{C}\mathrm{P}_{q}^{n-1}) C ( C P q n − 1 ) α ′ \alpha' α ′ α ′ ≤ α \alpha' \le\alpha α ′ ≤ α S q 2 n − 1 S_{q}^{2n-1} S q 2 n − 1 U q ( n ) / U q ( n − 1 ) U_q(n)/U_q(n-1) U q ( n ) / U q ( n − 1 ) C P q n − 1 \mathbb{C}\mathrm{P}_{q}^{n-1} C P q n − 1 x n n x_{nn} x nn C ( S q 2 n − 1 ) C(S_{q}^{2n-1}) C ( S q 2 n − 1 ) C ( C P q n − 1 ) C(\mathbb{C}\mathrm{P}_{q}^{n-1}) C ( C P q n − 1 ) δ > 0 \delta>0 δ > 0 S S S C ( S q 2 n − 1 ) C(S_{q}^{2n-1}) C ( S q 2 n − 1 ) 0 < t ≤ Λ 1 ( Q − α ′ ) 0<t\le\Lambda_1(Q^{-\alpha'}) 0 < t ≤ Λ 1 ( Q − α ′ ) ∥ S ( x n n ) − x n n ∥ ≥ δ \|S(x_{nn})-x_{nn}\|\ge\delta ∥ S ( x nn ) − x nn ∥ ≥ δ Q − α ′ j Q^{-\alpha'j} Q − α ′ j j ≥ 1 j\ge1 j ≥ 1 α ′ > 0 \alpha'>0 α ′ > 0
This is proved in Theorems 10.3, 10.5 and 10.8 . For n = d + 1 n=d+1 n = d + 1 α ′ < d + 2 \alpha'<d+2 α ′ < d + 2 O d \mathcal{O}_d O d L Q − 1 α ′ L_{Q^{-1}}^{\alpha'} L Q − 1 α ′ C P q d \mathbb{C}\mathrm{P}_q^d C P q d α ′ ≤ 1 \alpha'\le1 α ′ ≤ 1 n = 2 n=2 n = 2 α = d \alpha=d α = d Theorems 1.1 and 1.3 . For n = d + 1 n=d+1 n = d + 1 q = Q − 1 2 q=Q^{-\frac12} q = Q − 2 1 d d d d d d O d \mathcal{O}^d O d t = Q − 1 t=Q^{-1} t = Q − 1 [30 Section 3] .
The present paper is confined to the compact setting of the unit ball, compact quantum groups and their compact homogeneous spaces. The noncompact counterpart for the whole local field K d K^d K d E r ( 2 ) E_r(2) E r ( 2 ) [1 ] . None of the arguments below depends on that noncompact construction; the compact results established here instead provide its starting point.
Structure of the paper Section 2 gives the radial analysis of the Taibleson operator on the unit ball. Section 3 studies the little η \eta η Theorem 1.1 . Section 5 recalls the quantum group S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Theorem 1.2 . Section 7 constructs the quantum dynamics of Theorem 1.3 , and Section 8 proves Theorem 1.4 . Section 9 proves Theorem 1.5. Section 10 treats the quantum projective spaces and proves Theorem 1.6.
Radial analysis on the unit ball
Parameters, balls, and Fourier normalization
Let K K K O \mathcal{O} O π \pi π Q = # ( O / π O ) Q=\#(\mathcal{O}/\pi\mathcal{O}) Q = # ( O / π O ) ∣ π ∣ K = Q − 1 |\pi|_K=Q^{-1} ∣ π ∣ K = Q − 1 K d K^d K d d ≥ 1 d\ge1 d ≥ 1 d x dx d x μ ( O d ) = 1 \mu(\mathcal{O}^d)=1 μ ( O d ) = 1 χ \chi χ O \mathcal{O} O O \mathcal{O} O f ^ ( ξ ) = ∫ f ( x ) χ ( x ⋅ ξ ) ‾ d x \widehat{f}(\xi)=\int f(x)\overline{\chi(x\cdot\xi)}\,dx f ( ξ ) = ∫ f ( x ) χ ( x ⋅ ξ ) d x
Inner products are linear in the first variable. Throughout the paper, α > 0 \alpha>0 α > 0
τ = d e f Q − d , ρ = d e f Q α , r = d e f τ 1 2 , c = d e f ρ − 1 1 − τ ρ , κ = d e f 1 − τ 1 − τ ρ , λ n = d e f ρ n + 1 − κ . (6) \tau\mathrel{\mathop{=}^{\mathrm{def}}} Q^{-d},\qquad\rho\mathrel{\mathop{=}^{\mathrm{def}}} Q^\alpha,\qquad r \mathrel{\mathop{=}^{\mathrm{def}}} \tau^{\frac12},\qquad c \mathrel{\mathop{=}^{\mathrm{def}}} \frac{\rho-1}{1-\frac{\tau}{\rho}},\qquad\kappa\mathrel{\mathop{=}^{\mathrm{def}}} \frac{1-\tau}{1-\frac{\tau}{\rho}},\qquad\lambda_n \mathrel{\mathop{=}^{\mathrm{def}}} \rho^{n+1}-\kappa. \tag*{(6)} τ = def Q − d , ρ = def Q α , r = def τ 2 1 , c = def 1 − ρ τ ρ − 1 , κ = def 1 − ρ τ 1 − τ , λ n = def ρ n + 1 − κ . (6) In particular, 0 < τ < 1 0<\tau<1 0 < τ < 1 ρ > 1 \rho>1 ρ > 1 0 < κ < 1 0<\kappa<1 0 < κ < 1 λ 0 = c > 0 \lambda_0=c>0 λ 0 = c > 0 τ \tau τ ( 0 , 1 ) (0,1) ( 0 , 1 ) τ = Q − d \tau=Q^{-d} τ = Q − d p p p L p L^p L p
For any integer n ∈ Z n\in\mathbb{Z} n ∈ Z B n = d e f π n O d B_n\mathrel{\mathop{=}^{\mathrm{def}}}\pi^n\mathcal{O}^d B n = def π n O d S n = d e f B n ∖ B n + 1 S_n\mathrel{\mathop{=}^{\mathrm{def}}}B_n\setminus B_{n+1} S n = def B n ∖ B n + 1
μ ( B n ) = τ n , μ ( S n ) = w n = d e f ( 1 − τ ) τ n , 1 B n ^ = τ n 1 B − n . (7) \mu(B_n)=\tau^n,\qquad\mu(S_n)=w_n\mathrel{\mathop{=}^{\mathrm{def}}}(1-\tau)\tau^n,\qquad\widehat{1_{B_n}}=\tau^n1_{B_{-n}}. \tag*{(7)} μ ( B n ) = τ n , μ ( S n ) = w n = def ( 1 − τ ) τ n , 1 B n = τ n 1 B − n . (7) A radial function means a function of ∥ x ∥ K \lVert x\rVert_K ∥ x ∥ K S n S_n S n G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O ) S n S_n S n L 2 L^2 L 2 ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) ( f n ) n ≥ 0 (f_n)_{n\geq0} ( f n ) n ≥ 0 { 0 } \{0\} { 0 } ℓ N 0 ∞ \ell^\infty_{\mathbb{N}_0} ℓ N 0 ∞ O d \mathcal{O}^d O d C ( X τ ) C(X_\tau) C ( X τ ) f n = F ( τ n ) f_n=F(\tau^n) f n = F ( τ n ) X τ X_\tau X τ (2) .
Full-space, regional, and killed realizations
The full-space Taibleson operator is the nonnegative self-adjoint Fourier multiplier
D α = F − 1 M ∥ ξ ∥ K α F , D α f ( x ) = c ∫ K d f ( x ) − f ( y ) ∥ x − y ∥ K d + α d y . D^\alpha=\mathcal{F}^{-1}M_{\lVert\xi\rVert_K^\alpha}\mathcal{F},\qquad D^\alpha f(x)=c\int_{K^d}\frac{f(x)-f(y)}{\lVert x-y\rVert_K^{d+\alpha}}\,dy. D α = F − 1 M ∥ ξ ∥ K α F , D α f ( x ) = c ∫ K d ∥ x − y ∥ K d + α f ( x ) − f ( y ) d y . on Bruhat–Schwartz functions. The second formula has the normalization of [3 (1.2)–(1.4)] , with its numerator and integrand signs reversed simultaneously. For completeness, the normalization follows by testing on a character of unit frequency. The shell of radius Q Q Q ρ − 1 \rho^{-1} ρ − 1 ∫ ( 1 − χ ( y 1 ) ) ∥ y ∥ K − d − α d y \int(1-\chi(y_1))\lVert y\rVert_K^{-d-\alpha}\,dy ∫ ( 1 − χ ( y 1 )) ∥ y ∥ K − d − α d y Q j Q^j Q j j ≥ 2 j\geq2 j ≥ 2 ( 1 − τ ) ρ − j (1-\tau)\rho^{-j} ( 1 − τ ) ρ − j 1 − τ ρ − 1 = c − 1 \frac{1-\tau}{\rho-1}=c^{-1} ρ − 1 1 − τ = c − 1 G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O )
On O d \mathcal{O}^d O d regional operator on locally constant functions by
R α f ( x ) : = c ∫ O d f ( x ) − f ( y ) ∥ x − y ∥ K d + α d y . (8) R^\alpha f(x)\mathrel{\mathop:}=c\int_{\mathcal{O}^d}\frac{f(x)-f(y)}{\lVert x-y\rVert_K^{d+\alpha}}\,dy. \tag*{(8)} R α f ( x ) : = c ∫ O d ∥ x − y ∥ K d + α f ( x ) − f ( y ) d y . (8) These functions form a dense subspace of the Hilbert space L 2 ( O d ) L^2(\mathcal{O}^d) L 2 ( O d ) y = x y=x y = x
Proposition 2.1. The operator (8) is essentially self-adjoint and nonnegative on the locally constant functions. If f ~ \widetilde f f
( D α f ~ ) ∣ O d = R α f + κ f . (9) \left(D^\alpha\widetilde f\right)\big|_{\mathcal{O}^d}=R^\alpha f+\kappa f. \tag*{(9)} ( D α f ) O d = R α f + κ f . (9) On the characters of O d \mathcal{O}^d O d R α R^\alpha R α 0 0 0 ρ j − κ \rho^j-\kappa ρ j − κ j ≥ 1 j\geq1 j ≥ 1 M α M^\alpha M α O d \mathcal{O}^d O d ∥ ξ ∥ K α \lVert\xi\rVert_K^\alpha ∥ ξ ∥ K α M α = R α + κ ( I − E 0 ) M^\alpha=R^\alpha+\kappa(I-E_0) M α = R α + κ ( I − E 0 ) E 0 f = ∫ O d f E_0f=\int_{\mathcal{O}^d}f E 0 f = ∫ O d f
Proof: For x ∈ O d x\in\mathcal{O}^d x ∈ O d y ∉ O d y\notin\mathcal{O}^d y ∈ / O d ∥ x − y ∥ K = ∥ y ∥ K \lVert x-y\rVert_K=\lVert y\rVert_K ∥ x − y ∥ K = ∥ y ∥ K
c ∫ K d ∖ O d ∥ y ∥ K − d − α d y = c ( 1 − τ ) ∑ j ≥ 1 ρ − j = κ . c\int_{K^d\setminus\mathcal{O}^d}\lVert y\rVert_K^{-d-\alpha}\,dy=c(1-\tau)\sum_{j\geq1}\rho^{-j}=\kappa. c ∫ K d ∖ O d ∥ y ∥ K − d − α d y = c ( 1 − τ ) j ≥ 1 ∑ ρ − j = κ . This proves (9) . Alternatively, on a nontrivial character of conductor j j j ρ j − κ \rho^j-\kappa ρ j − κ L 2 ( O d ) L^2(\mathcal{O}^d) L 2 ( O d )
We keep the notation R α R^\alpha R α (8) .
Exact radial matrices and wavelets
Proposition 2.2. For a radial locally constant function on O d \mathcal{O}^d O d f n f_n f n S n S_n S n
( R r a d α f ) n = c ( 1 − τ ) ρ n [ ∑ j = 1 n ρ − j ( f n − f n − j ) + ∑ j ≥ 1 τ j ( f n − f n + j ) ] , n ≥ 0. (10) (R^\alpha_{\mathrm{rad}}f)_n=c(1-\tau)\rho^n\left[\sum_{j=1}^{n}\rho^{-j}(f_n-f_{n-j})+\sum_{j\ge1}\tau^j(f_n-f_{n+j})\right],\qquad n\ge0. \tag*{(10)} ( R rad α f ) n = c ( 1 − τ ) ρ n [ j = 1 ∑ n ρ − j ( f n − f n − j ) + j ≥ 1 ∑ τ j ( f n − f n + j ) ] , n ≥ 0. (10) Equivalently, the off-diagonal jump rates and the diagonal entries are
q n k = c w k ( ρ τ ) min ( n , k ) ( n ≠ k ) , (11) q_{nk}=c w_k\left(\frac{\rho}{\tau}\right)^{\min(n,k)}\qquad(n\ne k), \tag*{(11)} q nk = c w k ( τ ρ ) m i n ( n , k ) ( n = k ) , (11) R n n = c [ ( 1 − τ ) ρ n − 1 ρ − 1 + τ ρ n ] . (12) R_{nn}=c\left[(1-\tau)\frac{\rho^n-1}{\rho-1}+\tau\rho^n\right]. \tag*{(12)} R nn = c [ ( 1 − τ ) ρ − 1 ρ n − 1 + τ ρ n ] . (12) Detailed balance w n q n k = w k q k n w_nq_{nk}=w_kq_{kn} w n q nk = w k q k n
Proof: If x ∈ S n x\in S_n x ∈ S n y ∈ S k y\in S_k y ∈ S k k ≠ n k\ne n k = n ∥ x − y ∥ K = Q − min ( n , k ) \lVert x-y\rVert_K=Q^{-\min(n,k)} ∥ x − y ∥ K = Q − m i n ( n , k ) w k w_k w k (11) . The shell S n S_n S n k < n k<n k < n k > n k>n k > n (10) . Summation of the rates gives (12) . The balance identity follows from the symmetry of w n w k ( ρ τ ) min ( n , k ) w_nw_k\left(\frac{\rho}{\tau}\right)^{\min(n,k)} w n w k ( τ ρ ) m i n ( n , k )
For any integer n ∈ Z n\in\mathbb{Z} n ∈ Z
g n ≔ d e f 1 B n + 1 − τ 1 B n ( 1 − τ ) τ n + 1 . (13) g_n\mathrel{\mathop{\coloneqq}^{\mathrm{def}}}\frac{\mathbf{1}_{B_{n+1}}-\tau\mathbf{1}_{B_n}}{\sqrt{(1-\tau)\tau^{n+1}}}. \tag*{(13)} g n : = def ( 1 − τ ) τ n + 1 1 B n + 1 − τ 1 B n . (13) Theorem 2.3. On the unit ball, 1 , ( g n ) n ≥ 0 1,(g_n)_{n\ge0} 1 , ( g n ) n ≥ 0 ℓ 0 2 ( w ) \ell^2_0(w) ℓ 0 2 ( w )
R r a d α 1 = 0 , R r a d α g n = λ n g n , dom R r a d α = { f : ∑ n ≥ 0 λ n 2 ∣ ⟨ f , g n ⟩ ∣ 2 < ∞ } . (14) R^\alpha_{\mathrm{rad}}1=0,\qquad R^\alpha_{\mathrm{rad}}g_n=\lambda_ng_n,\qquad\operatorname{dom}R^\alpha_{\mathrm{rad}}=\left\{f:\sum_{n\ge0}\lambda_n^2|\langle f,g_n\rangle|^2<\infty\right\}. \tag*{(14)} R rad α 1 = 0 , R rad α g n = λ n g n , dom R rad α = { f : n ≥ 0 ∑ λ n 2 ∣ ⟨ f , g n ⟩ ∣ 2 < ∞ } . (14) The space C C C 1 , g 0 , g 1 , … 1,g_0,g_1,\ldots 1 , g 0 , g 1 , …
Proof: The numerator in (13) has integral zero and squared norm ( 1 − τ ) τ n + 1 (1-\tau)\tau^{n+1} ( 1 − τ ) τ n + 1 S 0 , … , S N − 1 S_0,\ldots,S_{N-1} S 0 , … , S N − 1 B N B_N B N ℓ 0 2 ( w ) \ell^2_0(w) ℓ 0 2 ( w ) (9) , applied to g n g_n g n n ≥ 0 n\ge0 n ≥ 0 (7) and the multiplier formula for D α D^\alpha D α (14) : the Fourier transform of the numerator in (13) is τ n + 1 1 S − n − 1 \tau^{n+1}\mathbf{1}_{S_{-n-1}} τ n + 1 1 S − n − 1 ∥ ξ ∥ K α \lVert\xi\rVert_K^\alpha ∥ ξ ∥ K α ρ n + 1 \rho^{n+1} ρ n + 1
Remark 2.4. Up to normalization, g n − 1 g_{n-1} g n − 1 0 0 0 p n τ , 0 ; 0 p_n^{\tau,0;0} p n τ , 0 ; 0 [48 Theorem 2.1] , which takes the values 0 0 0 − τ 1 − τ -\frac{\tau}{1-\tau} − 1 − τ τ 1 1 1 k < n − 1 k<n-1 k < n − 1 k = n − 1 k=n-1 k = n − 1 k ≥ n k\ge n k ≥ n G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O ) O d \mathcal{O}^d O d [48 Section 5.2] , after [38 ] ; for d = 1 d=1 d = 1 [26 ] .
The radial Dirichlet form
For any f ∈ ℓ N 0 2 ( w ) f\in\ell^2_{\mathbb{N}_0}(w) f ∈ ℓ N 0 2 ( w )
E R ( f ) ≔ ∑ 0 ≤ m < k w m q m k ∣ f m − f k ∣ 2 ∈ [ 0 , ∞ ] , dom E R ≔ { f ∈ ℓ N 0 2 ( w ) : E R ( f ) < ∞ } . (15) \mathcal{E}_R(f)\coloneqq\sum_{0\le m<k}w_mq_{mk}|f_m-f_k|^2\in[0,\infty],\qquad\operatorname{dom}\mathcal{E}_R\coloneqq\{f\in\ell^2_{\mathbb{N}_0}(w):\mathcal{E}_R(f)<\infty\}. \tag*{(15)} E R ( f ) : = 0 ≤ m < k ∑ w m q mk ∣ f m − f k ∣ 2 ∈ [ 0 , ∞ ] , dom E R : = { f ∈ ℓ N 0 2 ( w ) : E R ( f ) < ∞ } . (15) Since w m q m k = c ( 1 − τ ) 2 τ k ρ m w_mq_{mk}=c(1-\tau)^2\tau^k\rho^m w m q mk = c ( 1 − τ ) 2 τ k ρ m m < k m<k m < k k = m + j k=m+j k = m + j
E R ( f ) = c ( 1 − τ ) 2 ∑ j ≥ 1 τ − j ∑ m ≥ 0 Q ( α − d ) m ∣ f m + j − f m ∣ 2 . (16) \mathcal{E}_R(f)=c(1-\tau)^2\sum_{j\ge1}\tau^{-j}\sum_{m\ge0}Q^{(\alpha-d)m}|f_{m+j}-f_m|^2. \tag*{(16)} E R ( f ) = c ( 1 − τ ) 2 j ≥ 1 ∑ τ − j m ≥ 0 ∑ Q ( α − d ) m ∣ f m + j − f m ∣ 2 . (16) Lemma 2.5 The form (15) is the closed form of the nonnegative self-adjoint operator R r a d α R^\alpha_{\mathrm{rad}} R rad α dom E R = dom ( R r a d α ) 1 2 \operatorname{dom}\mathcal{E}_R=\operatorname{dom}(R^\alpha_{\mathrm{rad}})^{\frac12} dom E R = dom ( R rad α ) 2 1 C C C E R \mathcal{E}_R E R
E R ( f ) = ∑ n ≥ 0 λ n ∣ ⟨ f , g n ⟩ ∣ 2 , f ∈ ℓ N 0 2 ( w ) . \mathcal{E}_R(f)=\sum_{n\ge0}\lambda_n|\langle f,g_n\rangle|^2,\qquad f\in\ell^2_{\mathbb{N}_0}(w). E R ( f ) = n ≥ 0 ∑ λ n ∣ ⟨ f , g n ⟩ ∣ 2 , f ∈ ℓ N 0 2 ( w ) . Proof: We first show that E R \mathcal{E}_R E R ( f ( i ) ) (f^{(i)}) ( f ( i ) ) dom E R \operatorname{dom}\mathcal{E}_R dom E R ( E R ( ⋅ ) + ∥ ⋅ ∥ 2 ) 1 2 (\mathcal{E}_R(\cdot)+\|\cdot\|^2)^{\frac12} ( E R ( ⋅ ) + ∥ ⋅ ∥ 2 ) 2 1 ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) f f f
E R ( f ( i ) − f ) ≤ lim inf i ′ → ∞ E R ( f ( i ) − f ( i ′ ) ) . \mathcal{E}_R(f^{(i)}-f)\le\liminf_{i'\to\infty}\mathcal{E}_R(f^{(i)}-f^{(i')}). E R ( f ( i ) − f ) ≤ i ′ → ∞ lim inf E R ( f ( i ) − f ( i ′ ) ) . which is small for large i i i f ∈ dom E R f\in\operatorname{dom}\mathcal{E}_R f ∈ dom E R f ( i ) → f f^{(i)}\to f f ( i ) → f
Now, we consider an eventually constant sequence g ∈ C g\in C g ∈ C g k = g N g_k=g_N g k = g N k ≥ N k\ge N k ≥ N f ∈ dom E R f\in\operatorname{dom}\mathcal{E}_R f ∈ dom E R
E R ( f , g ) = ⟨ f , R r a d α g ⟩ ℓ N 0 2 ( w ) . (17) \mathcal{E}_R(f,g)=\langle f,R^\alpha_{\mathrm{rad}}g\rangle_{\ell^2_{\mathbb{N}_0}(w)}. \tag*{(17)} E R ( f , g ) = ⟨ f , R rad α g ⟩ ℓ N 0 2 ( w ) . (17) where E R ( ⋅ , ⋅ ) \mathcal{E}_R(\cdot,\cdot) E R ( ⋅ , ⋅ ) (15) . By symmetry of the kernel w n q n k w_nq_{nk} w n q nk E R ( f , g ) = 1 2 ∑ n ≠ k w n q n k ( f n − f k ) ( g n − g k ) \mathcal{E}_R(f,g)=\frac12\sum_{n\ne k}w_nq_{nk}(f_n-f_k)(g_n-g_k) E R ( f , g ) = 2 1 ∑ n = k w n q nk ( f n − f k ) ( g n − g k ) ∑ n ≠ k w n q n k ∣ f n ∣ ∣ g n − g k ∣ < ∞ \sum_{n\ne k}w_nq_{nk}|f_n||g_n-g_k|<\infty ∑ n = k w n q nk ∣ f n ∣∣ g n − g k ∣ < ∞ n , k ≥ N n,k\ge N n , k ≥ N n < N n<N n < N k k k 2 ∥ g ∥ ∞ w n ∣ f n ∣ R n n 2\|g\|_\infty w_n|f_n|R_{nn} 2∥ g ∥ ∞ w n ∣ f n ∣ R nn n ≥ N n\ge N n ≥ N k < N k<N k < N w n q n k = c w n w k ( ρ τ ) k w_nq_{nk}=cw_nw_k(\frac{\rho}{\tau})^k w n q nk = c w n w k ( τ ρ ) k 2 ∥ g ∥ ∞ c ∑ k < N w k ( ρ τ ) k ∑ n ≥ N w n ∣ f n ∣ 2\|g\|_\infty c\sum_{k<N}w_k(\frac{\rho}{\tau})^k\sum_{n\ge N}w_n|f_n| 2∥ g ∥ ∞ c ∑ k < N w k ( τ ρ ) k ∑ n ≥ N w n ∣ f n ∣ w w w
E R ( f , g ) = ∑ n w n f n ∑ k ≠ n q n k ( g n − g k ) = xref‘(2.5)‘ ⟨ f , R r a d α g ⟩ . \mathcal{E}_R(f,g)=\sum_n w_nf_n\sum_{k\ne n}q_{nk}(g_n-g_k)\overset{\text{\scriptsize{xref}`(2.5)`}}{=}\langle f,R^\alpha_{\mathrm{rad}}g\rangle. E R ( f , g ) = n ∑ w n f n k = n ∑ q nk ( g n − g k ) = xref ‘(2.5)‘ ⟨ f , R rad α g ⟩ . This proves (17) .
Suppose that f ∈ dom E R f\in\operatorname{dom}\mathcal{E}_R f ∈ dom E R E R ( f , g ) + ⟨ f , g ⟩ = 0 \mathcal{E}_R(f,g)+\langle f,g\rangle=0 E R ( f , g ) + ⟨ f , g ⟩ = 0 g ∈ C g\in C g ∈ C (17) , ⟨ f , ( R r a d α + I ) g ⟩ = 0 \langle f,(R^\alpha_{\mathrm{rad}}+I)g\rangle=0 ⟨ f , ( R rad α + I ) g ⟩ = 0 g ∈ C g\in C g ∈ C R r a d α + I R^\alpha_{\mathrm{rad}}+I R rad α + I C C C C C C (14) . Hence f f f C C C f = 0 f=0 f = 0 C C C E R \mathcal{E}_R E R (17) with f = g ∈ C f=g\in C f = g ∈ C E R \mathcal{E}_R E R C C C g ↦ ⟨ g , R r a d α g ⟩ = ∑ n λ n ∣ ⟨ g , g n ⟩ ∣ 2 g\mapsto\langle g,R^\alpha_{\mathrm{rad}}g\rangle=\sum_n\lambda_n|\langle g,g_n\rangle|^2 g ↦ ⟨ g , R rad α g ⟩ = ∑ n λ n ∣ ⟨ g , g n ⟩ ∣ 2 C C C C C C
Conditional expectations and an entrance law
For n ≥ 0 n \ge0 n ≥ 0 E n \mathbb{E}_n E n n n n
( E n f ) k = { f k , k < n , ( 1 − τ ) ∑ j ≥ 0 τ j f n + j , k ≥ n . (18) (\mathbb{E}_n f)_k =
\begin{cases}
f_k, & k<n,\\
(1-\tau)\sum_{j\ge0}\tau^j f_{n+j}, & k\ge n.
\end{cases} \tag*{(18)} ( E n f ) k = { f k , ( 1 − τ ) ∑ j ≥ 0 τ j f n + j , k < n , k ≥ n . (18) They are probability-preserving conditional expectations, E i E j = E min ( i , j ) \mathbb{E}_i\mathbb{E}_j=\mathbb{E}_{\min(i,j)} E i E j = E m i n ( i , j ) E 0 \mathbb{E}_0 E 0 g k g_k g k E n \mathbb{E}_n E n n ≤ k n\le k n ≤ k n > k n>k n > k
R r a d α = c ( I − E 0 ) + ( ρ − 1 ) ∑ n ≥ 1 ρ n ( I − E n ) R^\alpha_{\mathrm{rad}}=c(I-\mathbb{E}_0)+(\rho-1)\sum_{n\ge1}\rho^n(I-\mathbb{E}_n) R rad α = c ( I − E 0 ) + ( ρ − 1 ) n ≥ 1 ∑ ρ n ( I − E n ) on C \mathcal{C} C T t = e − t R r a d α T_t=e^{-tR^\alpha_{\mathrm{rad}}} T t = e − t R rad α
a 0 ( t ) : = 1 − e − t λ 0 , a n ( t ) : = e − t λ n − 1 − e − t λ n ( n ≥ 1 ) . (19) a_0(t)\mathrel{\mathop:}=1-e^{-t\lambda_0},\qquad a_n(t)\mathrel{\mathop:}=e^{-t\lambda_{n-1}}-e^{-t\lambda_n}\quad(n\ge1). \tag*{(19)} a 0 ( t ) : = 1 − e − t λ 0 , a n ( t ) : = e − t λ n − 1 − e − t λ n ( n ≥ 1 ) . (19) For t > 0 t>0 t > 0
T t = ∑ n ≥ 0 a n ( t ) E n . (20) T_t=\sum_{n\ge0}a_n(t)\mathbb{E}_n. \tag*{(20)} T t = n ≥ 0 ∑ a n ( t ) E n . (20) This identity follows on the orthonormal basis and then on bounded sequences by normality. It also proves conservativity and positivity directly. The operator E n \mathbb{E}_n E n Q − n Q^{-n} Q − n (20) is an instance of the description of isotropic Markov semigroups on ultrametric spaces as mixtures of such averaging operators [9 ] . Each E n \mathbb{E}_n E n E n f → f \mathbb{E}_n f\to f E n f → f f f f ( T t ) (T_t) ( T t ) C ( X τ ) C(X_\tau) C ( X τ )
Write P t ( n , k ) = ( T t 1 { k } ) n P_t(n,k)=(T_t\mathbf{1}_{\{k\}})_n P t ( n , k ) = ( T t 1 { k } ) n
ν j ( k ) : = ( 1 − τ ) τ k − j 1 { k ≥ j } , μ t ( k ) : = ∑ j ≥ 0 a j ( t ) ν j ( k ) . \nu_j(k)\mathrel{\mathop:}=(1-\tau)\tau^{k-j}\mathbf{1}_{\{k\ge j\}},\qquad\mu_t(k)\mathrel{\mathop:}=\sum_{j\ge0}a_j(t)\nu_j(k). ν j ( k ) : = ( 1 − τ ) τ k − j 1 { k ≥ j } , μ t ( k ) : = j ≥ 0 ∑ a j ( t ) ν j ( k ) . Lemma 2.6 For any t > 0 t>0 t > 0 P t ( n , ⋅ ) P_t(n,\cdot) P t ( n , ⋅ ) ℓ 1 \ell^1 ℓ 1 μ t \mu_t μ t n → ∞ n\to\infty n → ∞
∥ P t ( n , ⋅ ) − μ t ∥ ℓ 1 ≤ 2 e − t λ n . (21) \lVert P_t(n,\cdot)-\mu_t\rVert_{\ell^1}\le2e^{-t\lambda_n}. \tag*{(21)} ∥ P t ( n , ⋅ ) − μ t ∥ ℓ 1 ≤ 2 e − t λ n . (21) For t j = ρ − j t_j=\rho^{-j} t j = ρ − j j ≥ 1 j\ge1 j ≥ 1
μ t j ( j ) ≥ δ T , δ T : = ( 1 − τ ) ( e − 1 − e − ρ ) > 0. \mu_{t_j}(j)\ge\delta_T,\qquad\delta_T\mathrel{\mathop:}=(1-\tau)(e^{-1}-e^{-\rho})>0. μ t j ( j ) ≥ δ T , δ T : = ( 1 − τ ) ( e − 1 − e − ρ ) > 0. Proof. For ℓ ≤ n \ell\le n ℓ ≤ n n n n E ℓ \mathbb{E}_\ell E ℓ ν ℓ \nu_\ell ν ℓ ℓ > n \ell>n ℓ > n n n n ∑ ℓ > n a ℓ ( t ) = e − t λ n \sum_{\ell>n}a_\ell(t)=e^{-t\lambda_n} ∑ ℓ > n a ℓ ( t ) = e − t λ n (21) . Moreover μ t ( j ) ≥ ( 1 − τ ) a j ( t ) \mu_t(j)\ge(1-\tau)a_j(t) μ t ( j ) ≥ ( 1 − τ ) a j ( t ) a j ( t j ) = e κ t j ( e − 1 − e − ρ ) ≥ e − 1 − e − ρ a_j(t_j)=e^{\kappa t_j}(e^{-1}-e^{-\rho})\ge e^{-1}-e^{-\rho} a j ( t j ) = e κ t j ( e − 1 − e − ρ ) ≥ e − 1 − e − ρ
Little η \eta η
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 ≥ 0 k\ge0 k ≥ 0 ℓ 0 2 ( w ) \ell^2_0(w) ℓ 0 2 ( w )
Y η : = { η k : k ≥ 0 } ∪ { 0 } . Y_\eta\mathrel{\mathop:}=\{\eta^k:k\ge0\}\cup\{0\}. Y η : = { η k : k ≥ 0 } ∪ { 0 } . A function of k ∈ N 0 k \in\mathbb{N}_0 k ∈ N 0 η k \eta^k η k k ↦ p ( η k ) k \mapsto p(\eta^k) k ↦ p ( η k ) p ∈ C [ x ] p \in\mathbb{C}[x] p ∈ C [ x ] P n \mathcal{P}_n P n η k \eta^k η k n n n P \mathcal{P} P p p p Y η Y_\eta Y η p ( 0 ) p(0) p ( 0 ) p ( η k ) p(\eta^k) p ( η k ) k → ∞ k \to\infty k → ∞ q q q ( a ; η ) n : = ∏ i = 0 n − 1 ( 1 − a η i ) (a;\eta)_n \mathrel{\mathop:}= \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 ∪ { ∞ } [27 (17.2.1) and (17.2.4)] and [32 Section 1.2] .
The operator and its spectral decomposition
Put
β k : = τ η − k − 1 ( 1 − η k + 1 ) , δ k : = η − k ( 1 − η k ) , k ≥ 0. \beta_k \mathrel{\mathop:}= \tau\eta^{-k-1}(1-\eta^{k+1}), \qquad\delta_k \mathrel{\mathop:}= \eta^{-k}(1-\eta^k), \qquad k \ge0. β k : = τ η − k − 1 ( 1 − η k + 1 ) , δ k : = η − k ( 1 − η k ) , k ≥ 0. In particular δ 0 = 0 \delta_0=0 δ 0 = 0
w k β k = w k + 1 δ k + 1 = ( 1 − τ ) ( τ η ) k + 1 ( 1 − η k + 1 ) , k ≥ 0. (22) w_k\beta_k=w_{k+1}\delta_{k+1}=(1-\tau)\left(\frac{\tau}{\eta}\right)^{k+1}(1-\eta^{k+1}), \qquad k \ge0. \tag*{(22)} w k β k = w k + 1 δ k + 1 = ( 1 − τ ) ( η τ ) k + 1 ( 1 − η k + 1 ) , k ≥ 0. (22) holds. For any function f : N 0 → C f:\mathbb{N}_0 \to\mathbb{C} f : N 0 → C L η f L_\eta f L η f (3) , that is ( L η f ) ( k ) = β k ( f ( k ) − f ( k + 1 ) ) + δ k ( f ( k ) − f ( k − 1 ) ) (L_\eta f)(k)=\beta_k(f(k)-f(k+1))+\delta_k(f(k)-f(k-1)) ( L η f ) ( k ) = β k ( f ( k ) − f ( k + 1 )) + δ k ( f ( k ) − f ( k − 1 )) k = 0 k=0 k = 0 f ∈ ℓ N 0 2 ( w ) f \in\ell^2_{\mathbb{N}_0}(w) f ∈ ℓ N 0 2 ( w )
E η ( f ) : = ∑ k ≥ 0 w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 ∈ [ 0 , ∞ ] , dom E η : = { f ∈ ℓ N 0 2 ( w ) : E η ( f ) < ∞ } . (23) \mathcal{E}_\eta(f) \mathrel{\mathop:}= \sum_{k\ge0}w_k\beta_k\lvert f(k+1)-f(k)\rvert^2 \in[0,\infty], \qquad\operatorname{dom}\mathcal{E}_\eta\mathrel{\mathop:}= \{f\in\ell^2_{\mathbb{N}_0}(w):\mathcal{E}_\eta(f)<\infty\}. \tag*{(23)} E η ( f ) : = k ≥ 0 ∑ w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 ∈ [ 0 , ∞ ] , dom E η : = { f ∈ ℓ N 0 2 ( w ) : E η ( f ) < ∞ } . (23) Proposition 3.1. 1. For any integer n ≥ 0 n\ge0 n ≥ 0 L η L_\eta L η P n \mathcal{P}_n P n P n \mathcal{P}_n P n L η ( x n ) − Λ n ( η ) x n ∈ P n − 1 L_\eta(x^n)-\Lambda_n(\eta)x^n\in\mathcal{P}_{n-1} L η ( x n ) − Λ n ( η ) x n ∈ P n − 1 Λ n ( η ) \Lambda_n(\eta) Λ n ( η ) (4) and P − 1 = { 0 } \mathcal{P}_{-1}=\{0\} P − 1 = { 0 } ( Λ n ( η ) ) n ≥ 0 (\Lambda_n(\eta))_{n\ge0} ( Λ n ( η ) ) n ≥ 0 Λ 0 ( η ) = 0 \Lambda_0(\eta)=0 Λ 0 ( η ) = 0
The space P \mathcal{P} P ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) dom E η \operatorname{dom}\mathcal{E}_\eta dom E η f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_\eta f ∈ dom E η p ∈ P p\in\mathcal{P} p ∈ P
E η ( f , p ) = ⟨ f , L η p ⟩ ℓ N 0 2 ( w ) . (24) \mathcal{E}_\eta(f,p)=\langle f,L_\eta p\rangle_{\ell^2_{\mathbb{N}_0}(w)}. \tag*{(24)} E η ( f , p ) = ⟨ f , L η p ⟩ ℓ N 0 2 ( w ) . (24) The form E η \mathcal{E}_\eta E η P \mathcal{P} P E η \mathcal{E}_\eta E η L η L_\eta L η ( e n η ) n ≥ 0 (e^\eta_n)_{n\ge0} ( e n η ) n ≥ 0 1 , x , x 2 , … 1,x,x^2,\ldots 1 , x , x 2 , … L η e n η = Λ n ( η ) e n η L_\eta e^\eta_n=\Lambda_n(\eta)e^\eta_n L η e n η = Λ n ( η ) e n η dom E η = dom L η 1 / 2 \operatorname{dom}\mathcal{E}_\eta=\operatorname{dom}L_\eta^{1/2} dom E η = dom L η 1/2 E η ( f ) = ∑ n Λ n ( η ) ∣ ⟨ f , e n η ⟩ ∣ 2 \mathcal{E}_\eta(f)=\sum_n\Lambda_n(\eta)\lvert\langle f,e^\eta_n\rangle\rvert^2 E η ( f ) = ∑ n Λ n ( η ) ∣⟨ f , e n η ⟩ ∣ 2
For any n ≥ 0 n\ge0 n ≥ 0 e n η e^\eta_n e n η 0 0 0 φ n η : = ( e n η ( 0 ) ) − 1 e n η \varphi^\eta_n\mathrel{\mathop:}=(e^\eta_n(0))^{-1}e^\eta_n φ n η : = ( e n η ( 0 ) ) − 1 e n η n n n P n − 1 \mathcal{P}_{n-1} P n − 1 1 1 1 0 0 0 η \eta η p n ( x ; τ η − 1 , 1 ; η ) p_n(x;\tau\eta^{-1},1;\eta) p n ( x ; τ η − 1 , 1 ; η ) [44 Section 14.12] .
Proof. 1. Let f ( k ) = η n k = x n f(k)=\eta^{nk}=x^n f ( k ) = η nk = x n x = η k x=\eta^k x = η k k ≥ 0 k\ge0 k ≥ 0
( L η f ) ( k ) = τ η − k − 1 ( 1 − η k + 1 ) η n k ( 1 − η n ) + η − k ( 1 − η k ) η n k ( 1 − η − n ) = x n − 1 [ τ η ( 1 − η x ) ( 1 − η n ) + ( 1 − x ) ( 1 − η − n ) ] . \begin{aligned}
(L_\eta f)(k)=\tau\eta^{-k-1}(1-\eta^{k+1})\eta^{nk}(1-\eta^n)+\eta^{-k}(1-\eta^k)\eta^{nk}(1-\eta^{-n}) \\
&=x^{n-1}\left[\frac{\tau}{\eta}(1-\eta x)(1-\eta^n)+(1-x)(1-\eta^{-n})\right].
\end{aligned} ( L η f ) ( k ) = τ η − k − 1 ( 1 − η k + 1 ) η nk ( 1 − η n ) + η − k ( 1 − η k ) η nk ( 1 − η − n ) = x n − 1 [ η τ ( 1 − η x ) ( 1 − η n ) + ( 1 − x ) ( 1 − η − n ) ] . which is also valid for k = 0 k=0 k = 0 δ 0 = 0 \delta_0=0 δ 0 = 0 1 − x = 0 1-x=0 1 − x = 0 n = 0 n=0 n = 0 x n x^n x n − τ ( 1 − η n ) − ( 1 − η − n ) = η − n ( 1 − η n ) ( 1 − τ η n ) = Λ n ( η ) -\tau(1-\eta^n)-(1-\eta^{-n})=\eta^{-n}(1-\eta^n)(1-\tau\eta^n)=\Lambda_n(\eta) − τ ( 1 − η n ) − ( 1 − η − n ) = η − n ( 1 − η n ) ( 1 − τ η n ) = Λ n ( η )
Λ n + 1 ( η ) − Λ n ( η ) = ( 1 − η ) ( η − n − 1 − τ η n ) > 0 , (25) \Lambda_{n+1}(\eta)-\Lambda_n(\eta)=(1-\eta)(\eta^{-n-1}-\tau\eta^n)>0, \tag*{(25)} Λ n + 1 ( η ) − Λ n ( η ) = ( 1 − η ) ( η − n − 1 − τ η n ) > 0 , (25) since Λ n ( η ) = η − n − 1 − τ + τ η n \Lambda_n(\eta)=\eta^{-n}-1-\tau+\tau\eta^n Λ n ( η ) = η − n − 1 − τ + τ η n
The measure ∑ k w k δ η k \sum_k w_k\delta_{\eta^k} ∑ k w k δ η k Y η Y_\eta Y η C ( Y η ) C(Y_\eta) C ( Y η ) η k \eta^k η k ℓ 0 2 ( w ) \ell^2_0(w) ℓ 0 2 ( w ) P \mathcal{P} P p ∈ P p\in\mathcal{P} p ∈ P ∣ p ( η k + 1 ) − p ( η k ) ∣ ≤ ∥ p ′ ∥ L ∞ [ 0 , 1 ] η k |p(\eta^{k+1})-p(\eta^k)|\leq\|p'\|_{L^\infty[0,1]}\eta^k ∣ p ( η k + 1 ) − p ( η k ) ∣ ≤ ∥ p ′ ∥ L ∞ [ 0 , 1 ] η k (22)
E η ( p ) ≤ ( 1 − τ ) η − 1 ∥ p ′ ∥ L ∞ [ 0 , 1 ] 2 ∑ k ≥ 0 ( τ η ) k < ∞ . \mathcal{E}_\eta(p)\leq(1-\tau)\eta^{-1}\|p'\|_{L^\infty[0,1]}^2\sum_{k\geq0}(\tau\eta)^k<\infty. E η ( p ) ≤ ( 1 − τ ) η − 1 ∥ p ′ ∥ L ∞ [ 0 , 1 ] 2 k ≥ 0 ∑ ( τ η ) k < ∞. Now, let f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_\eta f ∈ dom E η p ∈ P p\in\mathcal{P} p ∈ P c k = w k β k = w k + 1 δ k + 1 c_k=w_k\beta_k=w_{k+1}\delta_{k+1} c k = w k β k = w k + 1 δ k + 1 N ≥ 1 N\geq1 N ≥ 1
∑ k = 0 N − 1 c k ( f ( k + 1 ) − f ( k ) ) ( p ( k + 1 ) − p ( k ) ) ‾ = ∑ k = 0 N w k f ( k ) ( L η p ) ( k ) ‾ + c N − 1 f ( N ) ( p ( N ) − p ( N − 1 ) ) ‾ . (26) \sum_{k=0}^{N-1}c_k\bigl(f(k+1)-f(k)\bigr)\overline{\bigl(p(k+1)-p(k)\bigr)}
=\sum_{k=0}^{N}w_kf(k)\overline{(L_\eta p)(k)}+c_{N-1}f(N)\overline{\bigl(p(N)-p(N-1)\bigr)}. \tag*{(26)} k = 0 ∑ N − 1 c k ( f ( k + 1 ) − f ( k ) ) ( p ( k + 1 ) − p ( k ) ) = k = 0 ∑ N w k f ( k ) ( L η p ) ( k ) + c N − 1 f ( N ) ( p ( N ) − p ( N − 1 ) ) . (26) Indeed, the coefficient of f ( k ) f(k) f ( k ) c k − 1 ( p ( k ) − p ( k − 1 ) ) + c k ( p ( k ) − p ( k + 1 ) ) c_{k-1}\bigl(p(k)-p(k-1)\bigr)+c_k\bigl(p(k)-p(k+1)\bigr) c k − 1 ( p ( k ) − p ( k − 1 ) ) + c k ( p ( k ) − p ( k + 1 ) ) 1 ≤ k ≤ N − 1 1\leq k\leq N-1 1 ≤ k ≤ N − 1 c 0 ( p ( 0 ) − p ( 1 ) ) c_0\bigl(p(0)-p(1)\bigr) c 0 ( p ( 0 ) − p ( 1 ) ) k = 0 k=0 k = 0 w k ( L η p ) ( k ) w_k(L_\eta p)(k) w k ( L η p ) ( k ) (22) . For the boundary term, we use ∣ f ( N ) ∣ ≤ w N − 1 / 2 ∥ f ∥ |f(N)|\leq w_N^{-1/2}\|f\| ∣ f ( N ) ∣ ≤ w N − 1/2 ∥ f ∥ c N − 1 ≤ ( τ η ) N c_{N-1}\leq\left(\frac{\tau}{\eta}\right)^N c N − 1 ≤ ( η τ ) N
∣ c N − 1 f ( N ) ( p ( N ) − p ( N − 1 ) ) ∣ ≤ ( τ η ) N ( 1 − τ ) − 1 / 2 τ − N / 2 ∥ f ∥ ∥ p ′ ∥ L ∞ [ 0 , 1 ] η N − 1 = ∥ f ∥ ∥ p ′ ∥ L ∞ [ 0 , 1 ] η ( 1 − τ ) 1 / 2 τ N / 2 , \left|c_{N-1}f(N)\bigl(p(N)-p(N-1)\bigr)\right|
\leq\left(\frac{\tau}{\eta}\right)^N(1-\tau)^{-1/2}\tau^{-N/2}\|f\|\|p'\|_{L^\infty[0,1]}\eta^{N-1}
=\frac{\|f\|\|p'\|_{L^\infty[0,1]}}{\eta(1-\tau)^{1/2}}\tau^{N/2}, c N − 1 f ( N ) ( p ( N ) − p ( N − 1 ) ) ≤ ( η τ ) N ( 1 − τ ) − 1/2 τ − N /2 ∥ f ∥∥ p ′ ∥ L ∞ [ 0 , 1 ] η N − 1 = η ( 1 − τ ) 1/2 ∥ f ∥∥ p ′ ∥ L ∞ [ 0 , 1 ] τ N /2 , which tends to zero. The left-hand side of (3.4) converges to E η ( f , p ) \mathcal{E}_\eta(f,p) E η ( f , p ) E η ( f , p ) \mathcal{E}_\eta(f,p) E η ( f , p ) ⟨ f , L η p ⟩ \langle f,L_\eta p\rangle ⟨ f , L η p ⟩ L η p ∈ P L_\eta p\in\mathcal{P} L η p ∈ P (24) .
The closedness follows by the Fatou argument of the proof of Lemma 2.5 . Suppose that f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_\eta f ∈ dom E η E η ( f , p ) + ⟨ f , p ⟩ = 0 \mathcal{E}_\eta(f,p)+\langle f,p\rangle=0 E η ( f , p ) + ⟨ f , p ⟩ = 0 p ∈ P p\in\mathcal{P} p ∈ P (24) , ⟨ f , ( L η + I ) p ⟩ = 0 \langle f,(L_\eta+I)p\rangle=0 ⟨ f , ( L η + I ) p ⟩ = 0 p ∈ P p\in\mathcal{P} p ∈ P L η + I L_\eta+I L η + I P n \mathcal{P}_n P n Λ m ( η ) + 1 \Lambda_m(\eta)+1 Λ m ( η ) + 1 P n \mathcal{P}_n P n f ⊥ P f\perp\mathcal{P} f ⊥ P f = 0 f=0 f = 0 P \mathcal{P} P E η \mathcal{E}_\eta E η
By (24) with f ∈ P f\in\mathcal{P} f ∈ P L η L_\eta L η P \mathcal{P} P E η ( p ) = ⟨ p , L η p ⟩ \mathcal{E}_\eta(p)=\langle p,L_\eta p\rangle E η ( p ) = ⟨ p , L η p ⟩ p ∈ P p\in\mathcal{P} p ∈ P L η L_\eta L η P n \mathcal{P}_n P n L η e n η L_\eta e_n^\eta L η e n η P n \mathcal{P}_n P n P n − 1 \mathcal{P}_{n-1} P n − 1 ⟨ L η e n η , p ⟩ = ⟨ e n η , L η p ⟩ = 0 \langle L_\eta e_n^\eta,p\rangle=\langle e_n^\eta,L_\eta p\rangle=0 ⟨ L η e n η , p ⟩ = ⟨ e n η , L η p ⟩ = 0 p ∈ P n − 1 p\in\mathcal{P}_{n-1} p ∈ P n − 1 L η e n η L_\eta e_n^\eta L η e n η e n η e_n^\eta e n η x n x^n x n Λ n ( η ) \Lambda_n(\eta) Λ n ( η ) ( e n η ) (e_n^\eta) ( e n η ) E η \mathcal{E}_\eta E η p ↦ ∑ n Λ n ( η ) ∣ ⟨ p , e n η ⟩ ∣ 2 p\mapsto\sum_n\Lambda_n(\eta)|\langle p,e_n^\eta\rangle|^2 p ↦ ∑ n Λ n ( η ) ∣ ⟨ p , e n η ⟩ ∣ 2 P \mathcal{P} P Λ n ( η ) \Lambda_n(\eta) Λ n ( η )
The measure ∑ k w k δ η k \sum_k w_k\delta_{\eta^k} ∑ k w k δ η k [ 0 , 1 ] [0,1] [ 0 , 1 ] [68 Theorem 3.3.1] , the zeros of e n η e_n^\eta e n η ( 0 , 1 ) (0,1) ( 0 , 1 ) e n η ( 0 ) ≠ 0 e_n^\eta(0)\neq0 e n η ( 0 ) = 0 P n − 1 \mathcal{P}_{n-1} P n − 1 P n \mathcal{P}_n P n p n ( x ; a , b ; η ) p_n(x;a,b;\eta) p n ( x ; a , b ; η ) [44 Section 14.12] are orthogonal with respect to the weights ( b η ; η ) k ( η ; η ) k ( a η ) k \frac{(b\eta;\eta)_k}{(\eta;\eta)_k}(a\eta)^k ( η ; η ) k ( b η ; η ) k ( a η ) k η k \eta^k η k a = η − 1 a=\eta^{-1} a = η − 1 b = 1 b=1 b = 1 τ k \tau^k τ k p n ( 0 ; a , b ; η ) = 1 p_n(0;a,b;\eta)=1 p n ( 0 ; a , b ; η ) = 1
For η = τ \eta=\tau η = τ a = b = 1 a=b=1 a = b = 1 φ n τ \varphi_n^\tau φ n τ τ \tau τ
φ 1 τ ( x ) = 1 − ( 1 + τ ) x , φ 2 τ ( x ) = 1 − ( 1 + τ ) ( 1 + τ + τ 2 ) τ x + ( 1 + τ + τ 2 ) ( 1 + τ 2 ) τ x 2 . (27) \varphi_1^\tau(x)=1-(1+\tau)x,\qquad
\varphi_2^\tau(x)=1-\frac{(1+\tau)(1+\tau+\tau^2)}{\tau}x+\frac{(1+\tau+\tau^2)(1+\tau^2)}{\tau}x^2. \tag*{(27)} φ 1 τ ( x ) = 1 − ( 1 + τ ) x , φ 2 τ ( x ) = 1 − τ ( 1 + τ ) ( 1 + τ + τ 2 ) x + τ ( 1 + τ + τ 2 ) ( 1 + τ 2 ) x 2 . (27) Indeed, the moments of the weights are ∑ k w k τ m k = 1 − τ 1 − τ m + 1 \sum_k w_k\tau^{mk}=\frac{1-\tau}{1-\tau^{m+1}} ∑ k w k τ mk = 1 − τ m + 1 1 − τ (27) are orthogonal to 1 1 1 1 1 1 x x x
Λ 1 ( τ ) = ( 1 − τ ) ( 1 − τ 2 ) τ , Λ 2 ( τ ) = ( 1 − τ 2 ) ( 1 − τ 3 ) τ 2 , 1 + τ + τ 2 τ Λ 1 ( τ ) = Λ 2 ( τ ) . (28) \Lambda_1(\tau)=\frac{(1-\tau)(1-\tau^2)}{\tau},\qquad\Lambda_2(\tau)=\frac{(1-\tau^2)(1-\tau^3)}{\tau^2},\qquad\frac{1+\tau+\tau^2}{\tau}\Lambda_1(\tau)=\Lambda_2(\tau). \tag*{(28)} Λ 1 ( τ ) = τ ( 1 − τ ) ( 1 − τ 2 ) , Λ 2 ( τ ) = τ 2 ( 1 − τ 2 ) ( 1 − τ 3 ) , τ 1 + τ + τ 2 Λ 1 ( τ ) = Λ 2 ( τ ) . (28) The Feller semigroup and its law at the origin
Proposition 3.3 Let ( T t η ) t ≥ 0 (T_t^\eta)_{t\geq0} ( T t η ) t ≥ 0 e − t L η e^{-tL_\eta} e − t L η ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w )
The semigroup is Markovian. More precisely, P t η ( n , k ) : = ( T t η 1 { k } ) ( n ) P_t^\eta(n,k)\mathrel{\mathop:}=(T_t^\eta1_{\{k\}})(n) P t η ( n , k ) : = ( T t η 1 { k } ) ( n ) N 0 \mathbb{N}_0 N 0 ( T t η f ) ( n ) = ∑ k P t η ( n , k ) f ( k ) (T_t^\eta f)(n)=\sum_k P_t^\eta(n,k)f(k) ( T t η f ) ( n ) = ∑ k P t η ( n , k ) f ( k ) f ∈ ℓ N 0 ∞ f\in\ell^\infty_{\mathbb{N}_0} f ∈ ℓ N 0 ∞
For any f ∈ C ( Y η ) f\in C(Y_\eta) f ∈ C ( Y η ) T t η f T_t^\eta f T t η f C ( Y η ) C(Y_\eta) C ( Y η ) C ( Y η ) C(Y_\eta) C ( Y η ) T t η φ n η = e − t Λ n ( η ) φ n η T_t^\eta\varphi_n^\eta=e^{-t\Lambda_n(\eta)}\varphi_n^\eta T t η φ n η = e − t Λ n ( η ) φ n η
The formula ∫ f d μ t η : = ( T t η f ) ( 0 ) \int f\,d\mu_t^\eta\mathrel{\mathop:}=(T_t^\eta f)(0) ∫ f d μ t η : = ( T t η f ) ( 0 ) μ t η \mu_t^\eta μ t η Y η Y_\eta Y η
∫ φ n η d μ t η = e − t Λ n ( η ) , n ≥ 0. (29) \int\varphi_n^\eta\,d\mu_t^\eta=e^{-t\Lambda_n(\eta)},\qquad n\geq0. \tag*{(29)} ∫ φ n η d μ t η = e − t Λ n ( η ) , n ≥ 0. (29) Suppose that sup n sup Y η ∣ φ n η ∣ < ∞ \sup_n\sup_{Y_\eta}|\varphi_n^\eta|<\infty sup n sup Y η ∣ φ n η ∣ < ∞ μ t η ( { 0 } ) = 0 \mu_t^\eta(\{0\})=0 μ t η ({ 0 }) = 0 t > 0 t>0 t > 0 P t η ( n , ⋅ ) → μ t η P_t^\eta(n,\cdot)\to\mu_t^\eta P t η ( n , ⋅ ) → μ t η ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 n → ∞ n\to\infty n → ∞
Proof. 1. Normal contractions operate on the kernel expression (23) : if T : C → C T:\mathbb{C}\to\mathbb{C} T : C → C T ( 0 ) = 0 T(0)=0 T ( 0 ) = 0 ∣ T ( a ) − T ( b ) ∣ ≤ ∣ a − b ∣ |T(a)-T(b)|\leq|a-b| ∣ T ( a ) − T ( b ) ∣ ≤ ∣ a − b ∣ E η ( T ∘ f ) ≤ E η ( f ) \mathcal{E}_\eta(T\circ f)\leq\mathcal{E}_\eta(f) E η ( T ∘ f ) ≤ E η ( f ) E η \mathcal{E}_\eta E η ( T t η ) (T_t^\eta) ( T t η ) ℓ N 0 ∞ \ell^\infty_{\mathbb{N}_0} ℓ N 0 ∞ [31 Theorem 1.4.1] . Since 1 ∈ P 1\in\mathcal{P} 1 ∈ P L η 1 = 0 L_\eta1=0 L η 1 = 0 T t η 1 = 1 T_t^\eta1=1 T t η 1 = 1 w w w f f f ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) ∑ k f ( k ) 1 { k } \sum_k f(k)1_{\{k\}} ∑ k f ( k ) 1 { k } T t η T_t^\eta T t η n n n T t η f T_t^\eta f T t η f f = 1 f=1 f = 1 ∑ k P t η ( n , k ) = 1 \sum_kP_t^\eta(n,k)=1 ∑ k P t η ( n , k ) = 1
For p ∈ P p\in\mathcal{P} p ∈ P T t η p = ∑ n e − t Λ n ( η ) ⟨ p , e n η ⟩ e n η T_t^\eta p=\sum_n e^{-t\Lambda_n(\eta)}\langle p,e_n^\eta\rangle e_n^\eta T t η p = ∑ n e − t Λ n ( η ) ⟨ p , e n η ⟩ e n η f ∈ C ( Y η ) f\in C(Y_\eta) f ∈ C ( Y η ) p i p_i p i f f f Y η Y_\eta Y η sup n ∣ ( T t η p i ) ( n ) − ( T t η f ) ( n ) ∣ ≤ sup k ∣ p i ( η k ) − f ( η k ) ∣ \sup_n|(T_t^\eta p_i)(n)-(T_t^\eta f)(n)|\leq\sup_k|p_i(\eta^k)-f(\eta^k)| sup n ∣ ( T t η p i ) ( n ) − ( T t η f ) ( n ) ∣ ≤ sup k ∣ p i ( η k ) − f ( η k ) ∣ T t η f T_t^\eta f T t η f N 0 \mathbb{N}_0 N 0 T t η f ∈ C ( Y η ) T_t^\eta f\in C(Y_\eta) T t η f ∈ C ( Y η ) p ∈ P p\in\mathcal{P} p ∈ P T t η p → p T_t^\eta p\to p T t η p → p t → 0 + t\to0^+ t → 0 + ( T t η ) (T_t^\eta) ( T t η ) C ( Y η ) C(Y_\eta) C ( Y η ) T t η φ n η = e − t Λ n ( η ) φ n η T_t^\eta\varphi_n^\eta=e^{-t\Lambda_n(\eta)}\varphi_n^\eta T t η φ n η = e − t Λ n ( η ) φ n η φ n η \varphi_n^\eta φ n η e n η e_n^\eta e n η
The functional f ↦ ( T t η f ) ( 0 ) f\mapsto(T_t^\eta f)(0) f ↦ ( T t η f ) ( 0 ) C ( Y η ) C(Y_\eta) C ( Y η ) T t η φ n η = e − t Λ n ( η ) φ n η T_t^\eta\varphi_n^\eta=e^{-t\Lambda_n(\eta)}\varphi_n^\eta T t η φ n η = e − t Λ n ( η ) φ n η 0 0 0 φ n η ( 0 ) = 1 \varphi_n^\eta(0)=1 φ n η ( 0 ) = 1 (29) .
Let C = sup n sup ∣ φ n η ∣ C=\sup_n\sup|\varphi_n^\eta| C = sup n sup ∣ φ n η ∣ 1 { k } 1_{\{k\}} 1 { k } ( e n η ) (e_n^\eta) ( e n η ) ∑ n ∣ e n η ( η k ) ∣ 2 = w k − 1 \sum_n|e_n^\eta(\eta^k)|^2=w_k^{-1} ∑ n ∣ e n η ( η k ) ∣ 2 = w k − 1 ∑ n ∣ φ n η ( η k ) ∣ 2 h n − 1 = w k − 1 \sum_n|\varphi_n^\eta(\eta^k)|^2h_n^{-1}=w_k^{-1} ∑ n ∣ φ n η ( η k ) ∣ 2 h n − 1 = w k − 1 h n : = ∥ φ n η ∥ 2 = e n η ( 0 ) − 2 h_n\mathrel{\mathop:}=\|\varphi_n^\eta\|^2=e_n^\eta(0)^{-2} h n : = ∥ φ n η ∥ 2 = e n η ( 0 ) − 2 h n ≤ C 2 h_n\leq C^2 h n ≤ C 2 φ n η ( η k ) → 0 \varphi_n^\eta(\eta^k)\to0 φ n η ( η k ) → 0 n → ∞ n\to\infty n → ∞ k k k (29) ,
e − t Λ n ( η ) = μ t η ( { 0 } ) + ∑ k ≥ 0 μ t η ( { η k } ) φ n η ( η k ) . e^{-t\Lambda_n(\eta)}=\mu_t^\eta(\{0\})+\sum_{k\geq0}\mu_t^\eta(\{\eta^k\})\varphi_n^\eta(\eta^k). e − t Λ n ( η ) = μ t η ({ 0 }) + k ≥ 0 ∑ μ t η ({ η k }) φ n η ( η k ) . Letting n → ∞ n\to\infty n → ∞ μ t η ( { 0 } ) = 0 \mu_t^\eta(\{0\})=0 μ t η ({ 0 }) = 0 1 { η k } 1_{\{\eta^k\}} 1 { η k } Y η Y_\eta Y η P t η ( n , k ) = P_t^\eta(n,k)= P t η ( n , k ) = ( T t η 1 { η k } ) ( n ) → μ t η ( { η k } ) (T_t^\eta1_{\{\eta^k\}})(n) \to\mu_t^\eta(\{\eta^k\}) ( T t η 1 { η k } ) ( n ) → μ t η ({ η k }) n → ∞ n \to\infty n → ∞ P t η ( n , ⋅ ) P_t^\eta(n,\cdot) P t η ( n , ⋅ ) μ t η \mu_t^\eta μ t η N 0 \mathbb{N}_0 N 0 ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1
In the sequel we identify μ t η \mu_t^\eta μ t η N 0 \mathbb{N}_0 N 0 μ t η ( { 0 } ) = 0 \mu_t^\eta(\{0\})=0 μ t η ({ 0 }) = 0 μ t η ( k ) \mu_t^\eta(k) μ t η ( k ) μ t η ( { η k } ) \mu_t^\eta(\{\eta^k\}) μ t η ({ η k })
Comparison of energies
We return to the local field, so that τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0
The operator L η L_\eta L η η = Q − α \eta=Q^{-\alpha} η = Q − α
We put
η ≡ d e f Q − α = ρ − 1 , γ α ≡ d e f α − d . \eta\mathrel{\mathop{\equiv}^{\mathrm{def}}} Q^{-\alpha}=\rho^{-1}, \qquad\gamma_\alpha\mathrel{\mathop{\equiv}^{\mathrm{def}}} \alpha-d. η ≡ def Q − α = ρ − 1 , γ α ≡ def α − d . Then τ η = Q γ α \frac{\tau}{\eta}=Q^{\gamma_\alpha} η τ = Q γ α (22) reads
w k β k = ( 1 − τ ) Q γ α ( k + 1 ) ( 1 − η k + 1 ) , k ≥ 0. (30) w_k\beta_k=(1-\tau)Q^{\gamma_\alpha(k+1)}(1-\eta^{k+1}), \qquad k\ge0. \tag*{(30)} w k β k = ( 1 − τ ) Q γ α ( k + 1 ) ( 1 − η k + 1 ) , k ≥ 0. (30) Theorem 4.1 The forms E R \mathcal{E}_R E R (15) and E η \mathcal{E}_\eta E η (23) satisfy, for any f ∈ ℓ N 0 2 ( w ) f\in\ell^2_{\mathbb{N}_0}(w) f ∈ ℓ N 0 2 ( w )
c 1 E η ( f ) ≤ E R ( f ) ≤ c 2 E η ( f ) , (31) c_1\mathcal{E}_\eta(f)\le\mathcal{E}_R(f)\le c_2\mathcal{E}_\eta(f), \tag*{(31)} c 1 E η ( f ) ≤ E R ( f ) ≤ c 2 E η ( f ) , (31) in [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] c 1 = ( ρ − 1 ) ( 1 − τ ) ρ − τ c_1=\frac{(\rho-1)(1-\tau)}{\rho-\tau} c 1 = ρ − τ ( ρ − 1 ) ( 1 − τ )
c 2 = { ρ ( 1 + τ ) ( ρ − τ ) ( 1 − τ ) 2 if α ≥ d , ( 1 − τ ) ρ 3 ( ρ − τ ) ( ρ − 1 ) 2 ( 1 − ρ τ ) if α < d . (32) c_2=
\begin{cases}
\frac{\rho(1+\tau)}{(\rho-\tau)(1-\tau)^2} & \text{if }\alpha\ge d,\\
\frac{(1-\tau)\rho^3}{(\rho-\tau)(\rho-1)^2(1-\rho\tau)} & \text{if }\alpha<d.
\end{cases} \tag*{(32)} c 2 = { ( ρ − τ ) ( 1 − τ ) 2 ρ ( 1 + τ ) ( ρ − τ ) ( ρ − 1 ) 2 ( 1 − ρ τ ) ( 1 − τ ) ρ 3 if α ≥ d , if α < d . (32) In particular, dom E R = dom E η \operatorname{dom}\mathcal{E}_R=\operatorname{dom}\mathcal{E}_\eta dom E R = dom E η Theorem 1.1 , and for α = d \alpha=d α = d 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
Proof. For the lower estimate, we keep only the terms j = 1 j=1 j = 1 (16) . Using 1 − η m + 1 ≤ 1 1-\eta^{m+1}\le1 1 − η m + 1 ≤ 1 (30) , we obtain
E R ( f ) ≥ c ( 1 − τ ) 2 ∑ m ≥ 0 Q γ α m ∣ f m + 1 − f m ∣ 2 ≥ c ( 1 − τ ) τ Q − γ α E η ( f ) . \mathcal{E}_R(f)\ge c(1-\tau)^2\sum_{m\ge0}Q^{\gamma_\alpha m}|f_{m+1}-f_m|^2\ge c(1-\tau)\tau Q^{-\gamma_\alpha}\mathcal{E}_\eta(f). E R ( f ) ≥ c ( 1 − τ ) 2 m ≥ 0 ∑ Q γ α m ∣ f m + 1 − f m ∣ 2 ≥ c ( 1 − τ ) τ Q − γ α E η ( f ) . Since τ Q − γ α = Q − α = ρ − 1 \tau Q^{-\gamma_\alpha}=Q^{-\alpha}=\rho^{-1} τ Q − γ α = Q − α = ρ − 1 c = ρ ( ρ − 1 ) ρ − τ c=\frac{\rho(\rho-1)}{\rho-\tau} c = ρ − τ ρ ( ρ − 1 ) c 1 c_1 c 1
For the upper estimate, let j ≥ 1 j\ge1 j ≥ 1 m ≥ 0 m\ge0 m ≥ 0
∣ f m + j − f m ∣ 2 ≤ j ∑ i = m m + j − 1 ∣ f i + 1 − f i ∣ 2 . |f_{m+j}-f_m|^2\le j\sum_{i=m}^{m+j-1}|f_{i+1}-f_i|^2. ∣ f m + j − f m ∣ 2 ≤ j i = m ∑ m + j − 1 ∣ f i + 1 − f i ∣ 2 . Exchanging the sums over m m m i i i
∑ m ≥ 0 Q γ α m ∣ f m + j − f m ∣ 2 ≤ j ∑ i ≥ 0 ∣ f i + 1 − f i ∣ 2 ∑ m = max ( 0 , i − j + 1 ) i Q γ α m ≤ j C j ∑ i ≥ 0 Q γ α i ∣ f i + 1 − f i ∣ 2 . (33) \sum_{m\ge0}Q^{\gamma_\alpha m}|f_{m+j}-f_m|^2\le j\sum_{i\ge0}|f_{i+1}-f_i|^2\sum_{m=\max(0,i-j+1)}^iQ^{\gamma_\alpha m}\le jC_j\sum_{i\ge0}Q^{\gamma_\alpha i}|f_{i+1}-f_i|^2. \tag*{(33)} m ≥ 0 ∑ Q γ α m ∣ f m + j − f m ∣ 2 ≤ j i ≥ 0 ∑ ∣ f i + 1 − f i ∣ 2 m = m a x ( 0 , i − j + 1 ) ∑ i Q γ α m ≤ j C j i ≥ 0 ∑ Q γ α i ∣ f i + 1 − f i ∣ 2 . (33) where C j = j C_j=j C j = j γ α ≥ 0 \gamma_\alpha\geq0 γ α ≥ 0 C j = Q − γ α ( j − 1 ) 1 − Q γ α C_j=\frac{Q^{-\gamma_\alpha(j-1)}}{1-Q^{\gamma_\alpha}} C j = 1 − Q γ α Q − γ α ( j − 1 ) γ α < 0 \gamma_\alpha<0 γ α < 0 j j j Q γ α i Q^{\gamma_\alpha i} Q γ α i γ α ≥ 0 \gamma_\alpha\geq0 γ α ≥ 0 ∑ m ≥ i − j + 1 Q γ α m = Q γ α ( i − j + 1 ) 1 − Q γ α \sum_{m\geq i-j+1}Q^{\gamma_\alpha m}=\frac{Q^{\gamma_\alpha(i-j+1)}}{1-Q^{\gamma_\alpha}} ∑ m ≥ i − j + 1 Q γ α m = 1 − Q γ α Q γ α ( i − j + 1 ) γ α < 0 \gamma_\alpha<0 γ α < 0 (30) gives Q γ α i ∣ f i + 1 − f i ∣ 2 ≤ Q − γ α ( 1 − τ ) ( 1 − η ) w i β i ∣ f i + 1 − f i ∣ 2 Q^{\gamma_\alpha i}|f_{i+1}-f_i|^2\leq\frac{Q^{-\gamma_\alpha}}{(1-\tau)(1-\eta)w_i\beta_i}|f_{i+1}-f_i|^2 Q γ α i ∣ f i + 1 − f i ∣ 2 ≤ ( 1 − τ ) ( 1 − η ) w i β i Q − γ α ∣ f i + 1 − f i ∣ 2 (16) , we obtain
E R ( f ) ≤ c ( 1 − τ ) Q − γ α 1 − η ( ∑ j ≥ 1 j C j τ j ) E η ( f ) . \mathcal{E}_{R}(f)\leq\frac{c(1-\tau)Q^{-\gamma_\alpha}}{1-\eta}\left(\sum_{j\geq1}jC_j\tau^j\right)\mathcal{E}_{\eta}(f). E R ( f ) ≤ 1 − η c ( 1 − τ ) Q − γ α ( j ≥ 1 ∑ j C j τ j ) E η ( f ) . If γ α ≥ 0 \gamma_\alpha\geq0 γ α ≥ 0 ∑ j j 2 τ j = τ ( 1 + τ ) ( 1 − τ ) 3 \sum_{j}j^2\tau^j=\frac{\tau(1+\tau)}{(1-\tau)^3} ∑ j j 2 τ j = ( 1 − τ ) 3 τ ( 1 + τ ) ρ τ Q − γ α = 1 \rho\tau Q^{-\gamma_\alpha}=1 ρ τ Q − γ α = 1 1 − η = ρ − 1 ρ 1-\eta=\frac{\rho-1}{\rho} 1 − η = ρ ρ − 1 (32) . If γ α < 0 \gamma_\alpha<0 γ α < 0 τ Q − γ α = Q − α = η \tau Q^{-\gamma_\alpha}=Q^{-\alpha}=\eta τ Q − γ α = Q − α = η
∑ j ≥ 1 j C j τ j = Q γ α 1 − Q γ α ∑ j ≥ 1 j η j = Q γ α 1 − Q γ α η ( 1 − η ) 2 , \sum_{j\geq1}jC_j\tau^j=\frac{Q^{\gamma_\alpha}}{1-Q^{\gamma_\alpha}}\sum_{j\geq1}j\eta^j=\frac{Q^{\gamma_\alpha}}{1-Q^{\gamma_\alpha}}\frac{\eta}{(1-\eta)^2}, j ≥ 1 ∑ j C j τ j = 1 − Q γ α Q γ α j ≥ 1 ∑ j η j = 1 − Q γ α Q γ α ( 1 − η ) 2 η , using Q γ α = ρ τ Q^{\gamma_\alpha}=\rho\tau Q γ α = ρ τ η = ρ − 1 \eta=\rho^{-1} η = ρ − 1 (32) . The values for α = d \alpha=d α = d ρ = τ − 1 \rho=\tau^{-1} ρ = τ − 1 (31) , (30) and the bounds 1 − η ≤ 1 − η k + 1 ≤ 1 1-\eta\leq1-\eta^{k+1}\leq1 1 − η ≤ 1 − η k + 1 ≤ 1
By Lemma 2.5 and Proposition 3.1, E R \mathcal{E}_{R} E R E η \mathcal{E}_{\eta} E η R r a d α R^\alpha_{\mathrm{rad}} R rad α L η L_\eta L η
Corollary 4.2 1. For any n ≥ 0 n\geq0 n ≥ 0 (5) holds. In particular, ∣ λ n − Λ n + 1 ( η ) ∣ ≤ 1 + τ |\lambda_n-\Lambda_{n+1}(\eta)|\leq1+\tau ∣ λ n − Λ n + 1 ( η ) ∣ ≤ 1 + τ λ n − Λ n + 1 ( η ) → 1 + τ − κ \lambda_n-\Lambda_{n+1}(\eta)\to1+\tau-\kappa λ n − Λ n + 1 ( η ) → 1 + τ − κ n → ∞ n\to\infty n → ∞
For any 0 ≤ θ ≤ 1 0\leq\theta\leq1 0 ≤ θ ≤ 1 dom ( R r a d α ) θ / 2 = dom L η θ / 2 \operatorname{dom}(R^\alpha_{\mathrm{rad}})^{\theta/2}=\operatorname{dom}L_\eta^{\theta/2} dom ( R rad α ) θ /2 = dom L η θ /2 f ↦ ∥ f ∥ + ∥ ( R r a d α ) θ / 2 f ∥ f\mapsto\|f\|+\|(R^\alpha_{\mathrm{rad}})^{\theta/2}f\| f ↦ ∥ f ∥ + ∥ ( R rad α ) θ /2 f ∥ f ↦ ∥ f ∥ + ∥ L η θ / 2 f ∥ f\mapsto\|f\|+\|L_\eta^{\theta/2}f\| f ↦ ∥ f ∥ + ∥ L η θ /2 f ∥
Proof. 1. Since η = ρ − 1 \eta=\rho^{-1} η = ρ − 1
Λ n + 1 ( η ) = ρ n + 1 ( 1 − ρ − n − 1 ) ( 1 − τ ρ − n − 1 ) = ρ n + 1 − 1 − τ + τ ρ − n − 1 . \Lambda_{n+1}(\eta)=\rho^{n+1}(1-\rho^{-n-1})(1-\tau\rho^{-n-1})=\rho^{n+1}-1-\tau+\tau\rho^{-n-1}. Λ n + 1 ( η ) = ρ n + 1 ( 1 − ρ − n − 1 ) ( 1 − τ ρ − n − 1 ) = ρ n + 1 − 1 − τ + τ ρ − n − 1 . Subtracting this from λ n = ρ n + 1 − κ \lambda_n=\rho^{n+1}-\kappa λ n = ρ n + 1 − κ (5) . Since 0 < κ < 1 0<\kappa<1 0 < κ < 1 0 < τ ρ − n − 1 < τ 0<\tau\rho^{-n-1}<\tau 0 < τ ρ − n − 1 < τ ( 0 , 1 + τ ) (0,1+\tau) ( 0 , 1 + τ )
By Theorem 4.1, the Hilbert spaces dom ( R r a d α ) 1 / 2 \operatorname{dom}(R^\alpha_{\mathrm{rad}})^{1/2} dom ( R rad α ) 1/2 dom L η 1 / 2 \operatorname{dom}L_\eta^{1/2} dom L η 1/2 A A A H H H A + I A+I A + I ∥ ( A + I ) i s ∥ ≤ 1 \|(A+I)^{is}\|\leq1 ∥ ( A + I ) i s ∥ ≤ 1 [36 Theorem 6.6.9] , the complex interpolation space [ H , dom A 1 / 2 ] θ [H,\operatorname{dom}A^{1/2}]_\theta [ H , dom A 1/2 ] θ dom A θ / 2 \operatorname{dom}A^{\theta/2} dom A θ /2 A = R r a d α A=R^\alpha_{\mathrm{rad}} A = R rad α A = L η A=L_\eta A = L η
Powers of the zonal operators
In this subsection, we fix a real number ϑ > 0 \vartheta> 0 ϑ > 0
η ≔ Q − ϑ \eta\coloneqq Q^{-\vartheta} η : = Q − ϑ in Section 3. The case ϑ = α \vartheta= \alpha ϑ = α ϑ = d \vartheta= d ϑ = d η = τ \eta= \tau η = τ L τ L_\tau L τ S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) r 2 = τ r^2 = \tau r 2 = τ n ≥ 2 n \ge2 n ≥ 2 ϑ = d n − 1 \vartheta= \frac{d}{n-1} ϑ = n − 1 d C P q n − 1 \mathbb{CP}^{n-1}_q CP q n − 1 q 2 = η q^2 = \eta q 2 = η Λ m ≔ Λ m ( η ) \Lambda_m \coloneqq\Lambda_m(\eta) Λ m : = Λ m ( η ) e m ≔ e m η e_m \coloneqq e_m^\eta e m : = e m η m ≥ 0 m \ge0 m ≥ 0 e 0 = 1 e_0 = 1 e 0 = 1 L η e m = Λ m e m L_\eta e_m = \Lambda_m e_m L η e m = Λ m e m e m = ± ∥ φ m η ∥ − 1 φ m η e_m = \pm\lVert\varphi_m^\eta\rVert^{-1}\varphi_m^\eta e m = ± ∥ φ m η ∥ − 1 φ m η s > 0 s > 0 s > 0 L η s L_\eta^s L η s f ↦ ∥ L η s 2 f ∥ 2 = ∑ m ≥ 1 Λ m s ∣ ⟨ f , e m ⟩ ∣ 2 f \mapsto\lVert L_\eta^{\frac{s}{2}}f\rVert^2 = \sum_{m\ge1}\Lambda_m^s|\langle f,e_m\rangle|^2 f ↦ ∥ L η 2 s f ∥ 2 = ∑ m ≥ 1 Λ m s ∣ ⟨ f , e m ⟩ ∣ 2 dom L η s 2 \operatorname{dom}L_\eta^{\frac{s}{2}} dom L η 2 s 0 < τ < 1 0 < \tau< 1 0 < τ < 1 0 < η < 1 0 < \eta< 1 0 < η < 1 Λ m ( η ) = η − m ( 1 − η m ) ( 1 − τ η m ) \Lambda_m(\eta) = \eta^{-m}(1-\eta^m)(1-\tau\eta^m) Λ m ( η ) = η − m ( 1 − η m ) ( 1 − τ η m )
c η η − m ≤ Λ m ( η ) ≤ η − m , m ≥ 1 , c_\eta\eta^{-m} \le\Lambda_m(\eta) \le\eta^{-m}, \qquad m \ge1, c η η − m ≤ Λ m ( η ) ≤ η − m , m ≥ 1 , where c η ≔ ( 1 − η ) ( 1 − τ η ) c_\eta\coloneqq(1-\eta)(1-\tau\eta) c η : = ( 1 − η ) ( 1 − τ η )
The following two lemmas hold for any 0 < τ < 1 0 < \tau< 1 0 < τ < 1 0 < η < 1 0 < \eta< 1 0 < η < 1
Lemma 4.4 Let m ≥ 1 m \ge1 m ≥ 1 φ m η ( x ) = ∑ j = 0 m c m , j x j \varphi_m^\eta(x) = \sum_{j=0}^m c_{m,j}x^j φ m η ( x ) = ∑ j = 0 m c m , j x j D ≔ ( τ ; η ) ∞ − 1 ( η ; η ) ∞ − 1 D \coloneqq(\tau;\eta)_\infty^{-1}(\eta;\eta)_\infty^{-1} D : = ( τ ; η ) ∞ − 1 ( η ; η ) ∞ − 1
∣ c m , j ∣ ≤ D η − m j + j ( j + 1 ) 2 , 0 ≤ j ≤ m . |c_{m,j}| \le D\eta^{-mj+\frac{j(j+1)}{2}}, \qquad0 \le j \le m. ∣ c m , j ∣ ≤ D η − mj + 2 j ( j + 1 ) , 0 ≤ j ≤ m . Consequently, ∣ ( φ m η ) ′ ( x ) ∣ ≤ A η − m |(\varphi_m^\eta)'(x)| \le A\eta^{-m} ∣ ( φ m η ) ′ ( x ) ∣ ≤ A η − m 0 ≤ x ≤ η m − 1 0 \le x \le\eta^{m-1} 0 ≤ x ≤ η m − 1 ∣ 1 − φ m η ( x ) ∣ ≤ A ′ η − m x |1-\varphi_m^\eta(x)| \le A'\eta^{-m}x ∣1 − φ m η ( x ) ∣ ≤ A ′ η − m x 0 ≤ x ≤ η m 0 \le x \le\eta^m 0 ≤ x ≤ η m
A ≔ η D ∑ j ≥ 1 j η j ( j − 1 ) 2 , A ′ ≔ D ∑ j ≥ 1 η j ( j + 1 ) 2 . A \coloneqq\eta D\sum_{j\ge1}j\eta^{\frac{j(j-1)}{2}}, \qquad A' \coloneqq D\sum_{j\ge1}\eta^{\frac{j(j+1)}{2}}. A : = η D j ≥ 1 ∑ j η 2 j ( j − 1 ) , A ′ : = D j ≥ 1 ∑ η 2 j ( j + 1 ) . Moreover, φ 1 η ( x ) = 1 − 1 − τ η 1 − τ x \varphi_1^\eta(x)=1-\frac{1-\tau\eta}{1-\tau}x φ 1 η ( x ) = 1 − 1 − τ 1 − τ η x
∥ φ m η ∥ 2 = ( 1 − τ ) τ m 1 − τ η 2 m ( ( η ; η ) m ( τ ; η ) m ) 2 . \lVert\varphi_m^\eta\rVert^2=\frac{(1-\tau)\tau^m}{1-\tau\eta^{2m}}\left(\frac{(\eta;\eta)_m}{(\tau;\eta)_m}\right)^2. ∥ φ m η ∥ 2 = 1 − τ η 2 m ( 1 − τ ) τ m ( ( τ ; η ) m ( η ; η ) m ) 2 . Proof : By part 4 of Proposition 3.1, φ m η \varphi_m^\eta φ m η η \eta η p m ( x ; τ η − 1 , 1 ; η ) p_m(x;\tau\eta^{-1},1;\eta) p m ( x ; τ η − 1 , 1 ; η ) [44 (14.12.1)] gives
c m , j = ( η − m ; η ) j ( τ η m ; η ) j ( τ ; η ) j ( η ; η ) j η j . c_{m,j}=\frac{(\eta^{-m};\eta)_j(\tau\eta^m;\eta)_j}{(\tau;\eta)_j(\eta;\eta)_j}\eta^j. c m , j = ( τ ; η ) j ( η ; η ) j ( η − m ; η ) j ( τ η m ; η ) j η j . For 1 ≤ j ≤ m 1 \le j \le m 1 ≤ j ≤ m ∣ ( η − m ; η ) j ∣ = ∏ i = 0 j − 1 η i − m ( 1 − η m − i ) ≤ η − m j + j ( j − 1 ) 2 |(\eta^{-m};\eta)_j|=\prod_{i=0}^{j-1}\eta^{i-m}(1-\eta^{m-i})\le\eta^{-mj+\frac{j(j-1)}{2}} ∣ ( η − m ; η ) j ∣ = ∏ i = 0 j − 1 η i − m ( 1 − η m − i ) ≤ η − mj + 2 j ( j − 1 ) 0 < ( τ η m ; η ) j ≤ 1 0<(\tau\eta^m;\eta)_j\le1 0 < ( τ η m ; η ) j ≤ 1 ( τ ; η ) j ≥ ( τ ; η ) ∞ (\tau;\eta)_j\ge(\tau;\eta)_\infty ( τ ; η ) j ≥ ( τ ; η ) ∞ ( η ; η ) j ≥ ( η ; η ) ∞ (\eta;\eta)_j\ge(\eta;\eta)_\infty ( η ; η ) j ≥ ( η ; η ) ∞ m = j = 1 m=j=1 m = j = 1 c 1 , 1 = − 1 − τ η 1 − τ c_{1,1}=-\frac{1-\tau\eta}{1-\tau} c 1 , 1 = − 1 − τ 1 − τ η 0 ≤ x ≤ η m − 1 0\le x\le\eta^{m-1} 0 ≤ x ≤ η m − 1
∣ ( φ m η ) ′ ( x ) ∣ ≤ ∑ j ≥ 1 j ∣ c m , j ∣ η ( m − 1 ) ( j − 1 ) ≤ D ∑ j ≥ 1 j η j ( j − 1 ) 2 − m + 1 = A η − m . |(\varphi_m^\eta)'(x)|\le\sum_{j\ge1}j|c_{m,j}|\eta^{(m-1)(j-1)}\le D\sum_{j\ge1}j\eta^{\frac{j(j-1)}{2}-m+1}=A\eta^{-m}. ∣ ( φ m η ) ′ ( x ) ∣ ≤ j ≥ 1 ∑ j ∣ c m , j ∣ η ( m − 1 ) ( j − 1 ) ≤ D j ≥ 1 ∑ j η 2 j ( j − 1 ) − m + 1 = A η − m . If 0 ≤ x ≤ η m 0\le x\le\eta^m 0 ≤ x ≤ η m x j ≤ η m ( j − 1 ) x x^j\le\eta^{m(j-1)}x x j ≤ η m ( j − 1 ) x j ≥ 1 j\ge1 j ≥ 1 c m , 0 = 1 c_{m,0}=1 c m , 0 = 1 ∣ 1 − φ m η ( x ) ∣ ≤ ∑ j ≥ 1 ∣ c m , j ∣ η m ( j − 1 ) x ≤ A ′ η − m x |1-\varphi_m^\eta(x)|\le\sum_{j\ge1}|c_{m,j}|\eta^{m(j-1)}x\le A'\eta^{-m}x ∣1 − φ m η ( x ) ∣ ≤ ∑ j ≥ 1 ∣ c m , j ∣ η m ( j − 1 ) x ≤ A ′ η − m x [44 (14.12.2)] with a = τ η − 1 a=\tau\eta^{-1} a = τ η − 1 b = 1 b=1 b = 1 1 − τ 1-\tau 1 − τ w k w_k w k
For n ≥ 1 n \ge1 n ≥ 1 ψ n ≔ g n − 1 \psi_n \coloneqq g_{n-1} ψ n : = g n − 1 R r a d α ψ n = λ n − 1 ψ n R_{\mathrm{rad}}^{\alpha}\psi_n=\lambda_{n-1}\psi_n R rad α ψ n = λ n − 1 ψ n τ = Q − d \tau=Q^{-d} τ = Q − d n , m ≥ 1 n,m\ge1 n , m ≥ 1
u n m ≔ ⟨ e m , ψ n ⟩ ℓ N 0 2 ( w ) . u_{nm}\coloneqq\langle e_m,\psi_n\rangle_{\ell^2_{N_0}(w)}. u nm : = ⟨ e m , ψ n ⟩ ℓ N 0 2 ( w ) . The families ( ψ n ) n ≥ 1 (\psi_n)_{n\ge1} ( ψ n ) n ≥ 1 ( e m ) m ≥ 1 (e_m)_{m\ge1} ( e m ) m ≥ 1 ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) Theorem 2.3 and Proposition 3.1. Thus ( u n m ) (u_{nm}) ( u nm ) ψ n \psi_n ψ n α \alpha α η \eta η
Lemma 4.5 For any integers n , m ≥ 1 n,m\ge1 n , m ≥ 1
∣ u n m ∣ ≤ C 1 ( τ 1 2 η ) n − m if n ≥ m , ∣ u n m ∣ ≤ C 2 B m − n η ( m − n ) 2 2 if m > n , (34) |u_{nm}|\le C_1\left(\tau^{\frac{1}{2}}\eta\right)^{n-m}\quad\text{if }n\ge m,\qquad|u_{nm}|\le C_2B^{m-n}\eta^{\frac{(m-n)^2}{2}}\quad\text{if }m>n, \tag*{(34)} ∣ u nm ∣ ≤ C 1 ( τ 2 1 η ) n − m if n ≥ m , ∣ u nm ∣ ≤ C 2 B m − n η 2 ( m − n ) 2 if m > n , (34) where
C 1 ≔ ( 1 − η ) A η ( 1 − τ η ) ( η ; η ) ∞ , C 2 ≔ 3 ( τ + η ) 4 τ η and B ≔ 4 η 1 2 c η . C_1\coloneqq\frac{(1-\eta)A}{\eta(1-\tau\eta)(\eta;\eta)_\infty},\qquad C_2\coloneqq\frac{\sqrt{3}(\tau+\eta)}{4\tau\eta}\quad\text{and}\quad B\coloneqq\frac{4\eta^{\frac{1}{2}}}{c_\eta}. C 1 : = η ( 1 − τ η ) ( η ; η ) ∞ ( 1 − η ) A , C 2 : = 4 τ η 3 ( τ + η ) and B : = c η 4 η 2 1 . Proof : 1. Let f ∈ ℓ N 0 2 ( w ) f\in\ell^2_{N_0}(w) f ∈ ℓ N 0 2 ( w ) n ≥ 1 n\ge1 n ≥ 1 τ w k = w k + 1 \tau w_k=w_{k+1} τ w k = w k + 1
⟨ f , 1 B n − τ 1 B n − 1 ⟩ = ∑ k ≥ n w k f ( k ) − ∑ k ≥ n − 1 w k + 1 f ( k ) = ∑ k ≥ n w k ( f ( k ) − f ( k − 1 ) ) , \left\langle f,1_{B_n}-\tau1_{B_{n-1}}\right\rangle=\sum_{k\ge n}w_kf(k)-\sum_{k\ge n-1}w_{k+1}f(k)=\sum_{k\ge n}w_k\bigl(f(k)-f(k-1)\bigr), ⟨ f , 1 B n − τ 1 B n − 1 ⟩ = k ≥ n ∑ w k f ( k ) − k ≥ n − 1 ∑ w k + 1 f ( k ) = k ≥ n ∑ w k ( f ( k ) − f ( k − 1 ) ) , the series being absolutely convergent. By (13) , we deduce that
⟨ f , ψ n ⟩ = ( ( 1 − τ ) τ n ) − 1 2 ∑ k ≥ n w k ( f ( k ) − f ( k − 1 ) ) . (35) \langle f,\psi_n\rangle=\bigl((1-\tau)\tau^n\bigr)^{-\frac{1}{2}}\sum_{k\ge n}w_k\bigl(f(k)-f(k-1)\bigr). \tag*{(35)} ⟨ f , ψ n ⟩ = ( ( 1 − τ ) τ n ) − 2 1 k ≥ n ∑ w k ( f ( k ) − f ( k − 1 ) ) . (35) Suppose that n ≥ m n\ge m n ≥ m k ≥ n k\ge n k ≥ n [ η k , η k − 1 ] [\eta^k,\eta^{k-1}] [ η k , η k − 1 ] [ 0 , η m − 1 ] [0,\eta^{m-1}] [ 0 , η m − 1 ] ( τ ; η ) m ≤ 1 (\tau;\eta)_m\le1 ( τ ; η ) m ≤ 1 ( η ; η ) m ≥ ( η ; η ) ∞ (\eta;\eta)_m\ge(\eta;\eta)_\infty ( η ; η ) m ≥ ( η ; η ) ∞ ∥ φ m ′ ∥ ∞ ≤ ( 1 − τ ) − 1 2 τ − m 2 ( η ; η ) ∞ − 1 \|\varphi_m'\|_\infty\le(1-\tau)^{-\frac{1}{2}}\tau^{-\frac{m}{2}}(\eta;\eta)_\infty^{-1} ∥ φ m ′ ∥ ∞ ≤ ( 1 − τ ) − 2 1 τ − 2 m ( η ; η ) ∞ − 1 Lemma 4.4 and the mean value theorem, we obtain
∣ e m ( k ) − e m ( k − 1 ) ∣ ≤ ( 1 − τ ) − 1 2 τ − m 2 ( η ; η ) ∞ − 1 A η − m ( 1 − η ) η k − 1 , k ≥ n . |e_m(k)-e_m(k-1)|\le(1-\tau)^{-\frac{1}{2}}\tau^{-\frac{m}{2}}(\eta;\eta)_\infty^{-1}A\eta^{-m}(1-\eta)\eta^{k-1},\qquad k\ge n. ∣ e m ( k ) − e m ( k − 1 ) ∣ ≤ ( 1 − τ ) − 2 1 τ − 2 m ( η ; η ) ∞ − 1 A η − m ( 1 − η ) η k − 1 , k ≥ n . Inserting this bound in (35) with f = e m f=e_m f = e m
∣ u n m ∣ ≤ ( 1 − η ) A η ( η ; η ) ∞ τ − n + m 2 η − m ∑ k ≥ n ( τ η ) k = C 1 ( τ 1 2 η ) n − m . |u_{nm}|\le\frac{(1-\eta)A}{\eta(\eta;\eta)_\infty}\tau^{-\frac{n+m}{2}}\eta^{-m}\sum_{k\ge n}(\tau\eta)^k=C_1\left(\tau^{\frac{1}{2}}\eta\right)^{n-m}. ∣ u nm ∣ ≤ η ( η ; η ) ∞ ( 1 − η ) A τ − 2 n + m η − m k ≥ n ∑ ( τ η ) k = C 1 ( τ 2 1 η ) n − m . Suppose that m > n m>n m > n g g g dom L η \operatorname{dom}L_\eta dom L η L η g L_\eta g L η g (3) . Indeed, g ∈ dom E η g\in\operatorname{dom}\mathcal{E}_\eta g ∈ dom E η f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_\eta f ∈ dom E η p p p g g g N N N L η g L_\eta g L η g (3) is finitely supported, letting N → ∞ N\to\infty N → ∞ E η ( f , g ) = ⟨ f , L η g ⟩ \mathcal{E}_\eta(f,g)=\langle f,L_\eta g\rangle E η ( f , g ) = ⟨ f , L η g ⟩ f ∈ dom E η f\in\operatorname{dom}\mathcal{E}_\eta f ∈ dom E η
In the orthonormal basis ( w k − 1 2 1 { k } ) k ≥ 0 (w_k^{-\frac{1}{2}}1_{\{k\}})_{k\ge0} ( w k − 2 1 1 { k } ) k ≥ 0 L η L_\eta L η (22) , we have β k = τ δ k + 1 \beta_k=\tau\delta_{k+1} β k = τ δ k + 1 β k + δ k \beta_k+\delta_k β k + δ k ( 1 + τ η − 1 ) η − k (1+\tau\eta^{-1})\eta^{-k} ( 1 + τ η − 1 ) η − k − β k δ k + 1 = − τ 1 2 δ k + 1 -\sqrt{\beta_k\delta_{k+1}}=-\tau^{\frac{1}{2}}\delta_{k+1} − β k δ k + 1 = − τ 2 1 δ k + 1 τ 1 2 η − 1 η − k \tau^{\frac{1}{2}}\eta^{-1}\eta^{-k} τ 2 1 η − 1 η − k 1 + τ η − 1 + 2 τ 1 2 η − 1 ≤ ( 1 + τ 1 2 ) 2 η − 1 ≤ 4 η − 1 1+\tau\eta^{-1}+2\tau^{\frac{1}{2}}\eta^{-1}\le(1+\tau^{\frac{1}{2}})^2\eta^{-1}\le4\eta^{-1} 1 + τ η − 1 + 2 τ 2 1 η − 1 ≤ ( 1 + τ 2 1 ) 2 η − 1 ≤ 4 η − 1 g g g { 0 , … , N } \{0,\ldots,N\} { 0 , … , N } L η g L_\eta g L η g { 0 , … , N + 1 } \{0,\ldots,N+1\} { 0 , … , N + 1 }
∥ L η g ∥ ≤ 4 η − N − 1 ∥ g ∥ . (36) \|L_\eta g\|\le4\eta^{-N-1}\|g\|. \tag*{(36)} ∥ L η g ∥ ≤ 4 η − N − 1 ∥ g ∥. (36) The sequence ψ n \psi_n ψ n { k ≤ n − 2 } \{k\le n-2\} { k ≤ n − 2 } { k ≥ n } \{k\ge n\} { k ≥ n } ( ( 1 − τ ) τ n ) − 1 2 \bigl((1-\tau)\tau^n\bigr)^{-\frac{1}{2}} ( ( 1 − τ ) τ n ) − 2 1 L η ψ n L_\eta\psi_n L η ψ n { n − 2 , n − 1 , n } \{n-2,n-1,n\} { n − 2 , n − 1 , n }
∣ L η ψ n ( k ) ∣ ≤ ( 1 + τ η − 1 ) η − k ( ( 1 − τ ) τ n ) − 1 2 , |L_\eta\psi_n(k)|\le(1+\tau\eta^{-1})\eta^{-k}\bigl((1-\tau)\tau^n\bigr)^{-\frac{1}{2}}, ∣ L η ψ n ( k ) ∣ ≤ ( 1 + τ η − 1 ) η − k ( ( 1 − τ ) τ n ) − 2 1 , so that
∥ L η ψ n ∥ 2 ≤ ( 1 + τ η − 1 ) 2 ∑ k = n − 2 n τ k − n η − 2 k ≤ 3 ( 1 + τ η − 1 ) 2 τ − 2 η − 2 n . \|L_\eta\psi_n\|^2\le(1+\tau\eta^{-1})^2\sum_{k=n-2}^{n}\tau^{k-n}\eta^{-2k}\le3(1+\tau\eta^{-1})^2\tau^{-2}\eta^{-2n}. ∥ L η ψ n ∥ 2 ≤ ( 1 + τ η − 1 ) 2 k = n − 2 ∑ n τ k − n η − 2 k ≤ 3 ( 1 + τ η − 1 ) 2 τ − 2 η − 2 n . Let K ≥ 1 K \ge1 K ≥ 1 1 ≤ i ≤ K 1 \le i \le K 1 ≤ i ≤ K L η i ψ n L_\eta^i\psi_n L η i ψ n { 0 , … , n + i − 1 } \{0,\ldots,n+i-1\} { 0 , … , n + i − 1 } (36) gives
∥ L η K ψ n ∥ ≤ 3 ( 1 + τ η − 1 ) τ − 1 η − n ∏ i = 1 K − 1 4 η − n − i = 3 ( 1 + τ η − 1 ) τ − 1 4 K − 1 η − K n − K ( K − 1 ) 2 . \left\|L_\eta^K\psi_n\right\| \le\sqrt{3}(1+\tau\eta^{-1})\tau^{-1}\eta^{-n}\prod_{i=1}^{K-1}4\eta^{-n-i}=\sqrt{3}(1+\tau\eta^{-1})\tau^{-1}4^{K-1}\eta^{-Kn-\frac{K(K-1)}{2}}. L η K ψ n ≤ 3 ( 1 + τ η − 1 ) τ − 1 η − n i = 1 ∏ K − 1 4 η − n − i = 3 ( 1 + τ η − 1 ) τ − 1 4 K − 1 η − K n − 2 K ( K − 1 ) . Since L η L_\eta L η L η e m = Λ m e m L_\eta e_m=\Lambda_m e_m L η e m = Λ m e m Λ m K u n m = ⟨ e m , L η K ψ n ⟩ \Lambda_m^K u_{nm}=\langle e_m,L_\eta^K\psi_n\rangle Λ m K u nm = ⟨ e m , L η K ψ n ⟩ K = m − n K=m-n K = m − n (33) , we obtain
∣ u n m ∣ ≤ c η − K η K m ∥ L η K ψ n ∥ ≤ 3 ( 1 + τ η − 1 ) 4 τ ( 4 c η ) K η K 2 2 + K 2 = C 2 B K η K 2 2 . |u_{nm}|\le c_\eta^{-K}\eta^{Km}\left\|L_\eta^K\psi_n\right\|\le\frac{\sqrt{3}(1+\tau\eta^{-1})}{4\tau}\left(\frac{4}{c_\eta}\right)^K\eta^{\frac{K^2}{2}+\frac{K}{2}}=C_2B^{K}\eta^{\frac{K^2}{2}}. ∣ u nm ∣ ≤ c η − K η K m L η K ψ n ≤ 4 τ 3 ( 1 + τ η − 1 ) ( c η 4 ) K η 2 K 2 + 2 K = C 2 B K η 2 K 2 . This proves (34) .
Theorem 4.6 Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0 ϑ > 0 \vartheta>0 ϑ > 0 η = Q − ϑ \eta=Q^{-\vartheta} η = Q − ϑ s ≔ α ϑ s\coloneqq\frac{\alpha}{\vartheta} s : = ϑ α
We have dom E R ⊂ dom L η s / 2 \operatorname{dom}\mathcal{E}_R\subset\operatorname{dom}L_\eta^{s/2} dom E R ⊂ dom L η s /2 c > 0 c>0 c > 0 c ∥ L η s / 2 f ∥ 2 ≤ E R ( f ) c\left\|L_\eta^{s/2}f\right\|^2\le\mathcal{E}_R(f) c L η s /2 f 2 ≤ E R ( f ) f ∈ dom E R f\in\operatorname{dom}\mathcal{E}_R f ∈ dom E R
If α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ dom E R = dom L η s / 2 \operatorname{dom}\mathcal{E}_R=\operatorname{dom}L_\eta^{s/2} dom E R = dom L η s /2 C C C E R ( f ) ≤ C ∥ L η s / 2 f ∥ 2 \mathcal{E}_R(f)\le C\left\|L_\eta^{s/2}f\right\|^2 E R ( f ) ≤ C L η s /2 f 2 f f f R r a d α R_{\mathrm{rad}}^\alpha R rad α L η s L_\eta^s L η s
If α ≥ d + 2 ϑ \alpha\ge d+2\vartheta α ≥ d + 2 ϑ
The constants c c c C C C Q Q Q d d d α \alpha α ϑ \vartheta ϑ ϑ = d \vartheta=d ϑ = d R r a d α R_{\mathrm{rad}}^\alpha R rad α L τ α / d L_\tau^{\alpha/d} L τ α / d α < 3 d \alpha<3d α < 3 d
Proof: Both quantities vanish on the constants and are unchanged if a constant is added to f f f f ⊥ 1 f\perp1 f ⊥ 1 n , m ≥ 1 n,m\ge1 n , m ≥ 1 a n ≔ ⟨ f , ψ n ⟩ a_n\coloneqq\langle f,\psi_n\rangle a n : = ⟨ f , ψ n ⟩ b m ≔ ⟨ f , e m ⟩ b_m\coloneqq\langle f,e_m\rangle b m : = ⟨ f , e m ⟩ a n = ∑ m u n m b m a_n=\sum_m u_{nm}b_m a n = ∑ m u nm b m b m = ∑ n u n m ‾ a n b_m=\sum_n\overline{u_{nm}}a_n b m = ∑ n u nm a n ∑ m ∣ u n m ∣ 2 = ∑ n ∣ u n m ∣ 2 = 1 \sum_m|u_{nm}|^2=\sum_n|u_{nm}|^2=1 ∑ m ∣ u nm ∣ 2 = ∑ n ∣ u nm ∣ 2 = 1 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
E R ( f ) = ∑ n ≥ 1 λ n − 1 ∣ a n ∣ 2 , ∥ L η s / 2 f ∥ 2 = ∑ m ≥ 1 Λ m s ∣ b m ∣ 2 . \mathcal{E}_R(f)=\sum_{n\ge1}\lambda_{n-1}|a_n|^2,\qquad\left\|L_\eta^{s/2}f\right\|^2=\sum_{m\ge1}\Lambda_m^s|b_m|^2. E R ( f ) = n ≥ 1 ∑ λ n − 1 ∣ a n ∣ 2 , L η s /2 f 2 = m ≥ 1 ∑ Λ m s ∣ b m ∣ 2 . Since η − s = Q α = ρ \eta^{-s}=Q^\alpha=\rho η − s = Q α = ρ 0 < κ < 1 0<\kappa<1 0 < κ < 1 (33) gives c η s ρ m ≤ Λ m s ≤ ρ m c_\eta^s\rho^m\le\Lambda_m^s\le\rho^m c η s ρ m ≤ Λ m s ≤ ρ m ( 1 − ρ − 1 ) ρ n ≤ λ n − 1 = ρ n − κ ≤ ρ n (1-\rho^{-1})\rho^n\le\lambda_{n-1}=\rho^n-\kappa\le\rho^n ( 1 − ρ − 1 ) ρ n ≤ λ n − 1 = ρ n − κ ≤ ρ n ( k i j ) i , j ≥ 1 (k_{ij})_{i,j\ge1} ( k ij ) i , j ≥ 1 ∣ k i j ∣ ≤ k ( i − j ) |k_{ij}|\le k(i-j) ∣ k ij ∣ ≤ k ( i − j ) k ∈ ℓ Z 1 k\in\ell^1_{\mathbb{Z}} k ∈ ℓ Z 1 ℓ N 2 \ell^2_{\mathbb{N}} ℓ N 2 ∥ k ∥ ℓ Z 1 \|k\|_{\ell^1_{\mathbb{Z}}} ∥ k ∥ ℓ Z 1
Suppose that E R ( f ) < ∞ \mathcal{E}_R(f)<\infty E R ( f ) < ∞ x n ≔ ρ n / 2 a n x_n\coloneqq\rho^{n/2}a_n x n : = ρ n /2 a n ρ m / 2 b m = ∑ n ρ ( m − n ) / 2 u n m x n \rho^{m/2}b_m=\sum_n\rho^{(m-n)/2}u_{nm}x_n ρ m /2 b m = ∑ n ρ ( m − n ) /2 u nm x n ρ ( m − n ) / 2 ∣ u n m ∣ ≤ k 1 ( m − n ) \rho^{(m-n)/2}|u_{nm}|\le k_1(m-n) ρ ( m − n ) /2 ∣ u nm ∣ ≤ k 1 ( m − n ) k 1 ( j ) ≔ C 2 ( B ρ 1 / 2 ) j η j 2 / 2 k_1(j)\coloneqq C_2(B\rho^{1/2})^j\eta^{j^2/2} k 1 ( j ) : = C 2 ( B ρ 1/2 ) j η j 2 /2 j > 0 j>0 j > 0 k 1 ( j ) ≔ C 1 ( τ η 2 ρ − 1 ) ∣ j ∣ / 2 k_1(j)\coloneqq C_1(\tau\eta^2\rho^{-1})^{|j|/2} k 1 ( j ) : = C 1 ( τ η 2 ρ − 1 ) ∣ j ∣/2 j ≤ 0 j\le0 j ≤ 0 k 1 ∈ ℓ Z 1 k_1\in\ell^1_{\mathbb{Z}} k 1 ∈ ℓ Z 1
∥ L η s / 2 f ∥ 2 ≤ ∑ m ≥ 1 ρ m ∣ b m ∣ 2 ≤ ∥ k 1 ∥ ℓ Z 2 2 ∑ n ≥ 1 ρ n ∣ a n ∣ 2 ≤ ∥ k 1 ∥ ℓ Z 2 2 1 − ρ − 1 E R ( f ) . \left\|L_\eta^{s/2}f\right\|^2\le\sum_{m\ge1}\rho^m|b_m|^2\le\|k_1\|_{\ell^2_{\mathbb{Z}}}^2\sum_{n\ge1}\rho^n|a_n|^2\le\frac{\|k_1\|_{\ell^2_{\mathbb{Z}}}^2}{1-\rho^{-1}}\mathcal{E}_R(f). L η s /2 f 2 ≤ m ≥ 1 ∑ ρ m ∣ b m ∣ 2 ≤ ∥ k 1 ∥ ℓ Z 2 2 n ≥ 1 ∑ ρ n ∣ a n ∣ 2 ≤ 1 − ρ − 1 ∥ k 1 ∥ ℓ Z 2 2 E R ( f ) . Suppose that α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ ρ τ η 2 = Q α − d − 2 ϑ < 1 \rho\tau\eta^2=Q^{\alpha-d-2\vartheta}<1 ρ τ η 2 = Q α − d − 2 ϑ < 1 ∥ L η s / 2 f ∥ < ∞ \left\|L_\eta^{s/2}f\right\|<\infty L η s /2 f < ∞ y m ≔ ρ m / 2 b m y_m\coloneqq\rho^{m/2}b_m y m : = ρ m /2 b m ρ n / 2 a n = ∑ m ρ ( n − m ) / 2 u n m y m \rho^{n/2}a_n=\sum_m\rho^{(n-m)/2}u_{nm}y_m ρ n /2 a n = ∑ m ρ ( n − m ) /2 u nm y m ρ ( n − m ) / 2 ∣ u n m ∣ ≤ k 2 ( n − m ) \rho^{(n-m)/2}|u_{nm}|\le k_2(n-m) ρ ( n − m ) /2 ∣ u nm ∣ ≤ k 2 ( n − m )
C 1 ( ρ τ η 2 ) j 2 C_{1}(\rho\tau\eta^{2})^{\frac{j}{2}} C 1 ( ρ τ η 2 ) 2 j j ≥ 0 j \ge0 j ≥ 0 k 2 ( j ) ≔ C 2 ( B ρ − 1 2 ) ∣ j ∣ η j 2 2 k_{2}(j) \coloneqq C_{2}(B\rho^{-\frac{1}{2}})^{|j|}\eta^{\frac{j^{2}}{2}} k 2 ( j ) : = C 2 ( B ρ − 2 1 ) ∣ j ∣ η 2 j 2 j < 0 j < 0 j < 0 ρ τ η 2 < 1 \rho\tau\eta^{2} < 1 ρ τ η 2 < 1 k 2 k_{2} k 2 ℓ Z 2 \ell^{2}_{\mathbb{Z}} ℓ Z 2
E R ( f ) ≤ ∑ n ≥ 1 ρ n ∣ a n ∣ 2 ≤ ∥ k 2 ∥ ℓ Z 2 2 ∑ m ≥ 1 ρ m ∣ b m ∣ 2 ≤ ∥ k 2 ∥ ℓ Z 2 2 c η s ∥ L η s 2 f ∥ 2 . \mathcal{E}_{R}(f) \leq\sum_{n \geq1} \rho^{n}|a_{n}|^{2} \leq\lVert k_{2}\rVert_{\ell^{2}_{\mathbb{Z}}}^{2}\sum_{m \geq1}\rho^{m}|b_{m}|^{2} \leq\frac{\lVert k_{2}\rVert_{\ell^{2}_{\mathbb{Z}}}^{2}}{c_{\eta}^{s}}\left\lVert L_{\eta}^{\frac{s}{2}}f\right\rVert^{2}. E R ( f ) ≤ n ≥ 1 ∑ ρ n ∣ a n ∣ 2 ≤ ∥ k 2 ∥ ℓ Z 2 2 m ≥ 1 ∑ ρ m ∣ b m ∣ 2 ≤ c η s ∥ k 2 ∥ ℓ Z 2 2 L η 2 s f 2 . This is Proposition 4.7 below. The last assertion is the case ϑ = d \vartheta= d ϑ = d
Proposition 4.7 Suppose that τ = Q − d \tau= Q^{-d} τ = Q − d η = Q − ϑ \eta= Q^{-\vartheta} η = Q − ϑ α ≥ d + 2 ϑ \alpha\geq d + 2\vartheta α ≥ d + 2 ϑ F : [ 0 , ∞ ) → [ 0 , ∞ ) F : [0,\infty) \to[0,\infty) F : [ 0 , ∞ ) → [ 0 , ∞ ) F ( L η ) F(L_{\eta}) F ( L η ) P P P k ↦ η k k \mapsto\eta^{k} k ↦ η k dom E R \operatorname{dom}\mathcal{E}_{R} dom E R F ( L η ) F(L_{\eta}) F ( L η ) R r a d α R_{\mathrm{rad}}^{\alpha} R rad α
Proof : The polynomials are finite linear combinations of eigenvectors of L η L_{\eta} L η F ( L η ) F(L_{\eta}) F ( L η ) f k = η k f_{k}=\eta^{k} f k = η k f f f (13) is the difference of two geometric tail moments,
∑ k ≥ n + 1 w k η k − τ ∑ k ≥ n w k η k = − ( 1 − τ ) ( 1 − η ) 1 − τ η τ n + 1 η n . \sum_{k \geq n+1} w_{k}\eta^{k}-\tau\sum_{k \geq n}w_{k}\eta^{k}=-\frac{(1-\tau)(1-\eta)}{1-\tau\eta}\tau^{n+1}\eta^{n}. k ≥ n + 1 ∑ w k η k − τ k ≥ n ∑ w k η k = − 1 − τ η ( 1 − τ ) ( 1 − η ) τ n + 1 η n . Hence ⟨ f , g n ⟩ = − ( 1 − τ ) 1 2 ( 1 − η ) 1 − τ η τ n + 1 2 η n \langle f,g_{n}\rangle=-\frac{(1-\tau)^{\frac{1}{2}}(1-\eta)}{1-\tau\eta}\tau^{\frac{n+1}{2}}\eta^{n} ⟨ f , g n ⟩ = − 1 − τ η ( 1 − τ ) 2 1 ( 1 − η ) τ 2 n + 1 η n E R ( f ) = ( 1 − τ ) ( 1 − η ) 2 ( 1 − τ η ) 2 ∑ n λ n τ n + 1 η 2 n \mathcal{E}_{R}(f)=\frac{(1-\tau)(1-\eta)^{2}}{(1-\tau\eta)^{2}}\sum_{n}\lambda_{n}\tau^{n+1}\eta^{2n} E R ( f ) = ( 1 − τ η ) 2 ( 1 − τ ) ( 1 − η ) 2 ∑ n λ n τ n + 1 η 2 n ( 1 − ρ − 1 ) ρ n + 1 ≤ λ n ≤ ρ n + 1 (1-\rho^{-1})\rho^{n+1}\leq\lambda_{n}\leq\rho^{n+1} ( 1 − ρ − 1 ) ρ n + 1 ≤ λ n ≤ ρ n + 1 ρ τ η 2 ≥ 1 \rho\tau\eta^{2}\geq1 ρ τ η 2 ≥ 1 α ≥ d + 2 ϑ \alpha\geq d+2\vartheta α ≥ d + 2 ϑ
Remark 4.8 The norm of the kernel k 2 k_{2} k 2 C 1 ( 1 − ( ρ τ η 2 ) 1 2 ) − 1 C_{1}(1-(\rho\tau\eta^{2})^{\frac{1}{2}})^{-1} C 1 ( 1 − ( ρ τ η 2 ) 2 1 ) − 1 C C C α → d + 2 ϑ \alpha\to d+2\vartheta α → d + 2 ϑ ( τ 1 2 η ) n − m (\tau^{\frac{1}{2}}\eta)^{n-m} ( τ 2 1 η ) n − m e m e_{m} e m 0 0 0 e 1 e_{1} e 1 η k \eta^{k} η k u n 1 u_{n1} u n 1 ( τ 1 2 η ) n (\tau^{\frac{1}{2}}\eta)^{n} ( τ 2 1 η ) n s s s α < d + 2 ϑ \alpha<d+2\vartheta α < d + 2 ϑ
s < 2 + d ϑ = 2 + log τ log η . s<2+\frac{d}{\vartheta}=2+\frac{\log\tau}{\log\eta}. s < 2 + ϑ d = 2 + log η log τ . For s ≤ 1 s\leq1 s ≤ 1 ϑ \vartheta ϑ θ = s \theta=s θ = s R r a d α R_{\mathrm{rad}}^{\alpha} R rad α ( R r a d ϑ ) s (R_{\mathrm{rad}}^{\vartheta})^{s} ( R rad ϑ ) s ( g n ) (g_{n}) ( g n ) ϑ < α < d + 2 ϑ \vartheta<\alpha<d+2\vartheta ϑ < α < d + 2 ϑ ϑ = d \vartheta=d ϑ = d α ≤ d \alpha\leq d α ≤ d s > 1 s>1 s > 1 L η L_{\eta} L η ϑ = d \vartheta=d ϑ = d ϑ = d n − 1 \vartheta=\frac{d}{n-1} ϑ = n − 1 d
The quantum group SUr(2) and the Podleś sphere
In this section 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^{2} τ = r 2
The concrete model
We use the standard realization of the compact quantum group S U r ( 2 ) \mathrm{SU}_{r}(2) SU r ( 2 ) [20 Section 5.3] . Its defining relations and coproduct are those of [54 Example 1.1.5, pp. 4–5] and [21 Section 11] , where our generator b b b α \alpha α ℓ N 0 2 ⊗ L 2 ( T ) \ell^{2}_{\mathbb{N}_{0}}\otimes L^{2}(\mathbb{T}) ℓ N 0 2 ⊗ L 2 ( T )
b ( e n ⊗ f ) = 1 − τ n e n − 1 ⊗ f , γ ( e n ⊗ f ) = r n e n ⊗ z f . b(e_{n}\otimes f)=\sqrt{1-\tau^{n}}e_{n-1}\otimes f,\qquad\gamma(e_{n}\otimes f)=r^{n}e_{n}\otimes zf. b ( e n ⊗ f ) = 1 − τ n e n − 1 ⊗ f , γ ( e n ⊗ f ) = r n e n ⊗ z f . Here e − 1 = 0 e_{-1}=0 e − 1 = 0 z z z T \mathbb{T} T C ( S U r ( 2 ) ) ≔ C ∗ ( b , γ ) C(\mathrm{SU}_{r}(2)) \coloneqq C^*(b,\gamma) C ( SU r ( 2 )) : = C ∗ ( b , γ ) M ≔ C ( S U r ( 2 ) ) ′ ′ \mathcal{M} \coloneqq C(\mathrm{SU}_{r}(2))^{\prime\prime} M : = C ( SU r ( 2 ) ) ′′
M = B ( ℓ N 0 2 ) ⊗ ‾ L ∞ ( T ) , h ( x ) = ( 1 − τ ) ∑ n ≥ 0 τ n ∫ T x n n ( z ) d m ( z ) . (37) \mathcal{M}=B(\ell^2_{\mathbb{N}_0})\overline{\otimes}L^\infty(\mathbb{T}),\qquad h(x)=(1-\tau)\sum_{n\geq0}\tau^n\int_{\mathbb{T}}x_{nn}(z)\,dm(z). \tag*{(37)} M = B ( ℓ N 0 2 ) ⊗ L ∞ ( T ) , h ( x ) = ( 1 − τ ) n ≥ 0 ∑ τ n ∫ T x nn ( z ) d m ( z ) . (37) The Haar formula is [20 (5.1)] . It defines a faithful normal state on M \mathcal{M} M b b b γ ∗ γ \gamma^*\gamma γ ∗ γ γ \gamma γ 1 ⊗ z 1\otimes z 1 ⊗ z h h h M \mathcal{M} M M \mathcal{M} M L ∞ ( S U r ( 2 ) ) L^\infty(\mathrm{SU}_{r}(2)) L ∞ ( SU r ( 2 )) h h h
We recall the relations and the coproduct, fixing all conventions:
b ∗ b + A = 1 , b b ∗ + τ A = 1 , γ γ ∗ = γ ∗ γ = A , b γ = r γ b , b γ ∗ = r γ ∗ b . (38) b^*b+A=1,\qquad bb^*+\tau A=1,\qquad\gamma\gamma^*=\gamma^*\gamma=A,\qquad b\gamma=r\gamma b,\qquad b\gamma^*=r\gamma^*b. \tag*{(38)} b ∗ b + A = 1 , b b ∗ + τ A = 1 , γ γ ∗ = γ ∗ γ = A , bγ = r γ b , b γ ∗ = r γ ∗ b . (38) Δ ( b ) = b ⊗ b − r γ ∗ ⊗ γ , Δ ( γ ) = γ ⊗ b + b ∗ ⊗ γ . (39) \Delta(b)=b\otimes b-r\gamma^*\otimes\gamma,\qquad\Delta(\gamma)=\gamma\otimes b+b^*\otimes\gamma. \tag*{(39)} Δ ( b ) = b ⊗ b − r γ ∗ ⊗ γ , Δ ( γ ) = γ ⊗ b + b ∗ ⊗ γ . (39) The fundamental representation is u ( 1 / 2 ) ≔ [ b − r γ ∗ γ b ∗ ] u^{(1/2)}\coloneqq\begin{bmatrix}b&-r\gamma^*\\ \gamma&b^*\end{bmatrix} u ( 1/2 ) : = [ b γ − r γ ∗ b ∗ ] ∗ * ∗ Δ : M → M ⊗ ‾ M \Delta:\mathcal{M}\to\mathcal{M}\overline{\otimes}\mathcal{M} Δ : M → M ⊗ M [54 Theorem 1.5.3] . We denote by Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_{r}(2)) Pol ( SU r ( 2 )) ∗ * ∗ b b b γ \gamma γ ε \varepsilon ε ε ( b ) = 1 \varepsilon(b)=1 ε ( b ) = 1 ε ( γ ) = 0 \varepsilon(\gamma)=0 ε ( γ ) = 0
Let V V V b = V ∗ ( 1 − A ) 1 2 b=V^*(1-A)^{\frac{1}{2}} b = V ∗ ( 1 − A ) 2 1 V ∗ = ( 1 − τ A ) − 1 2 b ∈ C ( S U r ( 2 ) ) V^*=(1-\tau A)^{-\frac{1}{2}}b\in C(\mathrm{SU}_{r}(2)) V ∗ = ( 1 − τ A ) − 2 1 b ∈ C ( SU r ( 2 )) e 00 ⊗ 1 = 1 − V V ∗ e_{00}\otimes1=1-VV^* e 00 ⊗ 1 = 1 − V V ∗ e i j ⊗ 1 = V i e 00 V ∗ j e_{ij}\otimes1=V^ie_{00}V^{*j} e ij ⊗ 1 = V i e 00 V ∗ j C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) e i j ⊗ z k ∈ C ( S U r ( 2 ) ) e_{ij}\otimes z^k\in C(\mathrm{SU}_{r}(2)) e ij ⊗ z k ∈ C ( SU r ( 2 )) k ∈ Z k\in\mathbb{Z} k ∈ Z p n γ = r n e n n ⊗ z p_n\gamma=r^ne_{nn}\otimes z p n γ = r n e nn ⊗ z p n = e n n ⊗ 1 p_n=e_{nn}\otimes1 p n = e nn ⊗ 1
With the semifinite trace Tr ⊗ m \operatorname{Tr}\otimes m Tr ⊗ m M \mathcal{M} M h h h D h = ( 1 − τ ) diag ( τ n ) ⊗ 1 D_h=(1-\tau)\operatorname{diag}(\tau^n)\otimes1 D h = ( 1 − τ ) diag ( τ n ) ⊗ 1
σ s h ( e i j ⊗ f ) = τ i s ( i − j ) e i j ⊗ f , σ s h ( b ) = τ − i s b , σ s h ( γ ) = γ . \sigma_s^h(e_{ij}\otimes f)=\tau^{is(i-j)}e_{ij}\otimes f,\qquad\sigma_s^h(b)=\tau^{-is}b,\qquad\sigma_s^h(\gamma)=\gamma. σ s h ( e ij ⊗ f ) = τ i s ( i − j ) e ij ⊗ f , σ s h ( b ) = τ − i s b , σ s h ( γ ) = γ . The formula follows from σ s h ( x ) = D h i s x D h − i s \sigma_s^h(x)=D_h^{is}xD_h^{-is} σ s h ( x ) = D h i s x D h − i s [54 Example 1.7.4, p. 31] .
Torus actions and the symbol map
For λ ∈ T \lambda\in\mathbb{T} λ ∈ T D λ D_\lambda D λ ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 D λ e n = λ − n e n D_\lambda e_n=\lambda^{-n}e_n D λ e n = λ − n e n ϑ λ \vartheta_\lambda ϑ λ L 2 ( T ) L^2(\mathbb{T}) L 2 ( T ) ( ϑ λ f ) ( ζ ) = f ( λ ζ ) (\vartheta_\lambda f)(\zeta)=f(\lambda\zeta) ( ϑ λ f ) ( ζ ) = f ( λ ζ ) λ , μ ∈ T \lambda,\mu\in\mathbb{T} λ , μ ∈ T
β λ , μ ( x ) ≔ ( D λ μ ⊗ ϑ λ − 1 μ ) x ( D λ μ ⊗ ϑ λ − 1 μ ) ∗ , x ∈ B ( ℓ N 0 2 ⊗ L 2 ( T ) ) . (40) \beta_{\lambda,\mu}(x)\coloneqq(D_{\lambda\mu}\otimes\vartheta_{\lambda^{-1}\mu})x(D_{\lambda\mu}\otimes\vartheta_{\lambda^{-1}\mu})^*,\qquad x\in B(\ell^2_{\mathbb{N}_0}\otimes L^2(\mathbb{T})). \tag*{(40)} β λ , μ ( x ) : = ( D λ μ ⊗ ϑ λ − 1 μ ) x ( D λ μ ⊗ ϑ λ − 1 μ ) ∗ , x ∈ B ( ℓ N 0 2 ⊗ L 2 ( T )) . (40) A direct computation gives
β λ , μ ( b ) = λ μ b , β λ , μ ( γ ) = λ − 1 μ γ , β λ , μ ( e i j ⊗ z k ) = ( λ μ ) j − i ( λ − 1 μ ) k e i j ⊗ z k . (41) \beta_{\lambda,\mu}(b)=\lambda\mu b,\qquad\beta_{\lambda,\mu}(\gamma)=\lambda^{-1}\mu\gamma,\qquad\beta_{\lambda,\mu}(e_{ij}\otimes z^k)=(\lambda\mu)^{j-i}(\lambda^{-1}\mu)^k e_{ij}\otimes z^k. \tag*{(41)} β λ , μ ( b ) = λ μ b , β λ , μ ( γ ) = λ − 1 μ γ , β λ , μ ( e ij ⊗ z k ) = ( λ μ ) j − i ( λ − 1 μ ) k e ij ⊗ z k . (41) Thus each β λ , μ \beta_{\lambda,\mu} β λ , μ ∗ * ∗ M \mathcal{M} M C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) h h h D λ μ D_{\lambda\mu} D λ μ D h D_h D h ϑ λ − 1 μ \vartheta_{\lambda^{-1}\mu} ϑ λ − 1 μ T \mathbb{T} T ( λ , μ ) ↦ β λ , μ ( x ) (\lambda,\mu)\mapsto\beta_{\lambda,\mu}(x) ( λ , μ ) ↦ β λ , μ ( x ) x ∈ C ( S U r ( 2 ) ) x\in C(\mathrm{SU}_{r}(2)) x ∈ C ( SU r ( 2 )) ∗ * ∗ b b b γ \gamma γ
Lemma 5.1 The C ∗ C^* C ∗ J ≔ K ( ℓ N 0 2 ) ⊗ min C ( T ) \mathcal{J}\coloneqq\mathcal{K}(\ell^2_{\mathbb{N}_0})\otimes_{\min}C(\mathbb{T}) J : = K ( ℓ N 0 2 ) ⊗ m i n C ( T ) C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) ∗ * ∗ σ : C ( S U r ( 2 ) ) → C ( T ) \sigma:C(\mathrm{SU}_{r}(2))\to C(\mathbb{T}) σ : C ( SU r ( 2 )) → C ( T ) J \mathcal{J} J σ ( b ) = z \sigma(b)=z σ ( b ) = z σ ( γ ) = 0 \sigma(\gamma)=0 σ ( γ ) = 0 λ ∈ T \lambda\in\mathbb{T} λ ∈ T ε λ ≔ ε λ ∘ σ \varepsilon_\lambda\coloneqq\varepsilon_\lambda\circ\sigma ε λ : = ε λ ∘ σ C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) ε λ ( b ) = λ \varepsilon_\lambda(b)=\lambda ε λ ( b ) = λ ε λ ( γ ) = 0 \varepsilon_\lambda(\gamma)=0 ε λ ( γ ) = 0 ε 1 \varepsilon_1 ε 1 ε \varepsilon ε
Proof. By the preceding subsection, J \mathcal{J} J C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) b ∈ B ( ℓ 2 ( N 0 ) ) ⊗ 1 b \in B(\ell^{2}(\mathbb{N}_{0})) \otimes1 b ∈ B ( ℓ 2 ( N 0 )) ⊗ 1 γ ∈ J \gamma\in\mathcal{J} γ ∈ J C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) B ( ℓ 2 ( N 0 ) ) ⊗ min C ( T ) B(\ell^{2}(\mathbb{N}_{0})) \otimes_{\min} C(\mathbb{T}) B ( ℓ 2 ( N 0 )) ⊗ m i n C ( T ) J \mathcal{J} J J \mathcal{J} J C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) A = diag ( τ n ) ⊗ 1 A=\operatorname{diag}(\tau^{n})\otimes1 A = diag ( τ n ) ⊗ 1 γ \gamma γ J \mathcal{J} J (38) , the image b ˙ \dot{b} b ˙ b b b b ˙ ∗ b ˙ = 1 = b ˙ b ˙ ∗ \dot{b}^{*}\dot{b}=1=\dot{b}\dot{b}^{*} b ˙ ∗ b ˙ = 1 = b ˙ b ˙ ∗ b ˙ \dot{b} b ˙ C ( Sp ( b ˙ ) ) C(\operatorname{Sp}(\dot{b})) C ( Sp ( b ˙ )) β 1 , μ \beta_{1,\mu} β 1 , μ J \mathcal{J} J (41) , hence induces an automorphism of the quotient sending b ˙ \dot{b} b ˙ μ b ˙ \mu\dot{b} μ b ˙ Sp ( b ˙ ) = μ Sp ( b ˙ ) \operatorname{Sp}(\dot{b})=\mu\operatorname{Sp}(\dot{b}) Sp ( b ˙ ) = μ Sp ( b ˙ ) μ ∈ T \mu\in\mathbb{T} μ ∈ T Sp ( b ˙ ) = T \operatorname{Sp}(\dot{b})=\mathbb{T} Sp ( b ˙ ) = T σ \sigma σ ε 1 \varepsilon_{1} ε 1 ε \varepsilon ε b b b γ \gamma γ
A direct computation on the generators, using (39) , ε λ ( γ ) = 0 \varepsilon_{\lambda}(\gamma)=0 ε λ ( γ ) = 0
β λ , μ = ( ε λ ⊗ Id ⊗ ε μ ) ∘ ( Δ ⊗ Id ) ∘ Δ on C ( S U r ( 2 ) ) . (42) \beta_{\lambda,\mu}=(\varepsilon_{\lambda}\otimes\operatorname{Id}\otimes\varepsilon_{\mu})\circ(\Delta\otimes\operatorname{Id})\circ\Delta\quad\text{on }C(\mathrm{SU}_{r}(2)). \tag*{(42)} β λ , μ = ( ε λ ⊗ Id ⊗ ε μ ) ∘ ( Δ ⊗ Id ) ∘ Δ on C ( SU r ( 2 )) . (42) Indeed, both sides are ∗ * ∗ ( Δ ⊗ Id ) Δ ( γ ) = ( γ ⊗ b + b ∗ ⊗ γ ) ⊗ b + ( b ∗ ⊗ b ∗ − r γ ⊗ γ ∗ ) ⊗ γ (\Delta\otimes\operatorname{Id})\Delta(\gamma)=(\gamma\otimes b+b^{*}\otimes\gamma)\otimes b+(b^{*}\otimes b^{*}-r\gamma\otimes\gamma^{*})\otimes\gamma ( Δ ⊗ Id ) Δ ( γ ) = ( γ ⊗ b + b ∗ ⊗ γ ) ⊗ b + ( b ∗ ⊗ b ∗ − r γ ⊗ γ ∗ ) ⊗ γ ε λ ⊗ Id ⊗ ε μ \varepsilon_{\lambda}\otimes\operatorname{Id}\otimes\varepsilon_{\mu} ε λ ⊗ Id ⊗ ε μ λ μ γ \lambda\mu\gamma λ μ γ β λ , 1 \beta_{\lambda,1} β λ , 1 β 1 , μ \beta_{1,\mu} β 1 , μ S U r ( 2 ) \mathrm{SU}_{r}(2) SU r ( 2 )
Matrix coefficients and zonal spherical functions
We use the irreducible unitary representations u ( l ) = ( u j k ( l ) ) j , k u^{(l)}=(u_{jk}^{(l)})_{j,k} u ( l ) = ( u j k ( l ) ) j , k l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_{0} l ∈ 2 1 N 0 2 l + 1 2l+1 2 l + 1 j , k ∈ { − l , − l + 1 , … , l } j,k\in\{-l,-l+1,\ldots,l\} j , k ∈ { − l , − l + 1 , … , l } [21 Section 11] ; for l = 1 2 l=\frac{1}{2} l = 2 1 u − 1 2 , − 1 2 ( 1 2 ) = b u_{-\frac{1}{2},-\frac{1}{2}}^{(\frac{1}{2})}=b u − 2 1 , − 2 1 ( 2 1 ) = b u 1 2 , 1 2 ( 1 2 ) = b ∗ u_{\frac{1}{2},\frac{1}{2}}^{(\frac{1}{2})}=b^{*} u 2 1 , 2 1 ( 2 1 ) = b ∗ Δ ( u j k ( l ) ) = ∑ m u j m ( l ) ⊗ u m k ( l ) \Delta(u_{jk}^{(l)})=\sum_{m}u_{jm}^{(l)}\otimes u_{mk}^{(l)} Δ ( u j k ( l ) ) = ∑ m u j m ( l ) ⊗ u mk ( l ) ε ( u j k ( l ) ) = δ j k \varepsilon(u_{jk}^{(l)})=\delta_{jk} ε ( u j k ( l ) ) = δ j k Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_{r}(2)) Pol ( SU r ( 2 )) [54 Example 2.4.8] and the orthogonality relations [54 Theorem 1.4.3] . Moreover, if Pol n \operatorname{Pol}_{n} Pol n n n n b , b ∗ , γ , γ ∗ b,b^{*},\gamma,\gamma^{*} b , b ∗ , γ , γ ∗ C ( l ) ≔ span { u j k ( l ) } j , k C^{(l)}\coloneqq\operatorname{span}\{u_{jk}^{(l)}\}_{j,k} C ( l ) : = span { u j k ( l ) } j , k
Pol n = ⨁ l ∈ 1 2 N 0 2 l ≤ n C ( l ) . (43) \operatorname{Pol}_{n}=\bigoplus_{\substack{l\in\frac{1}{2}\mathbb{N}_{0}\\2l\leq n}}C^{(l)}. \tag*{(43)} Pol n = l ∈ 2 1 N 0 2 l ≤ n ⨁ C ( l ) . (43) see the proof in [54 Example 2.4.8] . Finally, the Woronowicz characters f ζ f_{\zeta} f ζ ζ ∈ C \zeta\in\mathbb{C} ζ ∈ C Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_{r}(2)) Pol ( SU r ( 2 )) [54 Proposition 1.7.2] , and [21 (11.2)]
f ζ ( u j k ( l ) ) = δ j k r 2 j ζ . (44) f_{\zeta}(u_{jk}^{(l)})=\delta_{jk}r^{2j\zeta}. \tag*{(44)} f ζ ( u j k ( l ) ) = δ j k r 2 j ζ . (44) Lemma 5.2 For any l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_{0} l ∈ 2 1 N 0 j , k ∈ { − l , … , l } j,k\in\{-l,\ldots,l\} j , k ∈ { − l , … , l } λ , μ ∈ T \lambda,\mu\in\mathbb{T} λ , μ ∈ T
ε λ ( u j k ( l ) ) = δ j k λ − 2 j , β λ , μ ( u j k ( l ) ) = λ − 2 j μ − 2 k u j k ( l ) . (45) \varepsilon_{\lambda}(u_{jk}^{(l)})=\delta_{jk}\lambda^{-2j},\qquad\beta_{\lambda,\mu}(u_{jk}^{(l)})=\lambda^{-2j}\mu^{-2k}u_{jk}^{(l)}. \tag*{(45)} ε λ ( u j k ( l ) ) = δ j k λ − 2 j , β λ , μ ( u j k ( l ) ) = λ − 2 j μ − 2 k u j k ( l ) . (45) Consequently, the elements of Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_{r}(2)) Pol ( SU r ( 2 )) β λ , μ \beta_{\lambda,\mu} β λ , μ span { u 00 ( l ) : l ∈ N 0 } \operatorname{span}\{u_{00}^{(l)}:l\in\mathbb{N}_{0}\} span { u 00 ( l ) : l ∈ N 0 } β 1 , μ \beta_{1,\mu} β 1 , μ span { u j 0 ( l ) : l ∈ N 0 , ∣ j ∣ ≤ l } \operatorname{span}\{u_{j0}^{(l)}:l\in\mathbb{N}_{0},\ |j|\leq l\} span { u j 0 ( l ) : l ∈ N 0 , ∣ j ∣ ≤ l }
Proof: For ζ ∈ C ∖ { 0 } \zeta\in\mathbb{C}\setminus\{0\} ζ ∈ C ∖ { 0 } χ ζ : Pol ( S U r ( 2 ) ) → C \chi_{\zeta}:\operatorname{Pol}(\mathrm{SU}_{r}(2))\to\mathbb{C} χ ζ : Pol ( SU r ( 2 )) → C χ ζ ( b ) = ζ \chi_{\zeta}(b)=\zeta χ ζ ( b ) = ζ χ ζ ( b ∗ ) = ζ − 1 \chi_{\zeta}(b^{*})=\zeta^{-1} χ ζ ( b ∗ ) = ζ − 1 χ ζ ( γ ) = χ ζ ( γ ∗ ) = 0 \chi_{\zeta}(\gamma)=\chi_{\zeta}(\gamma^{*})=0 χ ζ ( γ ) = χ ζ ( γ ∗ ) = 0 (38) and their adjoints. For any noncommutative monomial x x x b , b ∗ , γ , γ ∗ b,b^{*},\gamma,\gamma^{*} b , b ∗ , γ , γ ∗ χ ζ ( x ) \chi_{\zeta}(x) χ ζ ( x ) 0 0 0 ζ m \zeta^{m} ζ m m m m ζ \zeta ζ ζ ↦ χ ζ ( x ) \zeta\mapsto\chi_{\zeta}(x) ζ ↦ χ ζ ( x ) x ∈ Pol ( S U r ( 2 ) ) x\in\operatorname{Pol}(\mathrm{SU}_{r}(2)) x ∈ Pol ( SU r ( 2 )) (44) and the labeling of u ( 1 2 ) u^{(\frac{1}{2})} u ( 2 1 ) f ζ f_{\zeta} f ζ r − ζ , r ζ , 0 , 0 r^{-\zeta},r^{\zeta},0,0 r − ζ , r ζ , 0 , 0 b , b ∗ , γ , γ ∗ b,b^{*},\gamma,\gamma^{*} b , b ∗ , γ , γ ∗ f ζ = χ r f_{\zeta}=\chi_{r} f ζ = χ r (44) , we deduce that χ x ( u j k ( l ) ) = δ j k x − 2 j \chi_x(u^{(l)}_{jk})=\delta_{jk}x^{-2j} χ x ( u j k ( l ) ) = δ j k x − 2 j x > 0 x>0 x > 0 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) x ∈ C ∖ { 0 } x\in\mathbb{C}\setminus\{0\} x ∈ C ∖ { 0 } λ ∈ T \lambda\in\mathbb{T} λ ∈ T χ λ \chi_\lambda χ λ ϵ λ \epsilon_\lambda ϵ λ b b b b ∗ b^* b ∗ γ \gamma γ γ ∗ \gamma^* γ ∗ (45) . The second follows from (42) and Δ ( 2 ) ( u j k ( l ) ) = ∑ a , c u j a ( l ) ⊗ u a c ( l ) ⊗ u c k ( l ) \Delta^{(2)}(u^{(l)}_{jk})=\sum_{a,c}u^{(l)}_{ja}\otimes u^{(l)}_{ac}\otimes u^{(l)}_{ck} Δ ( 2 ) ( u j k ( l ) ) = ∑ a , c u j a ( l ) ⊗ u a c ( l ) ⊗ u c k ( l ) − 2 j = 0 -2j=0 − 2 j = 0 j = 0 j=0 j = 0 l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0
Proposition 5.3 For any integer l ≥ 0 l\geq0 l ≥ 0 u 00 ( l ) = φ l τ ( A ) u^{(l)}_{00}=\varphi_l^{\tau}(A) u 00 ( l ) = φ l τ ( A ) φ l τ \varphi_l^{\tau} φ l τ η = τ \eta=\tau η = τ τ \tau τ φ l τ ( 0 ) = 1 \varphi_l^{\tau}(0)=1 φ l τ ( 0 ) = 1 ∣ φ l τ ( x ) ∣ ≤ 1 |\varphi_l^{\tau}(x)|\leq1 ∣ φ l τ ( x ) ∣ ≤ 1 x ∈ X τ x\in X_\tau x ∈ X τ
Proof : Let n ≥ 0 n\geq0 n ≥ 0 A m = ( γ ∗ γ ) m A^m=(\gamma^*\gamma)^m A m = ( γ ∗ γ ) m Pol 2 m \operatorname{Pol}_{2m} Pol 2 m β λ , μ \beta_{\lambda,\mu} β λ , μ (41) . By (43) and Lemma 5.2, the polynomials of degree at most n n n A A A span { u 00 ( l ) : l ∈ N 0 , l ≤ n } \operatorname{span}\{u^{(l)}_{00}:l\in\mathbb{N}_0,\ l\leq n\} span { u 00 ( l ) : l ∈ N 0 , l ≤ n } n + 1 n+1 n + 1 X τ X_\tau X τ A A A n n n u 00 ( n ) = p n ( A ) u^{(n)}_{00}=p_n(A) u 00 ( n ) = p n ( A ) p n p_n p n n n n
Recall that h ( F ( A ) ) = ∑ k w k F ( τ k ) h(F(A))=\sum_k w_kF(\tau^k) h ( F ( A )) = ∑ k w k F ( τ k ) (37) . By the orthogonality relations [54 Theorem 1.4.3 (ii)] , h ( ( u 00 ( l ) ) ∗ u 00 ( m ) ) = 0 h((u^{(l)}_{00})^*u^{(m)}_{00})=0 h (( u 00 ( l ) ) ∗ u 00 ( m ) ) = 0 l ≠ m l\ne m l = m p l p_l p l ℓ 2 ( N 0 , w ) \ell^2(\mathbb{N}_0,w) ℓ 2 ( N 0 , w ) τ k \tau^k τ k ϵ ( γ ) = 0 \epsilon(\gamma)=0 ϵ ( γ ) = 0 ϵ ( A ) = 0 \epsilon(A)=0 ϵ ( A ) = 0 ϵ ( p ( A ) ) = p ( 0 ) \epsilon(p(A))=p(0) ϵ ( p ( A )) = p ( 0 ) p p p p l ( 0 ) = ϵ ( u 00 ( l ) ) = 1 p_l(0)=\epsilon(u^{(l)}_{00})=1 p l ( 0 ) = ϵ ( u 00 ( l ) ) = 1 p l = φ l τ p_l=\varphi_l^\tau p l = φ l τ u ( l ) u^{(l)} u ( l ) ∑ j ( u j 0 ( l ) ) ∗ u j 0 ( l ) = 1 \sum_j (u^{(l)}_{j0})^*u^{(l)}_{j0}=1 ∑ j ( u j 0 ( l ) ) ∗ u j 0 ( l ) = 1 ∣ u 00 ( l ) ∣ ≤ 1 |u^{(l)}_{00}|\leq1 ∣ u 00 ( l ) ∣ ≤ 1 A A A X τ X_\tau X τ sup X τ ∣ φ l τ ∣ ≤ 1 \sup_{X_\tau}|\varphi_l^\tau|\leq1 sup X τ ∣ φ l τ ∣ ≤ 1
The identification of zonal matrix coefficients with little q q q [72 ] , [47 ] , [51 ] and the survey [46 ] ; the proof above only uses the facts recalled in this subsection.
Remark 5.4 We use the following conventions for U r s l 2 U_{r}\mathfrak{sl}_2 U r sl 2 [54 Section 2.4] :
K E K − 1 = r 2 E , K F K − 1 = r − 2 F , E F − F E = K − K − 1 r − r − 1 , KEK^{-1}=r^2E,\quad KFK^{-1}=r^{-2}F,\quad EF-FE=\frac{K-K^{-1}}{r-r^{-1}}, K E K − 1 = r 2 E , K F K − 1 = r − 2 F , E F − F E = r − r − 1 K − K − 1 , Δ ( K ) = K ⊗ K , Δ ( E ) = E ⊗ 1 + K ⊗ E , Δ ( F ) = F ⊗ K − 1 + 1 ⊗ F , K ∗ = K , E ∗ = F K , F ∗ = K − 1 E . \Delta(K)=K\otimes K,\quad\Delta(E)=E\otimes1+K\otimes E,\quad\Delta(F)=F\otimes K^{-1}+1\otimes F,\quad K^*=K,\quad E^*=FK,\quad F^*=K^{-1}E. Δ ( K ) = K ⊗ K , Δ ( E ) = E ⊗ 1 + K ⊗ E , Δ ( F ) = F ⊗ K − 1 + 1 ⊗ F , K ∗ = K , E ∗ = F K , F ∗ = K − 1 E . One checks on the generators that Δ \Delta Δ ∗ * ∗ [ x ] r ≔ r x − r − x r − r − 1 [x]_r\coloneqq\frac{r^x-r^{-x}}{r-r^{-1}} [ x ] r : = r − r − 1 r x − r − x s ∈ 1 2 N 0 s\in\frac{1}{2}\mathbb{N}_0 s ∈ 2 1 N 0 ∗ * ∗ V s V_s V s ( ξ i ) ∣ i ∣ ≤ s (\xi_i)_{|i|\leq s} ( ξ i ) ∣ i ∣ ≤ s
K ξ i = r 2 i ξ i , F ξ i = r − i [ s + i ] r [ s − i + 1 ] r ξ i − 1 , E ξ i = r i + 1 [ s − i ] r [ s + i + 1 ] r ξ i + 1 , (46) K\xi_i=r^{2i}\xi_i,\quad F\xi_i=r^{-i}\sqrt{[s+i]_r[s-i+1]_r}\,\xi_{i-1},\quad E\xi_i=r^{i+1}\sqrt{[s-i]_r[s+i+1]_r}\,\xi_{i+1}, \tag*{(46)} K ξ i = r 2 i ξ i , F ξ i = r − i [ s + i ] r [ s − i + 1 ] r ξ i − 1 , E ξ i = r i + 1 [ s − i ] r [ s + i + 1 ] r ξ i + 1 , (46) where ξ ± ( s + 1 ) = 0 \xi_{\pm(s+1)}=0 ξ ± ( s + 1 ) = 0 E = ( F K ) ∗ E=(FK)^* E = ( F K ) ∗ [ E , F ] ξ i = r 2 i − r − 2 i r − r − 1 ξ i [E,F]\xi_i=\frac{r^{2i}-r^{-2i}}{r-r^{-1}}\xi_i [ E , F ] ξ i = r − r − 1 r 2 i − r − 2 i ξ i
[ s + i ] r [ s − i + 1 ] r − [ s − i ] r [ s + i + 1 ] r = [ 2 i ] r . [s+i]_r[s-i+1]_r-[s-i]_r[s+i+1]_r=[2i]_r. [ s + i ] r [ s − i + 1 ] r − [ s − i ] r [ s + i + 1 ] r = [ 2 i ] r . Consider the Casimir element
Ω ≔ E F + r − 1 K + r K − 1 ( r − r − 1 ) 2 . \Omega\coloneqq EF+\frac{r^{-1}K+rK^{-1}}{(r-r^{-1})^2}. Ω : = E F + ( r − r − 1 ) 2 r − 1 K + r K − 1 . Since [ E , E F ] = E ( K − K − 1 ) r − r − 1 [E,EF]=\frac{E(K-K^{-1})}{r-r^{-1}} [ E , E F ] = r − r − 1 E ( K − K − 1 ) [ E , r − 1 K + r K − 1 ] = − ( r − r − 1 ) E ( K − K − 1 ) [E,r^{-1}K+rK^{-1}]=-(r-r^{-1})E(K-K^{-1}) [ E , r − 1 K + r K − 1 ] = − ( r − r − 1 ) E ( K − K − 1 ) [ E , Ω ] = 0 [E,\Omega]=0 [ E , Ω ] = 0 [ F , Ω ] = 0 [F,\Omega]=0 [ F , Ω ] = 0 Ω \Omega Ω V s V_s V s ξ s \xi_s ξ s E ξ s = 0 E\xi_s=0 E ξ s = 0
Ω ξ s = ( r 2 s − r − 2 s r − r − 1 + r 2 s − 1 + r 1 − 2 s ( r − r − 1 ) 2 ) ξ s = r 2 s + 1 + r − 2 s − 1 ( r − r − 1 ) 2 ξ s . \Omega\xi_s=\left(\frac{r^{2s}-r^{-2s}}{r-r^{-1}}+\frac{r^{2s-1}+r^{1-2s}}{(r-r^{-1})^2}\right)\xi_s
=\frac{r^{2s+1}+r^{-2s-1}}{(r-r^{-1})^2}\xi_s. Ω ξ s = ( r − r − 1 r 2 s − r − 2 s + ( r − r − 1 ) 2 r 2 s − 1 + r 1 − 2 s ) ξ s = ( r − r − 1 ) 2 r 2 s + 1 + r − 2 s − 1 ξ s . Thus Ω \Omega Ω V l V_l V l Ω l = r 2 l + 1 + r − 2 l − 1 ( r − r − 1 ) 2 \Omega_l=\frac{r^{2l+1}+r^{-2l-1}}{(r-r^{-1})^2} Ω l = ( r − r − 1 ) 2 r 2 l + 1 + r − 2 l − 1 l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_0 l ∈ 2 1 N 0
Λ l ≔ r − 2 l − 1 − r 2 + r 2 l + 2 = r ( r − r − 1 ) 2 ( Ω l − Ω 0 ) , (47) \Lambda_l\coloneqq r^{-2l}-1-r^2+r^{2l+2}=r(r-r^{-1})^2(\Omega_l-\Omega_0), \tag*{(47)} Λ l : = r − 2 l − 1 − r 2 + r 2 l + 2 = r ( r − r − 1 ) 2 ( Ω l − Ω 0 ) , (47) and Λ l = Λ l ( τ ) \Lambda_l=\Lambda_l(\tau) Λ l = Λ l ( τ ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 (4) . Through the duality between U r s l 2 U_{r}\mathfrak{sl}_2 U r sl 2 Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_r(2)) Pol ( SU r ( 2 )) x ↦ ( Id ⊗ Ω ) Δ ( x ) x\mapsto(\operatorname{Id}\otimes\Omega)\Delta(x) x ↦ ( Id ⊗ Ω ) Δ ( x ) C ( l ) C(l) C ( l ) Ω l \Omega_l Ω l L τ L_\tau L τ (47) , the restriction of this Casimir operator to the zonal polynomials φ l τ ( A ) \varphi_l^{\tau}(A) φ l τ ( A )
The Podleś sphere in the concrete model
We define
C ( S r 2 ) ≔ { x ∈ C ( S U r ( 2 ) ) : β 1 , μ ( x ) = x for any μ ∈ T } . C(S_r^2)\coloneqq\{x\in C(\mathrm{SU}_r(2)):\beta_{1,\mu}(x)=x\text{ for any }\mu\in\mathbb{T}\}. C ( S r 2 ) : = { x ∈ C ( SU r ( 2 )) : β 1 , μ ( x ) = x for any μ ∈ T } . and
L ∞ ( S r 2 ) ≔ { x ∈ M : β 1 , μ ( x ) = x for any μ ∈ T } . L^\infty(S_r^2)\coloneqq\{x\in\mathcal{M}:\beta_{1,\mu}(x)=x\text{ for any }\mu\in\mathbb{T}\}. L ∞ ( S r 2 ) : = { x ∈ M : β 1 , μ ( x ) = x for any μ ∈ T } . By (42) , C ( S r 2 ) C(S_r^2) C ( S r 2 ) C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) [61 ] . We put Pol ( S r 2 ) ≔ Pol ( S U r ( 2 ) ) ∩ C ( S r 2 ) \operatorname{Pol}(S_r^2)\coloneqq\operatorname{Pol}(\mathrm{SU}_r(2))\cap C(S_r^2) Pol ( S r 2 ) : = Pol ( SU r ( 2 )) ∩ C ( S r 2 ) span { u j 0 ( l ) : l ∈ N 0 , ∣ j ∣ ≤ l } \operatorname{span}\{u_{j0}^{(l)}:l\in\mathbb{N}_0,\ |j|\leq l\} span { u j 0 ( l ) : l ∈ N 0 , ∣ j ∣ ≤ l } W W W ℓ N 0 2 ⊗ L 2 ( T ) \ell^2_{\mathbb{N}_0}\otimes L^2(\mathbb{T}) ℓ N 0 2 ⊗ L 2 ( T ) W ( e n ⊗ f ) = e n ⊗ z n f W(e_n\otimes f)=e_n\otimes z^n f W ( e n ⊗ f ) = e n ⊗ z n f
θ ( x ) ≔ W ( x ⊗ 1 ) W ∗ , x ∈ B ( ℓ N 0 2 ) . \theta(x)\coloneqq W(x\otimes1)W^*,\qquad x\in B(\ell^2_{\mathbb{N}_0}). θ ( x ) : = W ( x ⊗ 1 ) W ∗ , x ∈ B ( ℓ N 0 2 ) . Finally, let ω τ \omega_\tau ω τ ω τ ( x ) ≔ Tr ( D x ) \omega_\tau(x)\coloneqq\operatorname{Tr}(Dx) ω τ ( x ) : = Tr ( D x ) B ( ℓ N 0 2 ) B(\ell^2_{\mathbb{N}_0}) B ( ℓ N 0 2 ) D ≔ ( 1 − τ ) diag ( τ n ) D\coloneqq(1-\tau)\operatorname{diag}(\tau^n) D : = ( 1 − τ ) diag ( τ n ) log 1 τ \log\frac{1}{\tau} log τ 1
Proposition 5.5 1. The map θ \theta θ ∗ * ∗ B ( ℓ N 0 2 ) B(\ell^2_{\mathbb{N}_0}) B ( ℓ N 0 2 ) L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) θ ( e i j ) = e i j ⊗ z i − j \theta(e_{ij})=e_{ij}\otimes z^{i-j} θ ( e ij ) = e ij ⊗ z i − j h ∘ θ = ω τ h\circ\theta=\omega_\tau h ∘ θ = ω τ
We have θ ( K ( ℓ N 0 2 ) + C 1 ) = C ( S r 2 ) \theta(K(\ell^2_{\mathbb{N}_0})+\mathbb{C}1)=C(S_r^2) θ ( K ( ℓ N 0 2 ) + C 1 ) = C ( S r 2 ) Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) C ( S r 2 ) C(S_r^2) C ( S r 2 )
The image under θ \theta θ ℓ N 0 ∞ \ell^\infty_{\mathbb{N}_0} ℓ N 0 ∞ W ∗ ( A ) W^*(A) W ∗ ( A ) M \mathcal{M} M β λ , μ \beta_{\lambda,\mu} β λ , μ W ∗ ( A ) W^*(A) W ∗ ( A ) L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) β λ , μ \beta_{\lambda,\mu} β λ , μ C ∗ ( A ) C^*(A) C ∗ ( A ) θ \theta θ
The averages E R ≔ ∫ T β 1 , μ d μ \mathbb{E}_{R}\coloneqq\int_{\mathbb{T}}\beta_{1,\mu}\,d\mu E R : = ∫ T β 1 , μ d μ E ≔ ∫ T 2 β λ , μ d λ d μ \mathbb{E}\coloneqq\int_{\mathbb{T}^2}\beta_{\lambda,\mu}\,d\lambda\,d\mu E : = ∫ T 2 β λ , μ d λ d μ h h h M \mathcal{M} M L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) W ∗ ( A ) W^*(A) W ∗ ( A ) C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) C ( S r 2 ) C(S_r^2) C ( S r 2 ) C ∗ ( A ) C^*(A) C ∗ ( A )
Proof. 1. We have W ∗ ( e n ⊗ f ) = e n ⊗ z − n f W^*(e_n\otimes f)=e_n\otimes z^{-n}f W ∗ ( e n ⊗ f ) = e n ⊗ z − n f W ( e i j ⊗ 1 ) W ∗ ( e n ⊗ f ) = δ j n e i ⊗ z i − j f W(e_{ij}\otimes1)W^*(e_n\otimes f)=\delta_{jn}e_i\otimes z^{i-j}f W ( e ij ⊗ 1 ) W ∗ ( e n ⊗ f ) = δ j n e i ⊗ z i − j f θ ( e i j ) = e i j ⊗ z i − j \theta(e_{ij})=e_{ij}\otimes z^{i-j} θ ( e ij ) = e ij ⊗ z i − j θ \theta θ ∗ * ∗ L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) (41) . Conversely, let y ∈ M y\in\mathcal{M} y ∈ M β 1 , μ \beta_{1,\mu} β 1 , μ y i j ∈ L ∞ ( T ) y_{ij}\in L^\infty(\mathbb{T}) y ij ∈ L ∞ ( T ) (41) , the Fourier coefficient of index k k k y i j y_{ij} y ij μ j − i + k \mu^{j-i+k} μ j − i + k β 1 , μ \beta_{1,\mu} β 1 , μ y i j = c i j z i − j y_{ij}=c_{ij}z^{i-j} y ij = c ij z i − j c i j ∈ C c_{ij}\in\mathbb{C} c ij ∈ C W ∗ y W W^*yW W ∗ y W c i j c_{ij} c ij W ∗ y W = c ⊗ 1 W^*yW=c\otimes1 W ∗ y W = c ⊗ 1 c ∈ B ( ℓ N 0 2 ) c\in B(\ell^2_{\mathbb{N}_0}) c ∈ B ( ℓ N 0 2 ) y = θ ( c ) y=\theta(c) y = θ ( c ) h ( θ ( x ) ) = ( 1 − τ ) ∑ n τ n x n n = ω τ ( x ) h(\theta(x))=(1-\tau)\sum_n\tau^n x_{nn}=\omega_\tau(x) h ( θ ( x )) = ( 1 − τ ) ∑ n τ n x nn = ω τ ( x ) (37) .
The elements θ ( e i j ) = e i j ⊗ z i − j \theta(e_{ij})=e_{ij}\otimes z^{i-j} θ ( e ij ) = e ij ⊗ z i − j C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) β 1 , μ \beta_{1,\mu} β 1 , μ θ ( K ( ℓ N 0 2 ) + C 1 ) ⊂ C ( S r 2 ) \theta(K(\ell^2_{\mathbb{N}_0})+\mathbb{C}1)\subset C(S_r^2) θ ( K ( ℓ N 0 2 ) + C 1 ) ⊂ C ( S r 2 ) x ∈ C ( S r 2 ) x\in C(S_r^2) x ∈ C ( S r 2 ) σ ∘ β 1 , μ = rot μ ∘ σ \sigma\circ\beta_{1,\mu}=\operatorname{rot}_\mu\circ\sigma σ ∘ β 1 , μ = rot μ ∘ σ ( rot μ g ) ( ζ ) = g ( μ ζ ) (\operatorname{rot}_\mu g)(\zeta)=g(\mu\zeta) ( rot μ g ) ( ζ ) = g ( μ ζ ) ∗ * ∗ b b b γ \gamma γ σ ( x ) \sigma(x) σ ( x ) c c c x − c 1 ∈ J x-c1\in\mathcal{J} x − c 1 ∈ J β 1 , μ \beta_{1,\mu} β 1 , μ μ \mu μ e i j ⊗ z k e_{ij}\otimes z^k e ij ⊗ z k J \mathcal{J} J e i j ⊗ z i − j e_{ij}\otimes z^{i-j} e ij ⊗ z i − j θ ( K ( ℓ N 0 2 ) ) \theta(K(\ell^2_{\mathbb{N}_0})) θ ( K ( ℓ N 0 2 )) x ∈ θ ( K ( ℓ N 0 2 ) + C 1 ) x\in\theta(K(\ell^2_{\mathbb{N}_0})+\mathbb{C}1) x ∈ θ ( K ( ℓ N 0 2 ) + C 1 ) E R \mathbb{E}_R E R C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) C ( S r 2 ) C(S_r^2) C ( S r 2 ) Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_r(2)) Pol ( SU r ( 2 )) C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) E R ( Pol ( S U r ( 2 ) ) ) = Pol ( S r 2 ) \mathbb{E}_R(\operatorname{Pol}(\mathrm{SU}_r(2)))=\operatorname{Pol}(S_r^2) E R ( Pol ( SU r ( 2 ))) = Pol ( S r 2 )
We have θ ( e n n ) = e n n ⊗ 1 = p n = 1 { τ n } ( A ) \theta(e_{nn})=e_{nn}\otimes1=p_n=1_{\{\tau^n\}}(A) θ ( e nn ) = e nn ⊗ 1 = p n = 1 { τ n } ( A ) θ ( ℓ N 0 ∞ ) = W ∗ ( A ) \theta(\ell^\infty_{N_0})=W^*(A) θ ( ℓ N 0 ∞ ) = W ∗ ( A ) (41) , an element of M \mathcal{M} M β λ , μ \beta_{\lambda,\mu} β λ , μ y i j y_{ij} y ij i ≠ j i\ne j i = j i = j i=j i = j B ( ℓ N 0 2 ) B(\ell^2_{N_0}) B ( ℓ N 0 2 ) x ∈ C ( S U r ( 2 ) ) x\in C(\mathrm{SU}_r(2)) x ∈ C ( SU r ( 2 )) β λ , μ \beta_{\lambda,\mu} β λ , μ σ ( x ) \sigma(x) σ ( x ) c c c x − c 1 x-c1 x − c 1 e n n ⊗ 1 e_{nn}\otimes1 e nn ⊗ 1 θ ( c 0 ) \theta(c_0) θ ( c 0 ) x = F ( A ) x=F(A) x = F ( A ) F ∈ C ( X τ ) F\in C(X_\tau) F ∈ C ( X τ )
Each β λ , μ \beta_{\lambda,\mu} β λ , μ h h h ( λ , μ ) ↦ β λ , μ ( x ) (\lambda,\mu)\mapsto\beta_{\lambda,\mu}(x) ( λ , μ ) ↦ β λ , μ ( x ) σ \sigma σ x ∈ M x\in\mathcal{M} x ∈ M x ∈ C ( S U r ( 2 ) ) x\in C(\mathrm{SU}_r(2)) x ∈ C ( SU r ( 2 )) h h h h h h
From now on, we identify C ∗ ( A ) C^*(A) C ∗ ( A ) C ( X τ ) C(X_\tau) C ( X τ ) F ( A ) ↔ F F(A)\leftrightarrow F F ( A ) ↔ F W ∗ ( A ) W^*(A) W ∗ ( A ) ℓ N 0 ∞ \ell^\infty_{N_0} ℓ N 0 ∞ τ = Q − d \tau=Q^{-d} τ = Q − d
Theorem 5.6 Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d ∗ * ∗
J : L ∞ ( O d ) r a d → W ∗ ( A ) , J ( F ( ∥ ⋅ ∥ K d ) ) = F ( A ) , J:L^\infty(\mathcal{O}^d)_{\mathrm{rad}}\to W^*(A),\qquad J\bigl(F(\|\cdot\|_K^d)\bigr)=F(A), J : L ∞ ( O d ) rad → W ∗ ( A ) , J ( F ( ∥ ⋅ ∥ K d ) ) = F ( A ) , where F F F X τ X_\tau X τ 1 S n 1_{S_n} 1 S n p n p_n p n 1 B n 1_{B_n} 1 B n P n : = ∑ k ≥ n p k P_n\mathrel{\mathop:}=\sum_{k\ge n}p_k P n : = ∑ k ≥ n p k C ∗ ( A ) C^*(A) C ∗ ( A ) J 2 J_2 J 2 ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) W ∗ ( A ) W^*(A) W ∗ ( A ) h h h
Proof. A radial bounded function is a sequence ( f n ) (f_n) ( f n ) J f = ∑ n f n p n Jf=\sum_n f_np_n J f = ∑ n f n p n p n p_n p n ∗ * ∗ W ∗ ( A ) W^*(A) W ∗ ( A ) (7) and (37) proves preservation of the state and the L 2 L^2 L 2 F F F 0 0 0
Boundary energies of random walks
In this section, we compare the boundary theory of the simple random walk on the tree of balls of O d \mathcal{O}^d O d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Theorem 1.2 . Throughout, 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 τ = Q − d \tau=Q^{-d} τ = Q − d
The tree of balls
Let T \mathbb{T} T n ≥ 0 n\ge0 n ≥ 0 x + π n O d x+\pi^n\mathcal{O}^d x + π n O d O d \mathcal{O}^d O d Q d Q^d Q d O d \mathcal{O}^d O d T \mathbb{T} T p T ( v , w ) p_{\mathbb{T}}(v,w) p T ( v , w ) m T ( v ) = deg ( v ) m_{\mathbb{T}}(v)=\deg(v) m T ( v ) = deg ( v ) O d \mathcal{O}^d O d F ∈ L 2 ( O d ) F\in L^2(\mathcal{O}^d) F ∈ L 2 ( O d ) H T F \mathcal{H}_{\mathbb{T}}F H T F F F F F F F
E T ( F ) : = ∑ v m T ( v ) ∑ w p T ( v , w ) ∣ H T F ( w ) − H T F ( v ) ∣ 2 ∈ [ 0 , ∞ ] . (48) \mathcal{E}_{\mathbb{T}}(F)\mathrel{\mathop:}=\sum_v m_{\mathbb{T}}(v)\sum_w p_{\mathbb{T}}(v,w)\left|\mathcal{H}_{\mathbb{T}}F(w)-\mathcal{H}_{\mathbb{T}}F(v)\right|^2\in[0,\infty]. \tag*{(48)} E T ( F ) : = v ∑ m T ( v ) w ∑ p T ( v , w ) ∣ H T F ( w ) − H T F ( v ) ∣ 2 ∈ [ 0 , ∞ ] . (48) Up to normalization, this is the Douglas integral of the trace of the walk on the boundary, see [43 ] and [75 ] . Recall that M d \mathcal{M}^{d} M d O d \mathcal{O}^{d} O d ∥ ξ ∥ K d \|\xi\|_{K}^{d} ∥ ξ ∥ K d Proposition 2.1 ).
Theorem 6.1 Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d F ∈ L 2 ( O d ) F\in L^{2}(\mathcal{O}^{d}) F ∈ L 2 ( O d )
E T ( F ) = 2 ( 1 − τ ) ⟨ F , M d F ⟩ L 2 ( O d ) . \mathcal{E}_{\mathsf{T}}(F)=2(1-\tau)\langle F,\mathcal{M}^{d}F\rangle_{L^{2}(\mathcal{O}^{d})}. E T ( F ) = 2 ( 1 − τ ) ⟨ F , M d F ⟩ L 2 ( O d ) . in [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] M d \mathcal{M}^{d} M d
Proof: We first consider a wavelet F = ∑ c f c 1 c F=\sum_{c}f_{c}\mathbf{1}_{c} F = ∑ c f c 1 c B B B n n n c c c Q d Q^{d} Q d n + 1 n+1 n + 1 B B B ∑ c f c = 0 \sum_{c}f_{c}=0 ∑ c f c = 0 H T F \mathcal{H}_{\mathsf{T}}F H T F B B B B B B c c c τ \tau τ H T F = f c ( 1 − τ j + 1 ) \mathcal{H}_{\mathsf{T}}F=f_{c}(1-\tau^{j+1}) H T F = f c ( 1 − τ j + 1 ) c c c j ≥ 0 j\geq0 j ≥ 0 c c c B B B c c c ( 1 − τ ) 2 ∣ f c ∣ 2 (1-\tau)^{2}|f_{c}|^{2} ( 1 − τ ) 2 ∣ f c ∣ 2 c c c
∑ j ≥ 0 τ − ( j + 1 ) ∣ f c ∣ 2 τ 2 ( j + 1 ) ( 1 − τ ) 2 = ( 1 − τ ) τ ∣ f c ∣ 2 . \sum_{j\geq0}\tau^{-(j+1)}|f_{c}|^{2}\tau^{2(j+1)}(1-\tau)^{2}=(1-\tau)\tau|f_{c}|^{2}. j ≥ 0 ∑ τ − ( j + 1 ) ∣ f c ∣ 2 τ 2 ( j + 1 ) ( 1 − τ ) 2 = ( 1 − τ ) τ ∣ f c ∣ 2 . Thus the edge energy of H T F \mathcal{H}_{\mathsf{T}}F H T F ( 1 − τ ) ∑ c ∣ f c ∣ 2 = ( 1 − τ ) τ − ( n + 1 ) ∥ F ∥ L 2 ( O d ) 2 (1-\tau)\sum_{c}|f_{c}|^{2}=(1-\tau)\tau^{-(n+1)}\|F\|_{L^{2}(\mathcal{O}^{d})}^{2} ( 1 − τ ) ∑ c ∣ f c ∣ 2 = ( 1 − τ ) τ − ( n + 1 ) ∥ F ∥ L 2 ( O d ) 2 m T ( v ) p T ( v , w ) = 1 m_{\mathsf{T}}(v)p_{\mathsf{T}}(v,w)=1 m T ( v ) p T ( v , w ) = 1 v , w v,w v , w (48) is twice the edge energy. On the other hand, the wavelets attached to the balls of level n n n n + 1 n+1 n + 1 M d \mathcal{M}^{d} M d Q d ( n + 1 ) = τ − ( n + 1 ) Q^{d(n+1)}=\tau^{-(n+1)} Q d ( n + 1 ) = τ − ( n + 1 )
Now, we consider wavelets F 1 F_{1} F 1 F 2 F_{2} F 2 B 1 B_{1} B 1 B 2 B_{2} B 2 B 1 ∩ B 2 = ∅ B_{1}\cap B_{2}=\varnothing B 1 ∩ B 2 = ∅ H T F 1 \mathcal{H}_{\mathsf{T}}F_{1} H T F 1 H T F 2 \mathcal{H}_{\mathsf{T}}F_{2} H T F 2 B 2 ⊂ B 1 B_{2}\subset B_{1} B 2 ⊂ B 1 H T F 1 \mathcal{H}_{\mathsf{T}}F_{1} H T F 1 B 2 B_{2} B 2 B 2 B_{2} B 2 H T F 2 \mathcal{H}_{\mathsf{T}}F_{2} H T F 2 F 2 F_{2} F 2 m ≥ 0 m\geq0 m ≥ 0 m m m m + 1 m+1 m + 1 L 2 ( O d ) L^{2}(\mathcal{O}^{d}) L 2 ( O d )
Finally, let F ∈ L 2 ( O d ) F\in L^{2}(\mathcal{O}^{d}) F ∈ L 2 ( O d ) F = c + ∑ ψ ψ F=c+\sum_{\psi}\psi F = c + ∑ ψ ψ v v v n n n v v v n n n m m m m m m m m m m + 1 m+1 m + 1 H T F \mathcal{H}_{\mathsf{T}}F H T F c c c H T ψ \mathcal{H}_{\mathsf{T}}\psi H T ψ ψ \psi ψ m m m H T F \mathcal{H}_{\mathsf{T}}F H T F m m m m + 1 m+1 m + 1 H T ψ \mathcal{H}_{\mathsf{T}}\psi H T ψ m m m
E T ( F ) = ∑ ψ E T ( ψ ) = 2 ( 1 − τ ) ∑ ψ ⟨ ψ , M d ψ ⟩ = 2 ( 1 − τ ) ⟨ F , M d F ⟩ . \mathcal{E}_{\mathsf{T}}(F)=\sum_{\psi}\mathcal{E}_{\mathsf{T}}(\psi)=2(1-\tau)\sum_{\psi}\langle\psi,\mathcal{M}^{d}\psi\rangle=2(1-\tau)\langle F,\mathcal{M}^{d}F\rangle. E T ( F ) = ψ ∑ E T ( ψ ) = 2 ( 1 − τ ) ψ ∑ ⟨ ψ , M d ψ ⟩ = 2 ( 1 − τ ) ⟨ F , M d F ⟩ . :::
On radial functions, Proposition 2.1 gives M r a d d = R r a d d + κ ( I − E 0 ) \mathcal{M}^{d}_{\mathrm{rad}}=R^{d}_{\mathrm{rad}}+\kappa(I-\mathbb{E}_{0}) M rad d = R rad d + κ ( I − E 0 ) κ = 1 1 + ρ \kappa=\frac{1}{1+\rho} κ = 1 + ρ 1 ρ = τ − 1 \rho=\tau^{-1} ρ = τ − 1 α = d \alpha=d α = d E T \mathcal{E}_{\mathsf{T}} E T 2 ( 1 − τ ) 2(1-\tau) 2 ( 1 − τ ) d d d κ ( I − E 0 ) \kappa(I-\mathbb{E}_{0}) κ ( I − E 0 )
The random walk on the dual of SUr _r r
Let G = S U r ( 2 ) G=\mathrm{SU}_r(2) G = SU r ( 2 ) ∗ * ∗ V s V_s V s s ∈ 1 2 N 0 s\in\frac{1}{2}\mathbb{N}_0 s ∈ 2 1 N 0 U r s l 2 U_r\mathfrak{sl}_2 U r sl 2 K ξ i = τ ξ i K\xi_i=\tau^{\xi_i} K ξ i = τ ξ i (46) holds. Since Δ \Delta Δ ∗ * ∗ ∗ * ∗ V r V_r V r ℓ ∞ ( G ^ ) = ⨁ s B ( V s ) \ell^\infty(\widehat{G})=\bigoplus_s B(V_s) ℓ ∞ ( G ) = ⨁ s B ( V s ) V s ⊗ V t V_s\otimes V_t V s ⊗ V t Δ ^ ( a ) = ∑ r ι r a ι r ∗ \widehat{\Delta}(a)=\sum_r \iota_r a\iota_r^* Δ ( a ) = ∑ r ι r a ι r ∗ ι r : V r → V s ⊗ V t \iota_r:V_r\to V_s\otimes V_t ι r : V r → V s ⊗ V t ϕ : = [ 2 ] r − 1 Tr ( K − 1 ⋅ ) \phi\mathrel{\mathop:}= [2]_r^{-1}\operatorname{Tr}(K^{-1}\mathord\cdot) ϕ : = [ 2 ] r − 1 Tr ( K − 1 ⋅ ) B ( V 1 2 ) B(V_{\frac{1}{2}}) B ( V 2 1 )
P : = ( Id ⊗ ϕ ) ∘ Δ ^ , h ^ ( a ) : = ∑ s [ 2 s + 1 ] r Tr ( K − 1 a s ) . (49) P\mathrel{\mathop:}= (\operatorname{Id}\otimes\phi)\circ\widehat{\Delta},\qquad\widehat{h}(a)\mathrel{\mathop:}=\sum_s [2s+1]_r\operatorname{Tr}(K^{-1}a_s). \tag*{(49)} P : = ( Id ⊗ ϕ ) ∘ Δ , h ( a ) : = s ∑ [ 2 s + 1 ] r Tr ( K − 1 a s ) . (49) On central elements, P P P [ 2 s ′ + 1 ] r [ 2 ] r [ 2 s + 1 ] r \frac{[2s'+1]_r}{[2]_r[2s+1]_r} [ 2 ] r [ 2 s + 1 ] r [ 2 s ′ + 1 ] r s s s s ′ = s ± 1 2 s'=s\pm\frac{1}{2} s ′ = s ± 2 1 Tr ( K − 1 ⋅ ) \operatorname{Tr}(K^{-1}\mathord\cdot) Tr ( K − 1 ⋅ ) U r s l 2 U_r\mathfrak{sl}_2 U r sl 2 B ( V s ) B(V_s) B ( V s ) P P P G G G ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) h ^ \widehat{h} h
We denote by H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) P P P ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C f ( A ) : = ∑ k f ( k ) p k ∈ W ∗ ( A ) f(A)\mathrel{\mathop:}=\sum_k f(k)p_k\in W^*(A) f ( A ) : = ∑ k f ( k ) p k ∈ W ∗ ( A ) p k = 1 { τ k } ( A ) p_k=1_{\{\tau^k\}}(A) p k = 1 { τ k } ( A ) F F F X τ X_\tau X τ k ↦ F ( τ k ) k\mapsto F(\tau^k) k ↦ F ( τ k ) N 0 \mathbb{N}_0 N 0 Θ : L ∞ ( S r 2 ) → H ∞ ( G ^ , P ) \Theta:L^\infty(S_r^2)\to H^\infty(\widehat{G},P) Θ : L ∞ ( S r 2 ) → H ∞ ( G , P ) [40 ] , see also [41 Section 2 and Theorem B] , where the convolution operator is written with the opposite convention: the map Θ \Theta Θ G G G G G G L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) ε ^ ∘ Θ = h ^ \widehat{\varepsilon}\circ\Theta=\widehat{h} ε ∘ Θ = h ε ^ \widehat{\varepsilon} ε
a ⋅ b : = s − l i m m → ∞ 1 m ∑ j = 0 m − 1 P j ( a b ) (50) a\mathbin{\cdot}b\mathrel{\mathop:}=\mathop{\mathrm{s-lim}}_{m\to\infty}\frac{1}{m}\sum_{j=0}^{m-1}P^j(ab) \tag*{(50)} a ⋅ b : = s − lim m → ∞ m 1 j = 0 ∑ m − 1 P j ( ab ) (50) of H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) ( G ^ , P ) (\widehat{G},P) ( G , P ) [53 ] . For r = 1 r=1 r = 1 S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) [12 ] , and for central transient random walks on the dual of S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) C ( S 2 ) C(S^2) C ( S 2 ) C ∗ ( S U ( 2 ) ) C^*(\mathrm{SU}(2)) C ∗ ( SU ( 2 )) [13 ] , see also [16 Sections 8 and 9] .
For any harmonic element a a a Γ ( a ) : = P ( a ∗ a ) − a ∗ a \Gamma(a)\mathrel{\mathop:}=P(a^*a)-a^*a Γ ( a ) : = P ( a ∗ a ) − a ∗ a a a a n = 2 s n=2s n = 2 s
E ( n ) ( a ) : = [ 2 s + 1 ] r Tr ( K − 1 Γ ( a ) s ) , E G ^ ( a ) : = ∑ n ≥ 0 E ( n ) ( a ) = h ^ ( Γ ( a ) ) ∈ [ 0 , ∞ ] . (51) \mathcal{E}^{(n)}(a)\mathrel{\mathop:}=[2s+1]_r\operatorname{Tr}(K^{-1}\Gamma(a)_s),\qquad\mathcal{E}_{\widehat{G}}(a)\mathrel{\mathop:}=\sum_{n\geq0}\mathcal{E}^{(n)}(a)=\widehat{h}(\Gamma(a))\in[0,\infty]. \tag*{(51)} E ( n ) ( a ) : = [ 2 s + 1 ] r Tr ( K − 1 Γ ( a ) s ) , E G ( a ) : = n ≥ 0 ∑ E ( n ) ( a ) = h ( Γ ( a )) ∈ [ 0 , ∞ ] . (51) For the simple random walk on T \mathbb{T} T m T m_{\mathbb{T}} m T h ^ \widehat{h} h (48) .
The diagonal chain. The diagonal elements of ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) P P P
P induces a Markov chain on the set of pairs ( s , i ) (s,i) ( s , i ) ∣ i ∣ ≤ s |i|\le s ∣ i ∣ ≤ s n = d e f 2 s n \mathrel{\mathop{=}\limits^{\mathrm{def}}} 2s n = def 2 s k = d e f s − i ∈ { 0 , … , n } k \mathrel{\mathop{=}\limits^{\mathrm{def}}} s-i\in\{0,\ldots,n\} k = def s − i ∈ { 0 , … , n }
D = d e f { ( n , k ) ∈ N 0 2 : k ≤ n } , \mathbb{D} \mathrel{\mathop{=}\limits^{\mathrm{def}}} \{(n,k)\in\mathbb{N}_0^2:k\le n\}, D = def {( n , k ) ∈ N 0 2 : k ≤ n } , and we say that ( n , k ) (n,k) ( n , k ) n n n P ( v , v ′ ) P(v,v') P ( v , v ′ ) ( P f ) ( v ) = ∑ v ′ P ( v , v ′ ) f ( v ′ ) (Pf)(v)=\sum_{v'}P(v,v')f(v') ( P f ) ( v ) = ∑ v ′ P ( v , v ′ ) f ( v ′ ) f f f D \mathbb{D} D
Proposition 6.2 For any ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D
( n , k ) → ( n + 1 , k ) : 1 − τ n − k + 1 ( 1 + τ ) ( 1 − τ n + 1 ) , ( n , k ) → ( n − 1 , k ) : τ ( 1 − τ n − k ) ( 1 + τ ) ( 1 − τ n + 1 ) , ( n , k ) → ( n + 1 , k + 1 ) : τ n − k + 1 ( 1 − τ k + 1 ) ( 1 + τ ) ( 1 − τ n + 1 ) , ( n , k ) → ( n − 1 , k − 1 ) : τ n − k + 1 ( 1 − τ k ) ( 1 + τ ) ( 1 − τ n + 1 ) . \begin{aligned}
(n,k)&\to(n+1,k):\frac{1-\tau^{n-k+1}}{(1+\tau)(1-\tau^{n+1})},\qquad&(n,k)&\to(n-1,k):\frac{\tau(1-\tau^{n-k})}{(1+\tau)(1-\tau^{n+1})},\\
(n,k)&\to(n+1,k+1):\frac{\tau^{n-k+1}(1-\tau^{k+1})}{(1+\tau)(1-\tau^{n+1})},\qquad&(n,k)&\to(n-1,k-1):\frac{\tau^{n-k+1}(1-\tau^k)}{(1+\tau)(1-\tau^{n+1})}.
\end{aligned} ( n , k ) ( n , k ) → ( n + 1 , k ) : ( 1 + τ ) ( 1 − τ n + 1 ) 1 − τ n − k + 1 , → ( n + 1 , k + 1 ) : ( 1 + τ ) ( 1 − τ n + 1 ) τ n − k + 1 ( 1 − τ k + 1 ) , ( n , k ) ( n , k ) → ( n − 1 , k ) : ( 1 + τ ) ( 1 − τ n + 1 ) τ ( 1 − τ n − k ) , → ( n − 1 , k − 1 ) : ( 1 + τ ) ( 1 − τ n + 1 ) τ n − k + 1 ( 1 − τ k ) . The chain is reversible for the measure
w ^ ( n , k ) = d e f [ n + 1 ] r r 2 k − n = 1 − τ n + 1 ( 1 − τ ) τ n − k . (52) \widehat{w}(n,k)\mathrel{\mathop{=}\limits^{\mathrm{def}}}[n+1]_r r^{2k-n}=\frac{1-\tau^{n+1}}{(1-\tau)\tau^{n-k}}. \tag*{(52)} w ( n , k ) = def [ n + 1 ] r r 2 k − n = ( 1 − τ ) τ n − k 1 − τ n + 1 . (52) which is the restriction of h ^ \widehat{h} h
Proof: We first compute the Clebsch–Gordan coefficients of V s ⊗ V 1 2 V_s\otimes V_{\frac{1}{2}} V s ⊗ V 2 1 s ≥ 1 2 s\ge\frac{1}{2} s ≥ 2 1 s = 0 s=0 s = 0 V 1 2 V_{\frac{1}{2}} V 2 1 − s − 1 ≤ i ≤ s -s-1\le i\le s − s − 1 ≤ i ≤ s
x i = d e f ( r i − s [ s + i + 1 ] r [ 2 s + 1 ] r ) 1 2 , y i = d e f ( r s + i + 1 [ s − i ] r [ 2 s + 1 ] r ) 1 2 . x_i\mathrel{\mathop{=}\limits^{\mathrm{def}}}\left(r^{i-s}\frac{[s+i+1]_r}{[2s+1]_r}\right)^{\frac{1}{2}},\qquad y_i\mathrel{\mathop{=}\limits^{\mathrm{def}}}\left(r^{s+i+1}\frac{[s-i]_r}{[2s+1]_r}\right)^{\frac{1}{2}}. x i = def ( r i − s [ 2 s + 1 ] r [ s + i + 1 ] r ) 2 1 , y i = def ( r s + i + 1 [ 2 s + 1 ] r [ s − i ] r ) 2 1 . Expanding the r r r r i − s [ s + i + 1 ] r + r s + i + 1 [ s − i ] r = [ 2 s + 1 ] r r^{i-s}[s+i+1]_r+r^{s+i+1}[s-i]_r=[2s+1]_r r i − s [ s + i + 1 ] r + r s + i + 1 [ s − i ] r = [ 2 s + 1 ] r x i 2 + y i 2 = 1 x_i^2+y_i^2=1 x i 2 + y i 2 = 1
ι s + 1 2 ξ i + 1 2 = x i ξ i ⊗ ξ 1 2 + y i ξ i + 1 ⊗ ξ − 1 2 , − s − 1 ≤ i ≤ s . (53) \iota_{s+\frac{1}{2}}\xi_{i+\frac{1}{2}}=x_i\xi_i\otimes\xi_{\frac{1}{2}}+y_i\xi_{i+1}\otimes\xi_{-\frac{1}{2}},\qquad-s-1\le i\le s. \tag*{(53)} ι s + 2 1 ξ i + 2 1 = x i ξ i ⊗ ξ 2 1 + y i ξ i + 1 ⊗ ξ − 2 1 , − s − 1 ≤ i ≤ s . (53) with ξ ± ( s + 1 ) = 0 \xi_{\pm(s+1)}=0 ξ ± ( s + 1 ) = 0 ξ s ⊗ ξ 1 2 \xi_s\otimes\xi_{\frac{1}{2}} ξ s ⊗ ξ 2 1 s + 1 2 s+\frac{1}{2} s + 2 1 V s + 1 2 V_{s+\frac{1}{2}} V s + 2 1 (53) for i = s i=s i = s (53) holds for some i > − s − 1 i>-s-1 i > − s − 1 Δ ( F ) = F ⊗ K − 1 + 1 ⊗ F \Delta(F)=F\otimes K^{-1}+1\otimes F Δ ( F ) = F ⊗ K − 1 + 1 ⊗ F (46) , we have F ξ 1 2 = r − 1 2 ξ − 1 2 F\xi_{\frac{1}{2}}=r^{-\frac{1}{2}}\xi_{-\frac{1}{2}} F ξ 2 1 = r − 2 1 ξ − 2 1 F ξ − 1 2 = 0 F\xi_{-\frac{1}{2}}=0 F ξ − 2 1 = 0 K − 1 ξ 1 2 = r − 1 2 ξ 1 2 K^{-1}\xi_{\frac{1}{2}}=r^{-\frac{1}{2}}\xi_{\frac{1}{2}} K − 1 ξ 2 1 = r − 2 1 ξ 2 1 ξ i − 1 ⊗ ξ 1 2 \xi_{i-1}\otimes\xi_{\frac{1}{2}} ξ i − 1 ⊗ ξ 2 1 Δ ( F ) ι s + 1 2 ξ i + 1 2 \Delta(F)\iota_{s+\frac{1}{2}}\xi_{i+\frac{1}{2}} Δ ( F ) ι s + 2 1 ξ i + 2 1 r − 1 r i − s [ s + i ] r [ s − i + 1 ] r x i r^{-1}r^{i-s}[s+i]_r[s-i+1]_r x_i r − 1 r i − s [ s + i ] r [ s − i + 1 ] r x i (46) in V s + 1 2 V_{s+\frac{1}{2}} V s + 2 1 Δ ( F ) ι s + 1 2 ξ i + 1 2 = r − i − 1 2 [ s + i + 1 ] r [ s − i + 1 ] r ι s + 1 2 ξ i − 1 2 \Delta(F)\iota_{s+\frac{1}{2}}\xi_{i+\frac{1}{2}}=r^{-i-\frac{1}{2}}[s+i+1]_r[s-i+1]_r\iota_{s+\frac{1}{2}}\xi_{i-\frac{1}{2}} Δ ( F ) ι s + 2 1 ξ i + 2 1 = r − i − 2 1 [ s + i + 1 ] r [ s − i + 1 ] r ι s + 2 1 ξ i − 2 1 ξ i − 1 ⊗ ξ 1 2 \xi_{i-1}\otimes\xi_{\frac{1}{2}} ξ i − 1 ⊗ ξ 2 1 ι s + 1 2 ξ i − 1 2 \iota_{s+\frac{1}{2}}\xi_{i-\frac{1}{2}} ι s + 2 1 ξ i − 2 1 ( r − 1 [ s + i ] r [ s + i + 1 ] r ) 1 2 x i = x i − 1 \left(r^{-1}\frac{[s+i]_r}{[s+i+1]_r}\right)^{\frac{1}{2}}x_i=x_{i-1} ( r − 1 [ s + i + 1 ] r [ s + i ] r ) 2 1 x i = x i − 1 ξ i − 1 ⊗ ξ 1 2 \xi_{i-1}\otimes\xi_{\frac{1}{2}} ξ i − 1 ⊗ ξ 2 1 ξ i ⊗ ξ − 1 2 \xi_i\otimes\xi_{-\frac{1}{2}} ξ i ⊗ ξ − 2 1 ( 1 − x i − 1 2 ) 1 2 = y i − 1 (1-x_{i-1}^2)^{\frac{1}{2}}=y_{i-1} ( 1 − x i − 1 2 ) 2 1 = y i − 1 (53) by induction. Finally, for ∣ i + 1 2 ∣ ≤ s − 1 2 |i+\frac{1}{2}|\le s-\frac{1}{2} ∣ i + 2 1 ∣ ≤ s − 2 1 i + 1 2 i+\frac{1}{2} i + 2 1 V s ⊗ V 1 2 V_s\otimes V_{\frac{1}{2}} V s ⊗ V 2 1 ξ i ⊗ ξ 1 2 \xi_i\otimes\xi_{\frac{1}{2}} ξ i ⊗ ξ 2 1 ξ i + 1 ⊗ ξ − 1 2 \xi_{i+1}\otimes\xi_{-\frac{1}{2}} ξ i + 1 ⊗ ξ − 2 1 ι s + 1 2 ξ i + 1 2 \iota_{s+\frac{1}{2}}\xi_{i+\frac{1}{2}} ι s + 2 1 ξ i + 2 1 ι s − 1 2 ξ i + 1 2 \iota_{s-\frac{1}{2}}\xi_{i+\frac{1}{2}} ι s − 2 1 ξ i + 2 1 ι s − 1 2 ξ i + 1 2 \iota_{s-\frac{1}{2}}\xi_{i+\frac{1}{2}} ι s − 2 1 ξ i + 2 1 y i y_i y i x i x_i x i i − 1 2 = ( i − 1 ) + 1 2 i-\frac{1}{2}=(i-1)+\frac{1}{2} i − 2 1 = ( i − 1 ) + 2 1
∣ ⟨ ι s + 1 2 ξ i + 1 2 , ξ i ⊗ ξ 1 2 ⟩ ∣ 2 = r i − s [ s + i + 1 ] r [ 2 s + 1 ] r , ∣ ⟨ ι s + 1 2 ξ i − 1 2 , ξ i ⊗ ξ − 1 2 ⟩ ∣ 2 = r s + i [ s − i + 1 ] r [ 2 s + 1 ] r , \left|\left\langle\iota_{s+\frac{1}{2}}\xi_{i+\frac{1}{2}},\xi_i\otimes\xi_{\frac{1}{2}}\right\rangle\right|^2=r^{i-s}\frac{[s+i+1]_r}{[2s+1]_r},\qquad\left|\left\langle\iota_{s+\frac{1}{2}}\xi_{i-\frac{1}{2}},\xi_i\otimes\xi_{-\frac{1}{2}}\right\rangle\right|^2=r^{s+i}\frac{[s-i+1]_r}{[2s+1]_r}, ⟨ ι s + 2 1 ξ i + 2 1 , ξ i ⊗ ξ 2 1 ⟩ 2 = r i − s [ 2 s + 1 ] r [ s + i + 1 ] r , ⟨ ι s + 2 1 ξ i − 2 1 , ξ i ⊗ ξ − 2 1 ⟩ 2 = r s + i [ 2 s + 1 ] r [ s − i + 1 ] r , ∣ ⟨ ι s − 1 2 ξ i + 1 2 , ξ i ⊗ ξ 1 2 ⟩ ∣ 2 = r s + i + 1 [ s − i ] r [ 2 s + 1 ] r , ∣ ⟨ ι s − 1 2 ξ i − 1 2 , ξ i ⊗ ξ − 1 2 ⟩ ∣ 2 = r i − s − 1 [ s + i ] r [ 2 s + 1 ] r . \left|\left\langle\iota_{s-\frac{1}{2}}\xi_{i+\frac{1}{2}},\xi_i\otimes\xi_{\frac{1}{2}}\right\rangle\right|^2=r^{s+i+1}\frac{[s-i]_r}{[2s+1]_r},\qquad\left|\left\langle\iota_{s-\frac{1}{2}}\xi_{i-\frac{1}{2}},\xi_i\otimes\xi_{-\frac{1}{2}}\right\rangle\right|^2=r^{i-s-1}\frac{[s+i]_r}{[2s+1]_r}. ⟨ ι s − 2 1 ξ i + 2 1 , ξ i ⊗ ξ 2 1 ⟩ 2 = r s + i + 1 [ 2 s + 1 ] r [ s − i ] r , ⟨ ι s − 2 1 ξ i − 2 1 , ξ i ⊗ ξ − 2 1 ⟩ 2 = r i − s − 1 [ 2 s + 1 ] r [ s + i ] r . Since ϕ \phi ϕ 1 1 + τ \frac{1}{1+\tau} 1 + τ 1 τ 1 + τ \frac{\tau}{1+\tau} 1 + τ τ C ξ 1 2 \mathbb{C}\xi_{\frac{1}{2}} C ξ 2 1 C ξ − 1 2 \mathbb{C}\xi_{-\frac{1}{2}} C ξ − 2 1 ι s ′ ξ i ′ \iota_{s'}\xi_{i'} ι s ′ ξ i ′ i ′ i' i ′ a a a
P ( a ) ( s , i ) = ∑ m = ± 1 2 τ 1 2 − m 1 + τ ∑ s ′ = s ± 1 2 ∣ ⟨ ι s ′ ξ i + m , ξ i ⊗ ξ m ⟩ ∣ 2 a ( s ′ , i + m ) . \mathrm{P}(a)(s,i)=\sum_{m=\pm\frac{1}{2}}\frac{\tau^{\frac{1}{2}-m}}{1+\tau}\sum_{s'=s\pm\frac{1}{2}}\left|\left\langle\iota_{s'}\xi_{i+m},\xi_i\otimes\xi_m\right\rangle\right|^2a(s',i+m). P ( a ) ( s , i ) = m = ± 2 1 ∑ 1 + τ τ 2 1 − m s ′ = s ± 2 1 ∑ ∣ ⟨ ι s ′ ξ i + m , ξ i ⊗ ξ m ⟩ ∣ 2 a ( s ′ , i + m ) . The formulas follow from [ a ] r / [ b ] r = r b − a 1 − τ a 1 − τ b [a]_r/[b]_r=r^{b-a}\frac{1-\tau^a}{1-\tau^b} [ a ] r / [ b ] r = r b − a 1 − τ b 1 − τ a s + i = n − k s+i=n-k s + i = n − k s − i = k s-i=k s − i = k K − 1 K^{-1} K − 1 ξ i \xi_i ξ i τ − i = r 2 k − n \tau^{-i}=r^{2k-n} τ − i = r 2 k − n (52) .
These transition probabilities were first computed by Biane Biane [15 Section 4] . He considers the quantum Bernoulli random walk on the dual of S U q ( 2 ) \mathrm{SU}_q(2) SU q ( 2 ) [10 ] , restricted to the subalgebra generated by the centre and by the generator of the torus, which is our diagonal algebra, for the state defined by the Woronowicz character, which is our state ϕ \phi ϕ [40 ] . With his conventions, the dictionary is q 2 = τ q^2=\tau q 2 = τ ( r , k ) = ( n , 2 k − n ) (r,k)=(n,2k-n) ( r , k ) = ( n , 2 k − n ) n = 2 s n=2s n = 2 s 1 2 Tr \frac{1}{2}\operatorname{Tr} 2 1 Tr ϕ \phi ϕ q → 0 q\to0 q → 0 ( 2 S − X , X ) (2S-X,X) ( 2 S − X , X ) X X X S S S [15 Sections 2 and 4] , see also [16 Section 13] . For the state ϕ \phi ϕ τ → 0 \tau\to0 τ → 0 Proposition 6.2 , the chain then moves deterministically from ( n , k ) (n,k) ( n , k ) ( n + 1 , k ) (n+1,k) ( n + 1 , k )
Recall that the closed form of L τ L_\tau L τ E τ ( f ) = ∑ k ≥ 0 w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 \mathcal{E}_\tau(f)=\sum_{k\geq0}w_k\beta_k|f(k+1)-f(k)|^2 E τ ( f ) = ∑ k ≥ 0 w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 w k β k = ( 1 − τ ) ( 1 − τ k + 1 ) w_k\beta_k=(1-\tau)(1-\tau^{k+1}) w k β k = ( 1 − τ ) ( 1 − τ k + 1 ) (22) and (23) with η = τ \eta=\tau η = τ
c τ : = 2 τ ( 1 + τ ) ( 1 − τ ) 2 . (54) c_\tau\mathrel{\mathop:}=\frac{2\tau}{(1+\tau)(1-\tau)^2}. \tag*{(54)} c τ : = ( 1 + τ ) ( 1 − τ ) 2 2 τ . (54) Proposition 6.4 For any ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D ( n , k ) (n,k) ( n , k ) ( n + 1 , k + 1 ) (n+1,k+1) ( n + 1 , k + 1 )
w ~ ( n , k ) P ( ( n , k ) , ( n + 1 , k + 1 ) ) = b k : = τ ( 1 − τ k + 1 ) ( 1 + τ ) ( 1 − τ ) = c τ 2 w k β k . \widetilde{w}(n,k)\mathrm{P}((n,k),(n+1,k+1))=b_k\mathrel{\mathop:}=\frac{\tau(1-\tau^{k+1})}{(1+\tau)(1-\tau)}=\frac{c_\tau}{2}w_k\beta_k. w ( n , k ) P (( n , k ) , ( n + 1 , k + 1 )) = b k : = ( 1 + τ ) ( 1 − τ ) τ ( 1 − τ k + 1 ) = 2 c τ w k β k . In particular, it does not depend on the level n n n ( n , k ) (n,k) ( n , k ) ( n + 1 , k ) (n+1,k) ( n + 1 , k )
w ~ ( n , k ) P ( ( n , k ) , ( n + 1 , k ) ) = a n − k , where a d : = 1 − τ d + 1 ( 1 − τ 2 ) τ d , d ∈ N 0 . \widetilde{w}(n,k)\mathrm{P}((n,k),(n+1,k))=a_{n-k},\qquad\text{where }a_d\mathrel{\mathop:}=\frac{1-\tau^{d+1}}{(1-\tau^2)\tau^d},\quad d\in\mathbb{N}_0. w ( n , k ) P (( n , k ) , ( n + 1 , k )) = a n − k , where a d : = ( 1 − τ 2 ) τ d 1 − τ d + 1 , d ∈ N 0 . Consequently, in the coordinates ( d , k ) = ( n − k , k ) ∈ N 0 2 (d,k)=(n-k,k)\in\mathbb{N}_0^2 ( d , k ) = ( n − k , k ) ∈ N 0 2 N 0 \mathbb{N}_0 N 0 ( a d ) (a_d) ( a d ) ( b k ) (b_k) ( b k )
Proof: By Proposition 6.2 and (52) , the first conductance is
1 − τ n + 1 ( 1 − τ ) τ n − k ⋅ τ n − k + 1 ( 1 − τ k + 1 ) ( 1 + τ ) ( 1 − τ n + 1 ) = τ ( 1 − τ k + 1 ) ( 1 + τ ) ( 1 − τ ) , \frac{1-\tau^{n+1}}{(1-\tau)\tau^{n-k}}\cdot\frac{\tau^{n-k+1}(1-\tau^{k+1})}{(1+\tau)(1-\tau^{n+1})}=\frac{\tau(1-\tau^{k+1})}{(1+\tau)(1-\tau)}, ( 1 − τ ) τ n − k 1 − τ n + 1 ⋅ ( 1 + τ ) ( 1 − τ n + 1 ) τ n − k + 1 ( 1 − τ k + 1 ) = ( 1 + τ ) ( 1 − τ ) τ ( 1 − τ k + 1 ) , and the second one is
1 − τ n + 1 ( 1 − τ ) τ n − k ⋅ 1 − τ n − k + 1 ( 1 + τ ) ( 1 − τ n + 1 ) = a n − k . \frac{1-\tau^{n+1}}{(1-\tau)\tau^{n-k}}\cdot\frac{1-\tau^{n-k+1}}{(1+\tau)(1-\tau^{n+1})}=a_{n-k}. ( 1 − τ ) τ n − k 1 − τ n + 1 ⋅ ( 1 + τ ) ( 1 − τ n + 1 ) 1 − τ n − k + 1 = a n − k . In the coordinates ( d , k ) (d,k) ( d , k ) ( n ± 1 , k ) (n\pm1,k) ( n ± 1 , k ) d d d ± 1 \pm1 ± 1 k k k ( n ± 1 , k ± 1 ) (n\pm1,k\pm1) ( n ± 1 , k ± 1 ) d d d k k k ± 1 \pm1 ± 1
The following lemma is the key algebraic property of the diagonal chain.
Lemma 6.5 Let j ≥ 0 j\geq0 j ≥ 0 g : N 0 → C g:\mathbb{N}_0\to\mathbb{C} g : N 0 → C f f f D \mathbb{D} D f ( n , k ) = g ( n ) τ j k f(n,k)=g(n)\tau^{jk} f ( n , k ) = g ( n ) τ j k n ≥ 1 n\geq1 n ≥ 1 0 ≤ k ≤ n 0\leq k\leq n 0 ≤ k ≤ n
( P f ) ( n , k ) = ( A j g ) ( n ) τ j k + ( B j g ) ( n ) τ ( j − 1 ) k , (55) (Pf)(n,k)=(A_jg)(n)\tau^{jk}+(B_jg)(n)\tau^{(j-1)k}, \tag*{(55)} ( P f ) ( n , k ) = ( A j g ) ( n ) τ j k + ( B j g ) ( n ) τ ( j − 1 ) k , (55) where
( A j g ) ( n ) ≔ ( 1 − τ n + j + 2 ) g ( n + 1 ) + ( τ − τ n − j + 1 ) g ( n − 1 ) ( 1 + τ ) ( 1 − τ n + 1 ) , (A_jg)(n)\coloneqq\frac{(1-\tau^{n+j+2})g(n+1)+(\tau-\tau^{n-j+1})g(n-1)}{(1+\tau)(1-\tau^{n+1})}, ( A j g ) ( n ) : = ( 1 + τ ) ( 1 − τ n + 1 ) ( 1 − τ n + j + 2 ) g ( n + 1 ) + ( τ − τ n − j + 1 ) g ( n − 1 ) , ( B j g ) ( n ) ≔ τ n + 1 [ ( τ j − 1 ) g ( n + 1 ) + ( τ − j − 1 ) g ( n − 1 ) ] ( 1 + τ ) ( 1 − τ n + 1 ) . (B_jg)(n)\coloneqq\frac{\tau^{n+1}\left[(\tau^j-1)g(n+1)+(\tau^{-j}-1)g(n-1)\right]}{(1+\tau)(1-\tau^{n+1})}. ( B j g ) ( n ) : = ( 1 + τ ) ( 1 − τ n + 1 ) τ n + 1 [ ( τ j − 1 ) g ( n + 1 ) + ( τ − j − 1 ) g ( n − 1 ) ] . Proof: Fix n ≥ 1 n\geq1 n ≥ 1 0 ≤ k ≤ n 0\leq k\leq n 0 ≤ k ≤ n y = τ k y=\tau^k y = τ k ( n − 1 , k ) (n-1,k) ( n − 1 , k ) k = n k=n k = n ( n − 1 , k − 1 ) (n-1,k-1) ( n − 1 , k − 1 ) k = 0 k=0 k = 0 f ( n , k ′ ) = g ( n ′ ) τ j k ′ f(n,k')=g(n')\tau^{jk'} f ( n , k ′ ) = g ( n ′ ) τ j k ′ ( P f ) ( n , k ) (Pf)(n,k) ( P f ) ( n , k ) Proposition 6.2 , the number ( 1 + τ ) ( 1 − τ n + 1 ) ( P f ) ( n , k ) (1+\tau)(1-\tau^{n+1})(Pf)(n,k) ( 1 + τ ) ( 1 − τ n + 1 ) ( P f ) ( n , k )
( 1 − τ n + 1 y − 1 ) g ( n + 1 ) y j + τ ( 1 − τ n y − 1 ) g ( n − 1 ) y j + τ n + 1 y − 1 ( 1 − τ y ) g ( n + 1 ) τ j y j + τ n + 1 y − 1 ( 1 − y ) g ( n − 1 ) τ − j y j . (1-\tau^{n+1}y^{-1})g(n+1)y^j+\tau(1-\tau^ny^{-1})g(n-1)y^j
+\tau^{n+1}y^{-1}(1-\tau y)g(n+1)\tau^jy^j+\tau^{n+1}y^{-1}(1-y)g(n-1)\tau^{-j}y^j. ( 1 − τ n + 1 y − 1 ) g ( n + 1 ) y j + τ ( 1 − τ n y − 1 ) g ( n − 1 ) y j + τ n + 1 y − 1 ( 1 − τ y ) g ( n + 1 ) τ j y j + τ n + 1 y − 1 ( 1 − y ) g ( n − 1 ) τ − j y j . Collecting the terms in y j y^j y j y j − 1 y^{j-1} y j − 1 (55) .
Note that B 0 = 0 B_0=0 B 0 = 0 A 0 A_0 A 0
1 − τ n + 2 ( 1 + τ ) ( 1 − τ n + 1 ) = [ n + 2 ] r [ 2 ] r [ n + 1 ] r \frac{1-\tau^{n+2}}{(1+\tau)(1-\tau^{n+1})}=\frac{[n+2]_r}{[2]_r[n+1]_r} ( 1 + τ ) ( 1 − τ n + 1 ) 1 − τ n + 2 = [ 2 ] r [ n + 1 ] r [ n + 2 ] r and
τ − τ n + 1 ( 1 + τ ) ( 1 − τ n + 1 ) = [ n ] r [ 2 ] r [ n + 1 ] r . \frac{\tau-\tau^{n+1}}{(1+\tau)(1-\tau^{n+1})}=\frac{[n]_r}{[2]_r[n+1]_r}. ( 1 + τ ) ( 1 − τ n + 1 ) τ − τ n + 1 = [ 2 ] r [ n + 1 ] r [ n ] r . In particular, the level of the diagonal chain is itself a Markov chain.
For any integer l ≥ 0 l\geq0 l ≥ 0 V l V_l V l f : D → C f:\mathbb{D}\to\mathbb{C} f : D → C n ≥ 0 n\geq0 n ≥ 0 k ↦ f ( n , k ) k\mapsto f(n,k) k ↦ f ( n , k ) { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } k ↦ p ( τ k ) k\mapsto p(\tau^k) k ↦ p ( τ k ) p p p l l l n < l n<l n < l n + 1 ≤ l n+1\leq l n + 1 ≤ l l − 1 l-1 l − 1 V l V_l V l f ( n , k ) = ∑ j = 0 l g j ( n ) τ j k f(n,k)=\sum_{j=0}^{l}g_j(n)\tau^{jk} f ( n , k ) = ∑ j = 0 l g j ( n ) τ j k Lemma 6.5 shows that P ( V l ) ⊂ V l P(V_l)\subset V_l P ( V l ) ⊂ V l
Remark 6.6 The restrictions to the level n = 2 s n=2s n = 2 s V l V_l V l ∑ j = 0 l a j K − j \sum_{j=0}^{l}a_jK^{-j} ∑ j = 0 l a j K − j V s V_s V s τ − i j = τ − s j τ j k \tau^{-ij}=\tau^{-sj}\tau^{jk} τ − ij = τ − s j τ j k ad ( E ) ( T ) = E T − K T K − 1 E \operatorname{ad}(E)(T)=ET-KTK^{-1}E ad ( E ) ( T ) = E T − K T K − 1 E ad ( F ) ( T ) = F T K − T F K \operatorname{ad}(F)(T)=FTK-TFK ad ( F ) ( T ) = F T K − T F K ad ( K ) ( T ) = K T K − 1 \operatorname{ad}(K)(T)=KTK^{-1} ad ( K ) ( T ) = K T K − 1 B ( V s ) \mathrm{B}(V_s) B ( V s ) ad ( E ) ( K − 1 ) = ( 1 − τ − 1 ) E K − 1 \operatorname{ad}(E)(K^{-1})=(1-\tau^{-1})EK^{-1} ad ( E ) ( K − 1 ) = ( 1 − τ − 1 ) E K − 1 ad ( E ) ( E K − 1 ) = 0 \operatorname{ad}(E)(EK^{-1})=0 ad ( E ) ( E K − 1 ) = 0 ad ( F ) ( K − 1 ) = ( 1 − τ ) F \operatorname{ad}(F)(K^{-1})=(1-\tau)F ad ( F ) ( K − 1 ) = ( 1 − τ ) F ad ( F ) ( F ) = 0 \operatorname{ad}(F)(F)=0 ad ( F ) ( F ) = 0 K − 1 K^{-1} K − 1 ad ( E ) 2 ( K − 1 ) = ad ( F ) 2 ( K − 1 ) = 0 \operatorname{ad}(E)^2(K^{-1})=\operatorname{ad}(F)^2(K^{-1})=0 ad ( E ) 2 ( K − 1 ) = ad ( F ) 2 ( K − 1 ) = 0 K − 1 K^{-1} K − 1 B ( V s ) \mathrm{B}(V_s) B ( V s ) 1 1 1 K − j K^{-j} K − j j ≤ l j\leq l j ≤ l l l l n ≥ l n\geq l n ≥ l l + 1 l+1 l + 1 V l V_l V l n n n l l l P ( V l ) ⊂ V l P(V_l)\subset V_l P ( V l ) ⊂ V l P P P
Harmonic functions of the diagonal chain
We denote by ( n t , k t ) t ≥ 0 (n_t,k_t)_{t\geq0} ( n t , k t ) t ≥ 0 P v \mathbf{P}_v P v E v \mathbf{E}_v E v v ∈ D v\in\mathbb{D} v ∈ D j t ≔ n t − k t j_t\coloneqq n_t-k_t j t : = n t − k t { ( n , n ) : n ≥ 0 } \{(n,n):n\geq0\} {( n , n ) : n ≥ 0 } k k k ( n , k ) → ( n ± 1 , k ± 1 ) (n,k)\to(n\pm1,k\pm1) ( n , k ) → ( n ± 1 , k ± 1 )
Lemma 6.7
For any ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D
E ( n , k ) [ ∑ t ≥ 0 r j t ] ≤ 1 + τ ( 1 − τ ) ( 1 − r ) 2 r n − k . \mathbf{E}_{(n,k)}\left[\sum_{t\geq0}r^{j_t}\right]\leq\frac{1+\tau}{(1-\tau)(1-r)^2}r^{n-k}. E ( n , k ) [ t ≥ 0 ∑ r j t ] ≤ ( 1 − τ ) ( 1 − r ) 2 1 + τ r n − k . Almost surely, the chain makes only finitely many horizontal moves, n t → ∞ n_t\to\infty n t → ∞ k t k_t k t k ∞ k_\infty k ∞
For any bounded function f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C H f \mathcal{H}f H f D \mathbb{D} D H f ( v ) ≔ E v [ f ( k ∞ ) ] \mathcal{H}f(v)\coloneqq\mathbf{E}_v[f(k_\infty)] H f ( v ) : = E v [ f ( k ∞ )] ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D
∣ H f ( n , k ) − f ( k ) ∣ ≤ 4 τ ( 1 − τ ) 2 ( 1 − r ) 2 ∥ f ∥ ∞ r n − k . (56) |\mathcal{H}f(n,k)-f(k)|\leq\frac{4\tau}{(1-\tau)^2(1-r)^2}\|f\|_\infty r^{n-k}. \tag*{(56)} ∣ H f ( n , k ) − f ( k ) ∣ ≤ ( 1 − τ ) 2 ( 1 − r ) 2 4 τ ∥ f ∥ ∞ r n − k . (56) If u u u D \mathbb{D} D g ( k ) ≔ lim n → ∞ u ( n , k ) g(k)\coloneqq\lim_{n\to\infty}u(n,k) g ( k ) : = lim n → ∞ u ( n , k ) k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 u = H g u=\mathcal{H}g u = H g
Proof. 1. From a state ( n , k ) (n,k) ( n , k ) j = n − k j=n-k j = n − k
p + ≔ 1 − τ j + 1 ( 1 + τ ) ( 1 − τ n + 1 ) , p_+\coloneqq\frac{1-\tau^{j+1}}{(1+\tau)(1-\tau^{n+1})}, p + : = ( 1 + τ ) ( 1 − τ n + 1 ) 1 − τ j + 1 , decreases by one with probability
p − ≔ τ ( 1 − τ j ) ( 1 + τ ) ( 1 − τ n + 1 ) , p_-\coloneqq\frac{\tau(1-\tau^j)}{(1+\tau)(1-\tau^{n+1})}, p − : = ( 1 + τ ) ( 1 − τ n + 1 ) τ ( 1 − τ j ) , and is unchanged otherwise. We have p − ≤ τ p + p_-\leq\tau p_+ p − ≤ τ p + p + ≥ 1 − τ 1 + τ p_+\geq\frac{1-\tau}{1+\tau} p + ≥ 1 + τ 1 − τ r − 1 − 1 > 0 r^{-1}-1>0 r − 1 − 1 > 0
E [ r j t + 1 − r j t ∣ ( n t , k t ) ] = r j t [ p + ( r − 1 ) + p − ( r − 1 − 1 ) ] ≤ r j t p + [ ( r − 1 ) + τ ( r − 1 − 1 ) ] = − r j t p + ( 1 − r ) 2 ≤ − ( 1 − τ ) ( 1 − r ) 2 1 + τ r j t . \begin{aligned}
\mathbf{E}\left[r^{j_{t+1}}-r^{j_t}\mid(n_t,k_t)\right]
&=r^{j_t}\left[p_+(r-1)+p_-(r^{-1}-1)\right]\\
&\leq r^{j_t}p_+\left[(r-1)+\tau(r^{-1}-1)\right]\\
&=-r^{j_t}p_+(1-r)^2
\leq-\frac{(1-\tau)(1-r)^2}{1+\tau}r^{j_t}.
\end{aligned} E [ r j t + 1 − r j t ∣ ( n t , k t ) ] = r j t [ p + ( r − 1 ) + p − ( r − 1 − 1 ) ] ≤ r j t p + [ ( r − 1 ) + τ ( r − 1 − 1 ) ] = − r j t p + ( 1 − r ) 2 ≤ − 1 + τ ( 1 − τ ) ( 1 − r ) 2 r j t . Taking expectations and summing over 0 ≤ t < T 0\leq t<T 0 ≤ t < T
( 1 − τ ) ( 1 − r ) 2 1 + τ ∑ t < T E ( n , k ) [ r j t ] ≤ r n − k . \frac{(1-\tau)(1-r)^2}{1+\tau}\sum_{t<T}\mathbf{E}_{(n,k)}[r^{j_t}]
\leq r^{n-k}. 1 + τ ( 1 − τ ) ( 1 − r ) 2 t < T ∑ E ( n , k ) [ r j t ] ≤ r n − k . Letting T → ∞ T\to\infty T → ∞
By Proposition 6.2 , the probability of a horizontal move from ( n , k ) (n,k) ( n , k )
r n − k + 1 ( 2 − τ k + 1 − τ k ) ( 1 + τ ) ( 1 − τ n + 1 ) ≤ 2 τ ( 1 + τ ) ( 1 − τ ) r n − k ≤ 2 τ ( 1 + τ ) ( 1 − τ ) r n − k . \frac{r^{n-k+1}(2-\tau^{k+1}-\tau^k)}{(1+\tau)(1-\tau^{n+1})}
\leq\frac{2\tau}{(1+\tau)(1-\tau)}r^{n-k}
\leq\frac{2\tau}{(1+\tau)(1-\tau)}r^{n-k}. ( 1 + τ ) ( 1 − τ n + 1 ) r n − k + 1 ( 2 − τ k + 1 − τ k ) ≤ ( 1 + τ ) ( 1 − τ ) 2 τ r n − k ≤ ( 1 + τ ) ( 1 − τ ) 2 τ r n − k . By the first part, the expected number of horizontal moves of the chain started at ( n , k ) (n,k) ( n , k )
2 τ ( 1 − τ ) 2 ( 1 − r ) 2 r n − k < ∞ . \frac{2\tau}{(1-\tau)^2(1-r)^2}r^{n-k}<\infty. ( 1 − τ ) 2 ( 1 − r ) 2 2 τ r n − k < ∞. Hence there are almost surely finitely many horizontal moves, and k t k_t k t k k k ∑ r j t < ∞ \sum r^{j_t}<\infty ∑ r j t < ∞ j t → ∞ j_t\to\infty j t → ∞ n t ≥ j t → ∞ n_t\geq j_t\to\infty n t ≥ j t → ∞
The function H f \mathcal{H}f H f ( n , k ) (n,k) ( n , k ) k ∞ = k k_\infty=k k ∞ = k ∣ H f ( n , k ) − f ( k ) ∣ |\mathcal{H}f(n,k)-f(k)| ∣ H f ( n , k ) − f ( k ) ∣ 2 ∥ f ∥ ∞ 2\|f\|_\infty 2∥ f ∥ ∞ (56) .
The process ( u ( n t , k t ) ) t ≥ 0 (u(n_t,k_t))_{t\geq0} ( u ( n t , k t ) ) t ≥ 0 k t = k ∞ k_t=k_\infty k t = k ∞ t t t n t → ∞ n_t\to\infty n t → ∞ u ( n t , k t ) → g ( k ∞ ) u(n_t,k_t)\to g(k_\infty) u ( n t , k t ) → g ( k ∞ ) g g g ∥ u ∥ ∞ \|u\|_\infty ∥ u ∥ ∞ u ( v ) = E v [ u ( n t , k t ) ] → E v [ g ( k ∞ ) ] = H g ( v ) u(v)=\mathbf{E}_v[u(n_t,k_t)]\to\mathbf{E}_v[g(k_\infty)]=\mathcal{H}g(v) u ( v ) = E v [ u ( n t , k t )] → E v [ g ( k ∞ )] = H g ( v ) v ∈ D v\in\mathbb{D} v ∈ D
For any n ∈ N 0 ∪ { ∞ } n\in\mathbb{N}_0\cup\{\infty\} n ∈ N 0 ∪ { ∞ } m n ≔ ∑ 0 ≤ k ≤ n τ k δ τ k m_n\coloneqq\sum_{0\leq k\leq n}\tau^k\delta_{\tau^k} m n : = ∑ 0 ≤ k ≤ n τ k δ τ k ( 1 − τ ) m ∞ (1-\tau)m_\infty ( 1 − τ ) m ∞ A A A h h h w ^ ( n , ⋅ ) \widehat{w}(n,\cdot) w ( n , ⋅ ) { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } m n m_n m n τ k ↦ k \tau^k \mapsto k τ k ↦ k 0 ≤ l ≤ n 0 \le l \le n 0 ≤ l ≤ n M l , n M_{l,n} M l , n l l l L 2 ( m n ) L^2(m_n) L 2 ( m n ) l − 1 l-1 l − 1 m n m_n m n n + 1 ≥ l + 1 n+1 \ge l+1 n + 1 ≥ l + 1 p ^ l \widehat p_l p l m ∞ m_\infty m ∞ η = τ \eta=\tau η = τ φ l τ = κ l p ^ l \varphi_l^\tau=\kappa_l\widehat p_l φ l τ = κ l p l κ l ≠ 0 \kappa_l\ne0 κ l = 0 φ l τ \varphi_l^\tau φ l τ l ≥ 1 l\ge1 l ≥ 1 ( ς l ( n ) ) n ≥ l (\varsigma_l(n))_{n\ge l} ( ς l ( n ) ) n ≥ l ς l ( l ) = 1 \varsigma_l(l)=1 ς l ( l ) = 1 ς l ( l − 1 ) = 0 \varsigma_l(l-1)=0 ς l ( l − 1 ) = 0 n ≥ l n\ge l n ≥ l
( 1 − τ n + l + 2 ) ς l ( n + 1 ) = ( 1 + τ ) ( 1 − τ n + 1 ) ς l ( n ) − ( τ − τ n − l + 1 ) ς l ( n − 1 ) . (57) (1-\tau^{n+l+2})\varsigma_l(n+1)=(1+\tau)(1-\tau^{n+1})\varsigma_l(n)-(\tau-\tau^{n-l+1})\varsigma_l(n-1). \tag*{(57)} ( 1 − τ n + l + 2 ) ς l ( n + 1 ) = ( 1 + τ ) ( 1 − τ n + 1 ) ς l ( n ) − ( τ − τ n − l + 1 ) ς l ( n − 1 ) . (57) The coefficient of ς l ( n − 1 ) \varsigma_l(n-1) ς l ( n − 1 ) n = l n=l n = l Lemma 6.5 , the relation (57) means ( A l ς l ) ( n ) = ς l ( n ) (A_l\varsigma_l)(n)=\varsigma_l(n) ( A l ς l ) ( n ) = ς l ( n )
Lemma 6.8 Put P τ ≔ ( τ 4 ; τ ) ∞ − 1 P_\tau\coloneqq(\tau^4;\tau)_\infty^{-1} P τ : = ( τ 4 ; τ ) ∞ − 1 M τ ≔ ∏ i ≥ 0 ( 1 + 4 P τ ( i + 1 ) τ i ) M_\tau\coloneqq\prod_{i\ge0}(1+4P_\tau(i+1)\tau^i) M τ : = ∏ i ≥ 0 ( 1 + 4 P τ ( i + 1 ) τ i ) l ≥ 1 l\ge1 l ≥ 1 n ≥ l − 1 n\ge l-1 n ≥ l − 1
∣ ς l ( n ) ∣ ≤ M τ , ∣ ς l ( n + 1 ) − ς l ( n ) ∣ ≤ 4 P τ M τ ( n − l + 2 ) τ n − l + 1 . (58) |\varsigma_l(n)|\le M_\tau,\qquad|\varsigma_l(n+1)-\varsigma_l(n)|\le4P_\tau M_\tau(n-l+2)\tau^{n-l+1}. \tag*{(58)} ∣ ς l ( n ) ∣ ≤ M τ , ∣ ς l ( n + 1 ) − ς l ( n ) ∣ ≤ 4 P τ M τ ( n − l + 2 ) τ n − l + 1 . (58) Consequently, the sequence ( ς l ( n ) ) n ≥ l (\varsigma_l(n))_{n\ge l} ( ς l ( n ) ) n ≥ l ς l ( ∞ ) \varsigma_l(\infty) ς l ( ∞ )
∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ 4 P τ M τ ( 1 − τ ) 2 ( n − l + 2 ) τ n − l + 1 |\varsigma_l(n)-\varsigma_l(\infty)|\le\frac{4P_\tau M_\tau}{(1-\tau)^2}(n-l+2)\tau^{n-l+1} ∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ ( 1 − τ ) 2 4 P τ M τ ( n − l + 2 ) τ n − l + 1 for any n ≥ l − 1 n\ge l-1 n ≥ l − 1
Proof. Put δ ( n ) ≔ ς l ( n + 1 ) − ς l ( n ) \delta(n)\coloneqq\varsigma_l(n+1)-\varsigma_l(n) δ ( n ) : = ς l ( n + 1 ) − ς l ( n ) n ≥ l − 1 n\ge l-1 n ≥ l − 1 δ ( l − 1 ) = 1 \delta(l-1)=1 δ ( l − 1 ) = 1 (57) gives, for any n ≥ l n\ge l n ≥ l
( 1 − τ n + l + 2 ) δ ( n ) = τ δ ( n − 1 ) + τ n [ ( τ l + 2 − τ − τ 2 ) ς l ( n ) + τ l − 1 ς l ( n − 1 ) ] . (59) (1-\tau^{n+l+2})\delta(n)=\tau\delta(n-1)+\tau^n\left[(\tau^{l+2}-\tau-\tau^2)\varsigma_l(n)+\tau^{l-1}\varsigma_l(n-1)\right]. \tag*{(59)} ( 1 − τ n + l + 2 ) δ ( n ) = τ δ ( n − 1 ) + τ n [ ( τ l + 2 − τ − τ 2 ) ς l ( n ) + τ l − 1 ς l ( n − 1 ) ] . (59) For n ≥ l − 1 n\ge l-1 n ≥ l − 1 M ( n ) ≔ max { 1 , ∣ ς l ( l − 1 ) ∣ , … , ∣ ς l ( n ) ∣ } M(n)\coloneqq\max\{1,|\varsigma_l(l-1)|,\ldots,|\varsigma_l(n)|\} M ( n ) : = max { 1 , ∣ ς l ( l − 1 ) ∣ , … , ∣ ς l ( n ) ∣ } ( M ( n ) ) (M(n)) ( M ( n )) M ( l − 1 ) = 1 M(l-1)=1 M ( l − 1 ) = 1 n ≥ l n\ge l n ≥ l τ n ∣ τ l + 2 − τ − τ 2 ∣ ≤ 2 τ n + 1 ≤ 2 τ n − l + 1 \tau^n|\tau^{l+2}-\tau-\tau^2|\le2\tau^{n+1}\le2\tau^{n-l+1} τ n ∣ τ l + 2 − τ − τ 2 ∣ ≤ 2 τ n + 1 ≤ 2 τ n − l + 1 1 − τ n + l + 2 ≥ 1 − τ n − l + 4 1-\tau^{n+l+2}\ge1-\tau^{n-l+4} 1 − τ n + l + 2 ≥ 1 − τ n − l + 4 l ≥ 1 l\ge1 l ≥ 1 (59) gives
∣ δ ( n ) ∣ ≤ τ ∣ δ ( n − 1 ) ∣ + 3 τ n − l + 1 M ( n ) 1 − τ n − l + 4 . |\delta(n)|\le\frac{\tau|\delta(n-1)|+3\tau^{n-l+1}M(n)}{1-\tau^{n-l+4}}. ∣ δ ( n ) ∣ ≤ 1 − τ n − l + 4 τ ∣ δ ( n − 1 ) ∣ + 3 τ n − l + 1 M ( n ) . Put a ( n ) ≔ ∣ δ ( n ) ∣ τ n − l + 1 M ( n ) a(n)\coloneqq\frac{|\delta(n)|}{\tau^{n-l+1}M(n)} a ( n ) : = τ n − l + 1 M ( n ) ∣ δ ( n ) ∣ n ≥ l − 1 n\ge l-1 n ≥ l − 1 a ( l − 1 ) = 1 a(l-1)=1 a ( l − 1 ) = 1 M ( n − 1 ) ≤ M ( n ) M(n-1)\le M(n) M ( n − 1 ) ≤ M ( n )
a ( n ) ≤ a ( n − 1 ) + 3 1 − τ n − l + 4 a(n)\le\frac{a(n-1)+3}{1-\tau^{n-l+4}} a ( n ) ≤ 1 − τ n − l + 4 a ( n − 1 ) + 3 for any n ≥ l n\ge l n ≥ l
a ( n ) ≤ ( 1 + 3 ( n − l + 1 ) ) ∏ i = 0 n − l 1 1 − τ i + 4 ≤ 4 P τ ( n − l + 2 ) , n ≥ l − 1. a(n)\le(1+3(n-l+1))\prod_{i=0}^{n-l}\frac{1}{1-\tau^{i+4}}\le4P_\tau(n-l+2),\qquad n\ge l-1. a ( n ) ≤ ( 1 + 3 ( n − l + 1 )) i = 0 ∏ n − l 1 − τ i + 4 1 ≤ 4 P τ ( n − l + 2 ) , n ≥ l − 1. Hence ∣ δ ( n ) ∣ ≤ 4 P τ ( n − l + 2 ) τ n − l + 1 M ( n ) |\delta(n)|\le4P_\tau(n-l+2)\tau^{n-l+1}M(n) ∣ δ ( n ) ∣ ≤ 4 P τ ( n − l + 2 ) τ n − l + 1 M ( n ) M ( n + 1 ) ≤ M ( n ) + ∣ δ ( n ) ∣ ≤ M ( n ) ( 1 + 4 P τ ( n − l + 2 ) τ n − l + 1 ) M(n+1)\le M(n)+|\delta(n)|\le M(n)(1+4P_\tau(n-l+2)\tau^{n-l+1}) M ( n + 1 ) ≤ M ( n ) + ∣ δ ( n ) ∣ ≤ M ( n ) ( 1 + 4 P τ ( n − l + 2 ) τ n − l + 1 ) n ≥ l − 1 n\ge l-1 n ≥ l − 1 M ( n ) ≤ M τ M(n)\le M_\tau M ( n ) ≤ M τ n ≥ l − 1 n\ge l-1 n ≥ l − 1 (58) . Finally, for any integer N ≥ 1 N\ge1 N ≥ 1
∑ p ≥ 0 ( N + p ) τ p = N 1 − τ + τ ( 1 − τ ) 2 ≤ N ( 1 − τ ) 2 . \sum_{p\ge0}(N+p)\tau^p=\frac{N}{1-\tau}+\frac{\tau}{(1-\tau)^2}\le\frac{N}{(1-\tau)^2}. p ≥ 0 ∑ ( N + p ) τ p = 1 − τ N + ( 1 − τ ) 2 τ ≤ ( 1 − τ ) 2 N . Thus the series ∑ i ≥ n ∣ δ ( i ) ∣ \sum_{i\ge n}|\delta(i)| ∑ i ≥ n ∣ δ ( i ) ∣ 4 P τ M τ ( 1 − τ ) 2 ( n − l + 2 ) τ n − l + 1 \frac{4P_\tau M_\tau}{(1-\tau)^2}(n-l+2)\tau^{n-l+1} ( 1 − τ ) 2 4 P τ M τ ( n − l + 2 ) τ n − l + 1
Proposition 6.9 Let l ≥ 1 l\ge1 l ≥ 1 h l h_l h l D \mathbb{D} D h l ( n , k ) = ς l ( n ) M l , n ( τ k ) h_l(n,k)=\varsigma_l(n)M_{l,n}(\tau^k) h l ( n , k ) = ς l ( n ) M l , n ( τ k ) n ≥ l n\ge l n ≥ l h l ( n , k ) = 0 h_l(n,k)=0 h l ( n , k ) = 0 n < l n<l n < l
The function h l h_l h l
The sequence ( ς l ( n ) ) (\varsigma_l(n)) ( ς l ( n )) ς l ( ∞ ) ≠ 0 \varsigma_l(\infty)\ne0 ς l ( ∞ ) = 0 C C C ∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ C ( n + 1 ) τ n |\varsigma_l(n)-\varsigma_l(\infty)|\le C(n+1)\tau^n ∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ C ( n + 1 ) τ n
We have H φ l τ = ς l ς l ( ∞ ) h l \mathcal{H}\varphi_l^{\tau} = \frac{\varsigma_l}{\varsigma_l(\infty)}h_l H φ l τ = ς l ( ∞ ) ς l h l H φ l τ \mathcal{H}\varphi_l^{\tau} H φ l τ n < l n < l n < l C l C_l C l ( n , k ) ∈ D (n,k) \in\mathbb{D} ( n , k ) ∈ D
∣ H φ l τ ( n , k ) − φ l ( τ k ) ∣ ≤ C l ( n + 1 ) τ n . (60) \left|\mathcal{H}\varphi_l^{\tau}(n,k)-\varphi_l(\tau^k)\right| \le C_l(n+1)\tau^n. \tag*{(60)} H φ l τ ( n , k ) − φ l ( τ k ) ≤ C l ( n + 1 ) τ n . (60) Proof. 1. For any n ∈ N 0 n \in\mathbb{N}_0 n ∈ N 0 W l ( n ) \mathcal{W}_l(n) W l ( n ) V l − 1 \mathcal{V}_{l-1} V l − 1 n n n V l \mathcal{V}_l V l n n n ∑ k w ^ ( n , k ) f ( k ) g ( k ) \sum_k \widehat{w}(n,k)f(k)g(k) ∑ k w ( n , k ) f ( k ) g ( k ) w ^ ( n , k ) \widehat{w}(n,k) w ( n , k ) τ − k \tau^{-k} τ − k W l ( n ) = C M l , n ( τ − ⋅ ) \mathcal{W}_l(n)=\mathbb{C}M_{l,n}(\tau^{-\,\cdot}) W l ( n ) = C M l , n ( τ − ⋅ ) n ≥ l n\ge l n ≥ l W l ( n ) = { 0 } \mathcal{W}_l(n)=\{0\} W l ( n ) = { 0 } n < l n<l n < l U l \mathcal{U}_l U l f f f D \mathbb{D} D f ( n , ⋅ ) ∈ W l ( n ) f(n,\cdot)\in\mathcal{W}_l(n) f ( n , ⋅ ) ∈ W l ( n ) n n n P ( U l ) ⊂ U l P(\mathcal{U}_l)\subset\mathcal{U}_l P ( U l ) ⊂ U l f ∈ U l f\in\mathcal{U}_l f ∈ U l P f ∈ V l Pf\in\mathcal{V}_l P f ∈ V l Lemma 6.5 . Let g ∈ V l − 1 g\in\mathcal{V}_{l-1} g ∈ V l − 1 P P P w ^ \widehat{w} w Proposition 6.2 , and since P g ∈ V l − 1 Pg\in\mathcal{V}_{l-1} P g ∈ V l − 1
∑ v ∈ D w ^ ( v ) ( P f ) ( v ) g ( v ) = ∑ v ∈ D w ^ ( v ) f ( v ) ( P g ) ( v ) = 0 , \sum_{v\in\mathbb{D}}\widehat{w}(v)(Pf)(v)g(v)=\sum_{v\in\mathbb{D}}\widehat{w}(v)f(v)(Pg)(v)=0, v ∈ D ∑ w ( v ) ( P f ) ( v ) g ( v ) = v ∈ D ∑ w ( v ) f ( v ) ( P g ) ( v ) = 0 , because f ( n ′ , ⋅ ) f(n',\cdot) f ( n ′ , ⋅ ) V l − 1 \mathcal{V}_{l-1} V l − 1 n ′ n' n ′ n ′ n' n ′
Now, let f ∈ U l f\in\mathcal{U}_l f ∈ U l f ( n , ⋅ ) = c ( n ) M l , n ( τ − ⋅ ) f(n,\cdot)=c(n)M_{l,n}(\tau^{-\,\cdot}) f ( n , ⋅ ) = c ( n ) M l , n ( τ − ⋅ ) n ≥ l n\ge l n ≥ l c ( l − 1 ) = 0 c(l-1)=0 c ( l − 1 ) = 0 n ≥ l n\ge l n ≥ l ( P f ) ( n , ⋅ ) (Pf)(n,\cdot) ( P f ) ( n , ⋅ ) W l ( n ) \mathcal{W}_l(n) W l ( n ) c ′ ( n ) M l , n ( τ − ⋅ ) c'(n)M_{l,n}(\tau^{-\,\cdot}) c ′ ( n ) M l , n ( τ − ⋅ ) c ′ ( n ) c'(n) c ′ ( n ) l l l ( P f ) ( n , ⋅ ) (Pf)(n,\cdot) ( P f ) ( n , ⋅ ) n + 1 > l n+1>l n + 1 > l Lemma 6.5 , the terms of degree at most l − 1 l-1 l − 1 f ( n ± 1 , ⋅ ) f(n\pm1,\cdot) f ( n ± 1 , ⋅ ) l − 1 l-1 l − 1 c ′ ( n ) = ( A l c ) ( n ) c'(n)=(A_lc)(n) c ′ ( n ) = ( A l c ) ( n ) n < l n<l n < l ( P f ) ( n , ⋅ ) ∈ W l ( n ) = { 0 } (Pf)(n,\cdot)\in\mathcal{W}_l(n)=\{0\} ( P f ) ( n , ⋅ ) ∈ W l ( n ) = { 0 } f f f ( A l c ) ( n ) = c ( n ) (A_lc)(n)=c(n) ( A l c ) ( n ) = c ( n ) n ≥ l n\ge l n ≥ l h l ∈ U l h_l\in\mathcal{U}_l h l ∈ U l A l ς l = ς l A_l\varsigma_l=\varsigma_l A l ς l = ς l h l h_l h l
By Lemma 6.8 , the sequence ( ς l ( n ) ) (\varsigma_l(n)) ( ς l ( n )) ∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ C ( n + 1 ) τ n |\varsigma_l(n)-\varsigma_l(\infty)|\le C(n+1)\tau^n ∣ ς l ( n ) − ς l ( ∞ ) ∣ ≤ C ( n + 1 ) τ n n ≥ l n\ge l n ≥ l C = 4 P τ M τ ( 1 − τ ) 2 τ − l − 1 C=\frac{4P_{\tau}M_{\tau}}{(1-\tau)^2}\tau^{-l-1} C = ( 1 − τ ) 2 4 P τ M τ τ − l − 1 n − l + 2 ≤ n + 1 n-l+2\le n+1 n − l + 2 ≤ n + 1
The moments of the measures m n \mathfrak{m}_n m n ∣ ∫ x i d m n − ∫ x i d m ∞ ∣ ≤ τ n + 1 1 − τ \left|\int x^i\,d\mathfrak{m}_n-\int x^i\,d\mathfrak{m}_{\infty}\right|\le\frac{\tau^{n+1}}{1-\tau} ∫ x i d m n − ∫ x i d m ∞ ≤ 1 − τ τ n + 1 i ≥ 0 i\ge0 i ≥ 0 M l , n M_{l,n} M l , n m n \mathfrak{m}_n m n 2 l − 2 2l-2 2 l − 2 m ∞ \mathfrak{m}_{\infty} m ∞ m ∞ \mathfrak{m}_{\infty} m ∞ M l , n M_{l,n} M l , n p ^ l \widehat{p}_l p l O ( τ n ) O(\tau^n) O ( τ n ) sup n ≥ l sup x ∈ [ 0 , 1 ] ∣ M l , n ( x ) ∣ < ∞ \sup_{n\ge l}\sup_{x\in[0,1]}|M_{l,n}(x)|<\infty sup n ≥ l sup x ∈ [ 0 , 1 ] ∣ M l , n ( x ) ∣ < ∞ sup x ∈ [ 0 , 1 ] ∣ M l , n ( x ) − p ^ l ( x ) ∣ = O ( τ n ) \sup_{x\in[0,1]}|M_{l,n}(x)-\widehat{p}_l(x)|=O(\tau^n) sup x ∈ [ 0 , 1 ] ∣ M l , n ( x ) − p l ( x ) ∣ = O ( τ n )
Consequently, the function h l h_l h l k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 lim n → ∞ h l ( n , k ) = ς l ( ∞ ) p ^ l ( τ k ) \lim_{n\to\infty}h_l(n,k)=\varsigma_l(\infty)\widehat{p}_l(\tau^k) lim n → ∞ h l ( n , k ) = ς l ( ∞ ) p l ( τ k ) ς l ( ∞ ) = 0 \varsigma_l(\infty)=0 ς l ( ∞ ) = 0 Lemma 6.7 gives h l = H 0 = 0 h_l=\mathcal{H}0=0 h l = H 0 = 0 h l ( l , ⋅ ) = M l , l ( τ − ⋅ ) h_l(l,\cdot)=M_{l,l}(\tau^{-\,\cdot}) h l ( l , ⋅ ) = M l , l ( τ − ⋅ ) l l l l + 1 l+1 l + 1 τ − k \tau^{-k} τ − k 0 ≤ k ≤ l 0\le k\le l 0 ≤ k ≤ l ς l ( ∞ ) ≠ 0 \varsigma_l(\infty)\ne0 ς l ( ∞ ) = 0
Put u : = ς l ς l ( ∞ ) h l u\mathrel{\mathop:}=\frac{\varsigma_l}{\varsigma_l(\infty)}h_l u : = ς l ( ∞ ) ς l h l u u u lim n → ∞ u ( n , k ) = p ^ l ( τ k ) = φ l ( τ k ) \lim_{n\to\infty}u(n,k)=\widehat{p}_l(\tau^k)=\varphi_l(\tau^k) lim n → ∞ u ( n , k ) = p l ( τ k ) = φ l ( τ k ) k k k Lemma 6.7 , we have u = H φ l τ u=\mathcal{H}\varphi_l^{\tau} u = H φ l τ n ≥ l n\ge l n ≥ l x = τ k x=\tau^k x = τ k
H φ l τ ( n , k ) − φ l ( τ k ) = H l [ ( ς l ( n ) ς l ( ∞ ) − 1 ) M l , n ( x ) + M l , n ( x ) − p ^ l ( x ) ] , \mathcal{H}\varphi_l^{\tau}(n,k)-\varphi_l(\tau^k)=\mathcal{H}_l\left[\left(\frac{\varsigma_l(n)}{\varsigma_l(\infty)}-1\right)M_{l,n}(x)+M_{l,n}(x)-\widehat{p}_l(x)\right], H φ l τ ( n , k ) − φ l ( τ k ) = H l [ ( ς l ( ∞ ) ς l ( n ) − 1 ) M l , n ( x ) + M l , n ( x ) − p l ( x ) ] , and the second part gives (60) . The finitely many levels n < l n<l n < l C l C_l C l
Theorem 6.10
Theorem 6.10 The harmonic measure of the diagonal chain seen from the trivial representation is the Aref’eva–Volovich law: for any k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0
P ( 0 , 0 ) ( k ∞ = k ) = ( 1 − τ ) τ k . \mathbf{P}_{(0,0)}(k_{\infty}=k)=(1-\tau)\tau^k. P ( 0 , 0 ) ( k ∞ = k ) = ( 1 − τ ) τ k . Proof. Let μ \mu μ τ k ∞ \tau^{k_\infty} τ k ∞ P ( 0 , 0 ) \mathbf{P}_{(0,0)} P ( 0 , 0 ) l ≥ 1 l \ge1 l ≥ 1 Proposition 6.9 gives
∫ φ l τ d μ = E ( 0 , 0 ) [ φ l τ ( τ k ∞ ) ] = H φ l τ ( 0 , 0 ) = 0 , \int\varphi_l^\tau\,\mathrm{d}\mu= \mathbf{E}_{(0,0)}[\varphi_l^\tau(\tau^{k_\infty})] = \mathcal{H}\varphi_l^\tau(0,0) = 0, ∫ φ l τ d μ = E ( 0 , 0 ) [ φ l τ ( τ k ∞ )] = H φ l τ ( 0 , 0 ) = 0 , since H φ l τ \mathcal{H}\varphi_l^\tau H φ l τ 0 < l 0 < l 0 < l ( 1 − τ ) m ∞ (1-\tau)m_\infty ( 1 − τ ) m ∞ φ l τ \varphi_l^\tau φ l τ φ 0 τ = 1 \varphi_0^\tau= 1 φ 0 τ = 1 ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) ( φ l τ ) l ≥ 0 (\varphi_l^\tau)_{l\ge0} ( φ l τ ) l ≥ 0 μ \mu μ ( 1 − τ ) m ∞ (1-\tau)m_\infty ( 1 − τ ) m ∞ [ 0 , 1 ] [0,1] [ 0 , 1 ]
The following proposition gives a second proof of Theorem 6.10, which does not use the spin harmonic functions, and a stronger statement. It is the analogue, for the state φ \varphi φ [15 Section 4] .
Proposition 6.11 For ( n , k ) ∈ D (n,k) \in\mathbb{D} ( n , k ) ∈ D Λ ( n , k ) ≔ ( 1 − τ ) τ k 1 − τ n + 1 \Lambda(n,k) \coloneqq\frac{(1-\tau)\tau^k}{1-\tau^{n+1}} Λ ( n , k ) : = 1 − τ n + 1 ( 1 − τ ) τ k Λ ( n , ⋅ ) \Lambda(n,\cdot) Λ ( n , ⋅ ) { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } Q Q Q N 0 \mathbb{N}_0 N 0
Q ( n , n + 1 ) = 1 − τ n + 2 ( 1 + τ ) ( 1 − τ n + 1 ) , Q ( n , n − 1 ) = τ ( 1 − τ n ) ( 1 + τ ) ( 1 − τ n + 1 ) . Q(n,n+1)=\frac{1-\tau^{n+2}}{(1+\tau)(1-\tau^{n+1})}, \qquad Q(n,n-1)=\frac{\tau(1-\tau^n)}{(1+\tau)(1-\tau^{n+1})}. Q ( n , n + 1 ) = ( 1 + τ ) ( 1 − τ n + 1 ) 1 − τ n + 2 , Q ( n , n − 1 ) = ( 1 + τ ) ( 1 − τ n + 1 ) τ ( 1 − τ n ) . Under any P v \mathbf{P}_v P v ( n t − 2 k t ) t ≥ 0 (n_t-2k_t)_{t\ge0} ( n t − 2 k t ) t ≥ 0 Z \mathbb{Z} Z + 1 +1 + 1 − 1 -1 − 1 1 1 + τ \frac{1}{1+\tau} 1 + τ 1 τ 1 + τ \frac{\tau}{1+\tau} 1 + τ τ Q Q Q − 1 -1 − 1 h ( n ) = 1 − τ n + 1 h(n)=1-\tau^{n+1} h ( n ) = 1 − τ n + 1 − 1 -1 − 1
For any n , m ∈ N 0 n,m\in\mathbb{N}_0 n , m ∈ N 0 ∣ n − m ∣ = 1 |n-m|=1 ∣ n − m ∣ = 1 0 ≤ k ′ ≤ m 0\le k'\le m 0 ≤ k ′ ≤ m
∑ k = 0 n Λ ( n , k ) P ( ( n , k ) , ( m , k ′ ) ) = Q ( n , m ) Λ ( m , k ′ ) . \sum_{k=0}^{n}\Lambda(n,k)P((n,k),(m,k'))=Q(n,m)\Lambda(m,k'). k = 0 ∑ n Λ ( n , k ) P (( n , k ) , ( m , k ′ )) = Q ( n , m ) Λ ( m , k ′ ) . For any integer N ≥ 1 N\ge1 N ≥ 1 m 1 , … , m N ∈ N 0 m_1,\ldots,m_N\in\mathbb{N}_0 m 1 , … , m N ∈ N 0 0 ≤ j ≤ m N 0\le j\le m_N 0 ≤ j ≤ m N m 0 = 0 m_0=0 m 0 = 0
P ( 0 , 0 ) ( n 1 = m 1 , … , n N = m N , k N = j ) = Q ( m 0 , m 1 ) ⋯ Q ( m N − 1 , m N ) Λ ( m N , j ) . \mathbf{P}_{(0,0)}(n_1=m_1,\ldots,n_N=m_N,k_N=j)=Q(m_0,m_1)\cdots Q(m_{N-1},m_N)\Lambda(m_N,j). P ( 0 , 0 ) ( n 1 = m 1 , … , n N = m N , k N = j ) = Q ( m 0 , m 1 ) ⋯ Q ( m N − 1 , m N ) Λ ( m N , j ) . In particular, under P ( 0 , 0 ) \mathbf{P}_{(0,0)} P ( 0 , 0 ) ( n t ) t ≥ 0 (n_t)_{t\ge0} ( n t ) t ≥ 0 Q Q Q k N k_N k N ( n 0 , … , n N ) (n_0,\ldots,n_N) ( n 0 , … , n N ) Λ ( n N , ⋅ ) \Lambda(n_N,\cdot) Λ ( n N , ⋅ )
Under P ( 0 , 0 ) \mathbf{P}_{(0,0)} P ( 0 , 0 ) k ∞ k_\infty k ∞ ( n t ) t ≥ 0 (n_t)_{t\ge0} ( n t ) t ≥ 0 ( 1 − τ ) τ k (1-\tau)\tau^k ( 1 − τ ) τ k k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0
Proof. Put D n ≔ ( 1 + τ ) ( 1 − τ n + 1 ) D_n \coloneqq(1+\tau)(1-\tau^{n+1}) D n : = ( 1 + τ ) ( 1 − τ n + 1 )
The quantity n − 2 k n-2k n − 2 k 1 1 1 ( n + 1 , k ) (n+1,k) ( n + 1 , k ) ( n − 1 , k − 1 ) (n-1,k-1) ( n − 1 , k − 1 ) Proposition 6.2 , the sum of their probabilities is D n − 1 ( ( 1 − τ n − k + 1 ) + τ n − k + 1 ( 1 − τ k ) ) = 1 − τ n + 1 D n = 1 1 + τ D_n^{-1}\bigl((1-\tau^{n-k+1})+\tau^{n-k+1}(1-\tau^k)\bigr)=\frac{1-\tau^{n+1}}{D_n}=\frac{1}{1+\tau} D n − 1 ( ( 1 − τ n − k + 1 ) + τ n − k + 1 ( 1 − τ k ) ) = D n 1 − τ n + 1 = 1 + τ 1 ( n , k ) (n,k) ( n , k ) ( n t − 2 k t ) (n_t-2k_t) ( n t − 2 k t ) n ≥ 0 n\ge0 n ≥ 0 1 1 + τ h ( n + 1 ) + τ 1 + τ h ( n − 1 ) = h ( n ) \frac{1}{1+\tau}h(n+1)+\frac{\tau}{1+\tau}h(n-1)=h(n) 1 + τ 1 h ( n + 1 ) + 1 + τ τ h ( n − 1 ) = h ( n ) h ( − 1 ) = 0 h(-1)=0 h ( − 1 ) = 0 h ( n + 1 ) ( 1 + τ ) h ( n ) = Q ( n , n + 1 ) \frac{h(n+1)}{(1+\tau)h(n)}=Q(n,n+1) ( 1 + τ ) h ( n ) h ( n + 1 ) = Q ( n , n + 1 ) τ h ( n − 1 ) ( 1 + τ ) h ( n ) = Q ( n , n − 1 ) \frac{\tau h(n-1)}{(1+\tau)h(n)}=Q(n,n-1) ( 1 + τ ) h ( n ) τ h ( n − 1 ) = Q ( n , n − 1 ) h ( n ) h(n) h ( n ) n n n − 1 -1 − 1 − 1 -1 − 1 τ n + 1 \tau^{n+1} τ n + 1
Suppose that m = n + 1 m=n+1 m = n + 1 ( n + 1 , k ′ ) (n+1,k') ( n + 1 , k ′ ) ( n , k ′ ) (n,k') ( n , k ′ ) k ′ ≤ n k'\le n k ′ ≤ n ( n , k ′ − 1 ) (n,k'-1) ( n , k ′ − 1 ) k ′ ≥ 1 k'\ge1 k ′ ≥ 1 Proposition 6.2 , the left-hand side is equal to
1 − τ ( 1 − τ n + 1 ) D n [ τ k ′ ( 1 − τ n − k ′ + 1 ) + τ k ′ − 1 τ n − k ′ + 2 ( 1 − τ k ′ ) ] = ( 1 − τ ) τ k ′ D n . \frac{1-\tau}{(1-\tau^{n+1})D_n}\left[\tau^{k'}(1-\tau^{n-k'+1})+\tau^{k'-1}\tau^{n-k'+2}(1-\tau^{k'})\right]=\frac{(1-\tau)\tau^{k'}}{D_n}. ( 1 − τ n + 1 ) D n 1 − τ [ τ k ′ ( 1 − τ n − k ′ + 1 ) + τ k ′ − 1 τ n − k ′ + 2 ( 1 − τ k ′ ) ] = D n ( 1 − τ ) τ k ′ . where the first term vanishes for k ′ = n + 1 k'=n+1 k ′ = n + 1 k ′ = 0 k'=0 k ′ = 0 Q ( n , n + 1 ) Λ ( n + 1 , k ′ ) Q(n,n+1)\Lambda(n+1,k') Q ( n , n + 1 ) Λ ( n + 1 , k ′ ) m = n − 1 m=n-1 m = n − 1 ( n − 1 , k ′ ) (n-1,k') ( n − 1 , k ′ ) ( n , k ′ ) (n,k') ( n , k ′ ) ( n , k ′ + 1 ) (n,k'+1) ( n , k ′ + 1 )
1 − τ ( 1 − τ n + 1 ) D n [ τ k ′ τ ( 1 − τ n − k ′ ) + τ k ′ + 1 τ n − k ′ ( 1 − τ k ′ + 1 ) ] = ( 1 − τ ) τ k ′ + 1 D n , \frac{1-\tau}{(1-\tau^{n+1})D_n}\left[\tau^{k'}\tau(1-\tau^{n-k'})+\tau^{k'+1}\tau^{n-k'}(1-\tau^{k'+1})\right]=\frac{(1-\tau)\tau^{k'+1}}{D_n}, ( 1 − τ n + 1 ) D n 1 − τ [ τ k ′ τ ( 1 − τ n − k ′ ) + τ k ′ + 1 τ n − k ′ ( 1 − τ k ′ + 1 ) ] = D n ( 1 − τ ) τ k ′ + 1 , which is Q ( n , n − 1 ) Λ ( n − 1 , k ′ ) Q(n,n-1)\Lambda(n-1,k') Q ( n , n − 1 ) Λ ( n − 1 , k ′ )
We argue by induction on N N N N = 1 N=1 N = 1 n = 0 n=0 n = 0 Λ ( 0 , ⋅ ) = δ 0 \Lambda(0,\cdot)=\delta_0 Λ ( 0 , ⋅ ) = δ 0 N N N ± 1 \pm1 ± 1 { n 1 = m 1 , … , n N + 1 = m N + 1 , k N + 1 = j } \{n_1=m_1,\ldots,n_{N+1}=m_{N+1},k_{N+1}=j\} { n 1 = m 1 , … , n N + 1 = m N + 1 , k N + 1 = j }
∑ k = 0 m N Q ( m 0 , m 1 ) ⋯ Q ( m N − 1 , m N ) Λ ( m N , k ) P ( ( m N , k ) , ( m N + 1 , j ) ) , \sum_{k=0}^{m_N}Q(m_0,m_1)\cdots Q(m_{N-1},m_N)\Lambda(m_N,k)P((m_N,k),(m_{N+1},j)), k = 0 ∑ m N Q ( m 0 , m 1 ) ⋯ Q ( m N − 1 , m N ) Λ ( m N , k ) P (( m N , k ) , ( m N + 1 , j )) , which is Q ( m 0 , m 1 ) ⋯ Q ( m N , m N + 1 ) Λ ( m N + 1 , j ) Q(m_0,m_1)\cdots Q(m_N,m_{N+1})\Lambda(m_{N+1},j) Q ( m 0 , m 1 ) ⋯ Q ( m N , m N + 1 ) Λ ( m N + 1 , j ) j j j ( n 1 , … , n N ) (n_1,\ldots,n_N) ( n 1 , … , n N )
Let M ∈ N 0 M\in\mathbb{N}_0 M ∈ N 0 A A A σ \sigma σ n 0 , … , n M n_0,\ldots,n_M n 0 , … , n M N ≥ M N\ge M N ≥ M j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0 P ( 0 , 0 ) ( A ∩ { k N = j } ) = E ( 0 , 0 ) [ 1 A Λ ( n N , j ) ] \mathbb{P}_{(0,0)}(A\cap\{k_N=j\})=\mathbb{E}_{(0,0)}[\mathbf{1}_A\Lambda(n_N,j)] P ( 0 , 0 ) ( A ∩ { k N = j }) = E ( 0 , 0 ) [ 1 A Λ ( n N , j )] Λ ( n , j ) = 0 \Lambda(n,j)=0 Λ ( n , j ) = 0 j > n j>n j > n Lemma 6.7 , almost surely n N → ∞ n_N\to\infty n N → ∞ k N = k ∞ k_N=k_\infty k N = k ∞ N N N Λ ( n N , j ) → ( 1 − τ ) τ j \Lambda(n_N,j)\to(1-\tau)\tau^j Λ ( n N , j ) → ( 1 − τ ) τ j P ( 0 , 0 ) ( A ∩ { k ∞ = j } ) = ( 1 − τ ) τ j P ( 0 , 0 ) ( A ) \mathbb{P}_{(0,0)}(A\cap\{k_\infty=j\})=(1-\tau)\tau^j\mathbb{P}_{(0,0)}(A) P ( 0 , 0 ) ( A ∩ { k ∞ = j }) = ( 1 − τ ) τ j P ( 0 , 0 ) ( A ) A A A π \pi π σ \sigma σ A A A k ∞ k_\infty k ∞
Part 3 is an instance of the criterion of Rogers and Pitman for functions of Markov chains [63 ] , with the intertwining kernel Λ \Lambda Λ P P P K K K Z \mathbb{Z} Z P P P S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) S U ( n ) \mathrm{SU}(n) SU ( n ) [11 ] , see also [15 Section 4] and [23 ] .
In the variable n − k n-k n − k m n m_n m n τ − ( n − k ) \tau^{-(n-k)} τ − ( n − k ) τ \tau τ α = β = 1 \alpha=\beta=1 α = β = 1 N = n N=n N = n [44 Section 14.6] and [45 (3.6.1)–(3.6.2)] . Hence M l , n ( τ k ) M_{l,n}(\tau^k) M l , n ( τ k ) Q l ( τ − ( n − k ) ; 1 , 1 , n ∣ τ ) Q_l(\tau^{-(n-k)};1,1,n\mid\tau) Q l ( τ − ( n − k ) ; 1 , 1 , n ∣ τ ) l l l τ k \tau^k τ k Proposition 6.9 describes the Poisson integral of the zonal spherical function φ l ( A ) \varphi_l^{(A)} φ l ( A ) q q q χ l ( n ) / s l ( ∞ ) \chi_l(n)/s_l(\infty) χ l ( n ) / s l ( ∞ ) s l ( n ) s_l(n) s l ( n )
Proposition 6.12 Let l ≥ 1 l\ge1 l ≥ 1
For any x ∈ C x\in\mathbb{C} x ∈ C ∣ x ∣ < 1 |x|<1 ∣ x ∣ < 1
∑ n ≥ l s l ( n ) x n − l = ( 1 − τ 2 l + 1 ) ∑ i ≥ 0 ( τ − 1 x ; τ ) i ( τ − 1 x ; τ ) i ( x ; τ ) i + 1 ( x ; τ ) i + 1 τ ( 2 l + 1 ) i . (61) \sum_{n\ge l}s_l(n)x^{n-l}=(1-\tau^{2l+1})\sum_{i\ge0}\frac{(\tau^{-1}x;\tau)_i(\tau^{-1}x;\tau)_i}{(x;\tau)_{i+1}(x;\tau)_{i+1}}\tau^{(2l+1)i}. \tag*{(61)} n ≥ l ∑ s l ( n ) x n − l = ( 1 − τ 2 l + 1 ) i ≥ 0 ∑ ( x ; τ ) i + 1 ( x ; τ ) i + 1 ( τ − 1 x ; τ ) i ( τ − 1 x ; τ ) i τ ( 2 l + 1 ) i . (61) We have ς l ( ∞ ) = ( τ l + 2 ; τ ) l ( τ ; τ ) l \varsigma_l(\infty)=\frac{(\tau^{l+2};\tau)_l}{(\tau;\tau)_l} ς l ( ∞ ) = ( τ ; τ ) l ( τ l + 2 ; τ ) l [ 2 l + 1 l ] τ \begin{bmatrix}2l+1\\l\end{bmatrix}_{\tau} [ 2 l + 1 l ] τ [27 (17.2.29)] . In particular, ς l ( ∞ ) ≥ 1 \varsigma_l(\infty)\geq1 ς l ( ∞ ) ≥ 1
There exists a constant B l B_l B l
ς l ( n ) = ς l ( ∞ ) − τ Λ l ( τ ) ς l ( ∞ ) 1 − τ ( n − l + 1 ) τ n + B l τ n + O ( τ 3 n 2 ) . \varsigma_l(n)=\varsigma_l(\infty)-\frac{\tau\Lambda_l(\tau)\varsigma_l(\infty)}{1-\tau}(n-l+1)\tau^n+B_l\tau^n+O\left(\tau^{\frac{3n}{2}}\right). ς l ( n ) = ς l ( ∞ ) − 1 − τ τ Λ l ( τ ) ς l ( ∞ ) ( n − l + 1 ) τ n + B l τ n + O ( τ 2 3 n ) . In particular, the factor n + 1 n+1 n + 1
Proof: 1. By Lemma 6.8, the power series G ( x ) ≔ ∑ m ≥ 0 ς l ( l + m ) x m G(x)\coloneqq\sum_{m\geq0}\varsigma_l(l+m)x^m G ( x ) : = ∑ m ≥ 0 ς l ( l + m ) x m ∣ x ∣ < 1 |x|<1 ∣ x ∣ < 1 (57) with n = l + m n=l+m n = l + m x m + 1 x^{m+1} x m + 1 m ≥ 0 m\geq0 m ≥ 0 ς l ( l − 1 ) = 0 \varsigma_l(l-1)=0 ς l ( l − 1 ) = 0 ς l ( l ) = 1 \varsigma_l(l)=1 ς l ( l ) = 1
∑ m ≥ 0 ( 1 − τ 2 l + 2 + m ) ς l ( l + m + 1 ) x m + 1 = G ( x ) − 1 − τ 2 l + 1 ( G ( τ x ) − 1 ) , \sum_{m\geq0}(1-\tau^{2l+2+m})\varsigma_l(l+m+1)x^{m+1}=G(x)-1-\tau^{2l+1}(G(\tau x)-1), m ≥ 0 ∑ ( 1 − τ 2 l + 2 + m ) ς l ( l + m + 1 ) x m + 1 = G ( x ) − 1 − τ 2 l + 1 ( G ( τ x ) − 1 ) , ∑ m ≥ 0 ( 1 + τ ) ( 1 − τ l + 1 + m ) ς l ( l + m ) x m + 1 = ( 1 + τ ) x ( G ( x ) − τ l + 1 G ( τ x ) ) , \sum_{m\geq0}(1+\tau)(1-\tau^{l+1+m})\varsigma_l(l+m)x^{m+1}=(1+\tau)x\bigl(G(x)-\tau^{l+1}G(\tau x)\bigr), m ≥ 0 ∑ ( 1 + τ ) ( 1 − τ l + 1 + m ) ς l ( l + m ) x m + 1 = ( 1 + τ ) x ( G ( x ) − τ l + 1 G ( τ x ) ) , ∑ m ≥ 0 τ ( 1 − τ m ) ς l ( l + m − 1 ) x m + 1 = τ x 2 ( G ( x ) − τ G ( τ x ) ) . \sum_{m\geq0}\tau(1-\tau^m)\varsigma_l(l+m-1)x^{m+1}=\tau x^2\bigl(G(x)-\tau G(\tau x)\bigr). m ≥ 0 ∑ τ ( 1 − τ m ) ς l ( l + m − 1 ) x m + 1 = τ x 2 ( G ( x ) − τ G ( τ x ) ) . We deduce that, for any ∣ x ∣ < 1 |x|<1 ∣ x ∣ < 1
( 1 − x ) ( 1 − τ x ) G ( x ) = ( τ l − τ x ) ( τ l + 1 − τ x ) G ( τ x ) + 1 − τ 2 l + 1 . (1-x)(1-\tau x)G(x)=(\tau^l-\tau x)(\tau^{l+1}-\tau x)G(\tau x)+1-\tau^{2l+1}. ( 1 − x ) ( 1 − τ x ) G ( x ) = ( τ l − τ x ) ( τ l + 1 − τ x ) G ( τ x ) + 1 − τ 2 l + 1 . Put R ( x ) ≔ τ 2 l + 1 ( 1 − τ − l x ) ( 1 − τ 1 − l x ) ( 1 − x ) ( 1 − τ x ) R(x)\coloneqq\frac{\tau^{2l+1}(1-\tau^{-l}x)(1-\tau^{1-l}x)}{(1-x)(1-\tau x)} R ( x ) : = ( 1 − x ) ( 1 − τ x ) τ 2 l + 1 ( 1 − τ − l x ) ( 1 − τ 1 − l x ) T i ( x ) T_i(x) T i ( x ) i i i (61) . Since ∏ r = 0 i − 1 R ( τ r x ) = ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i ( x ; τ ) i ( τ x ; τ ) i τ ( 2 l + 1 ) i \prod_{r=0}^{i-1}R(\tau^r x)=\frac{(\tau^{-l}x;\tau)_i(\tau^{1-l}x;\tau)_i}{(x;\tau)_i(\tau x;\tau)_i}\tau^{(2l+1)i} ∏ r = 0 i − 1 R ( τ r x ) = ( x ; τ ) i ( τ x ; τ ) i ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i τ ( 2 l + 1 ) i N N N
G ( x ) = ( 1 − τ 2 l + 1 ) ∑ i = 0 N − 1 T i ( x ) + ∏ r = 0 N − 1 R ( τ r x ) ⋅ G ( τ N x ) . G(x)=(1-\tau^{2l+1})\sum_{i=0}^{N-1}T_i(x)+\prod_{r=0}^{N-1}R(\tau^r x)\cdot G(\tau^N x). G ( x ) = ( 1 − τ 2 l + 1 ) i = 0 ∑ N − 1 T i ( x ) + r = 0 ∏ N − 1 R ( τ r x ) ⋅ G ( τ N x ) . Since R ( τ r x ) → τ 2 l + 1 R(\tau^r x)\to\tau^{2l+1} R ( τ r x ) → τ 2 l + 1 r → ∞ r\to\infty r → ∞ G ( τ N x ) → G ( 0 ) = 1 G(\tau^N x)\to G(0)=1 G ( τ N x ) → G ( 0 ) = 1 N → ∞ N\to\infty N → ∞ (61) .
Let ρ > 0 \rho>0 ρ > 0 ∣ x ∣ ≤ ρ |x|\leq\rho ∣ x ∣ ≤ ρ i ≥ 0 i\geq0 i ≥ 0 ∣ ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i ∣ ≤ K ρ ≔ ( − τ − l ρ ; τ ) ∞ 2 |(\tau^{-l}x;\tau)_i(\tau^{1-l}x;\tau)_i|\leq K_\rho\coloneqq(-\tau^{-l}\rho;\tau)_\infty^2 ∣ ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i ∣ ≤ K ρ : = ( − τ − l ρ ; τ ) ∞ 2 0 ≤ x < 1 0\leq x<1 0 ≤ x < 1
∣ ( 1 − x ) T i ( x ) ∣ = ∣ ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i ∣ ( τ x ; τ ) i ( τ x ; τ ) i + 1 τ ( 2 l + 1 ) i ≤ K 1 ( τ ; τ ) ∞ 2 τ ( 2 l + 1 ) i . |(1-x)T_i(x)|=\frac{|(\tau^{-l}x;\tau)_i(\tau^{1-l}x;\tau)_i|}{(\tau x;\tau)_i(\tau x;\tau)_{i+1}}\tau^{(2l+1)i}\leq\frac{K_1}{(\tau;\tau)_\infty^2}\tau^{(2l+1)i}. ∣ ( 1 − x ) T i ( x ) ∣ = ( τ x ; τ ) i ( τ x ; τ ) i + 1 ∣ ( τ − l x ; τ ) i ( τ 1 − l x ; τ ) i ∣ τ ( 2 l + 1 ) i ≤ ( τ ; τ ) ∞ 2 K 1 τ ( 2 l + 1 ) i . By Lemma 6.8 and Abel’s theorem, ( 1 − x ) G ( x ) → ς l ( ∞ ) (1-x)G(x)\to\varsigma_l(\infty) ( 1 − x ) G ( x ) → ς l ( ∞ ) x → 1 − x\to1^- x → 1 −
ς l ( ∞ ) = ( 1 − τ 2 l + 1 ) ∑ i ≥ 0 ( τ − l ; τ ) i ( τ 1 − l ; τ ) i ( τ ; τ ) i ( τ ; τ ) i + 1 τ ( 2 l + 1 ) i = 1 − τ 2 l + 1 1 − τ 2 ϕ 1 ( τ 1 − l , τ − l τ 2 ; τ , τ 2 l + 1 ) . \varsigma_l(\infty)=(1-\tau^{2l+1})\sum_{i\geq0}\frac{(\tau^{-l};\tau)_i(\tau^{1-l};\tau)_i}{(\tau;\tau)_i(\tau;\tau)_{i+1}}\tau^{(2l+1)i}
=\frac{1-\tau^{2l+1}}{1-\tau}\,{}_2\phi_1\left(\begin{matrix}\tau^{1-l},\tau^{-l}\\ \tau^2\end{matrix};\tau,\tau^{2l+1}\right). ς l ( ∞ ) = ( 1 − τ 2 l + 1 ) i ≥ 0 ∑ ( τ ; τ ) i ( τ ; τ ) i + 1 ( τ − l ; τ ) i ( τ 1 − l ; τ ) i τ ( 2 l + 1 ) i = 1 − τ 1 − τ 2 l + 1 2 ϕ 1 ( τ 1 − l , τ − l τ 2 ; τ , τ 2 l + 1 ) . where we used ( τ ; τ ) i + 1 = ( 1 − τ ) ( τ 2 ; τ ) i (\tau;\tau)_{i+1}=(1-\tau)(\tau^2;\tau)_i ( τ ; τ ) i + 1 = ( 1 − τ ) ( τ 2 ; τ ) i i ≥ l i\geq l i ≥ l ( τ 1 − l ; τ ) i = 0 (\tau^{1-l};\tau)_i=0 ( τ 1 − l ; τ ) i = 0 q q q [32 (II.7)] , applied with q − n = τ 1 − l q^{-n}=\tau^{1-l} q − n = τ 1 − l a = τ − l a=\tau^{-l} a = τ − l c = τ 2 c=\tau^2 c = τ 2 c q n a = τ 2 l + 1 \frac{cq^n}{a}=\tau^{2l+1} a c q n = τ 2 l + 1 2 ϕ 1 {}_2\phi_1 2 ϕ 1 ( τ l + 2 ; τ ) l − 1 ( τ 2 ; τ ) l − 1 \frac{(\tau^{l+2};\tau)_{l-1}}{(\tau^2;\tau)_{l-1}} ( τ 2 ; τ ) l − 1 ( τ l + 2 ; τ ) l − 1 ( 1 − τ 2 l + 1 ) ( τ l + 2 ; τ ) l − 1 = ( τ l + 2 ; τ ) l (1-\tau^{2l+1})(\tau^{l+2};\tau)_{l-1}=(\tau^{l+2};\tau)_l ( 1 − τ 2 l + 1 ) ( τ l + 2 ; τ ) l − 1 = ( τ l + 2 ; τ ) l ( 1 − τ ) ( τ 2 ; τ ) l − 1 = ( τ ; τ ) l (1-\tau)(\tau^2;\tau)_{l-1}=(\tau;\tau)_l ( 1 − τ ) ( τ 2 ; τ ) l − 1 = ( τ ; τ ) l ς l ( ∞ ) \varsigma_l(\infty) ς l ( ∞ ) ( τ l + 2 ; τ ) l ( τ ; τ ) l = ∏ i = 1 l 1 − τ l + 1 + i 1 − τ i ≥ 1 \frac{(\tau^{l+2};\tau)_l}{(\tau;\tau)_l}=\prod_{i=1}^{l}\frac{1-\tau^{l+1+i}}{1-\tau^i}\geq1 ( τ ; τ ) l ( τ l + 2 ; τ ) l = ∏ i = 1 l 1 − τ i 1 − τ l + 1 + i ≥ 1
Let ρ > 0 \rho>0 ρ > 0 K K K { ∣ x ∣ ≤ ρ } \{|x|\leq\rho\} { ∣ x ∣ ≤ ρ } { τ − r : r ∈ N 0 } \{\tau^{-r}:r\in\mathbb{N}_0\} { τ − r : r ∈ N 0 } ( x ; τ ) ∞ (x;\tau)_\infty ( x ; τ ) ∞ ( τ x ; τ ) ∞ (\tau x;\tau)_\infty ( τ x ; τ ) ∞ K K K ∣ ( x ; τ ) i + 1 ∣ ≥ ∣ ( x ; τ ) ∞ ∣ ( − ρ ; τ ) ∞ |(x;\tau)_{i+1}|\geq\frac{|(x;\tau)_\infty|}{(-\rho;\tau)_\infty} ∣ ( x ; τ ) i + 1 ∣ ≥ ( − ρ ; τ ) ∞ ∣ ( x ; τ ) ∞ ∣ ∣ ( τ x ; τ ) i + 1 ∣ ≥ ∣ ( τ x ; τ ) ∞ ∣ ( − ρ ; τ ) ∞ |(\tau x;\tau)_{i+1}|\geq\frac{|(\tau x;\tau)_\infty|}{(-\rho;\tau)_\infty} ∣ ( τ x ; τ ) i + 1 ∣ ≥ ( − ρ ; τ ) ∞ ∣ ( τ x ; τ ) ∞ ∣ i i i ∣ T i ∣ ≤ K ′ τ ( 2 l + 1 ) i |T_i|\leq K'\tau^{(2l+1)i} ∣ T i ∣ ≤ K ′ τ ( 2 l + 1 ) i K K K K ′ K' K ′ (61) converges uniformly on K K K G G G C \mathbb{C} C { τ − r : r ∈ N 0 } \{\tau^{-r}:r\in\mathbb{N}_0\} { τ − r : r ∈ N 0 } { ∣ x ∣ < τ − 2 } \{|x|<\tau^{-2}\} { ∣ x ∣ < τ − 2 } T i T_i T i x = 1 x=1 x = 1 1 − x 1-x 1 − x ( x ; τ ) i + 1 (x;\tau)_{i+1} ( x ; τ ) i + 1 x = τ − 1 x=\tau^{-1} x = τ − 1 1 − τ x 1-\tau x 1 − τ x ( τ x ; τ ) i + 1 (\tau x;\tau)_{i+1} ( τ x ; τ ) i + 1 i ≥ 1 i\geq1 i ≥ 1 ( x ; τ ) i + 1 (x;\tau)_{i+1} ( x ; τ ) i + 1 G G G x = 1 x=1 x = 1 − ς l ( ∞ ) -\varsigma_l(\infty) − ς l ( ∞ ) T 0 T_0 T 0 τ − 1 \tau^{-1} τ − 1 i ≥ 1 i\geq1 i ≥ 1 ( 1 − τ x ) − 2 ( x ; τ ) i + 1 ( τ x ; τ ) i + 1 (1-\tau x)^{-2}(x;\tau)_{i+1}(\tau x;\tau)_{i+1} ( 1 − τ x ) − 2 ( x ; τ ) i + 1 ( τ x ; τ ) i + 1 ( 1 − τ − 1 ) ( τ ; τ ) i − 1 ( τ ; τ ) i (1-\tau^{-1})(\tau;\tau)_{i-1}(\tau;\tau)_i ( 1 − τ − 1 ) ( τ ; τ ) i − 1 ( τ ; τ ) i x = τ − 1 x=\tau^{-1} x = τ − 1
lim x → τ − 1 ( 1 − τ x ) 2 G ( x ) = ( 1 − τ 2 l + 1 ) ∑ i = 1 l ( τ − l − 1 ; τ ) i ( τ − l ; τ ) i ( 1 − τ − 1 ) ( τ ; τ ) i − 1 ( τ ; τ ) i τ ( 2 l + 1 ) i . \lim_{x\to\tau^{-1}}(1-\tau x)^2G(x)=(1-\tau^{2l+1})\sum_{i=1}^{l}\frac{(\tau^{-l-1};\tau)_i(\tau^{-l};\tau)_i}{(1-\tau^{-1})(\tau;\tau)_{i-1}(\tau;\tau)_i}\tau^{(2l+1)i}. x → τ − 1 lim ( 1 − τ x ) 2 G ( x ) = ( 1 − τ 2 l + 1 ) i = 1 ∑ l ( 1 − τ − 1 ) ( τ ; τ ) i − 1 ( τ ; τ ) i ( τ − l − 1 ; τ ) i ( τ − l ; τ ) i τ ( 2 l + 1 ) i . the terms with i ≥ l + 1 i\geq l+1 i ≥ l + 1 ( τ − l ; τ ) i = 0 (\tau^{-l};\tau)_i=0 ( τ − l ; τ ) i = 0 i = p + 1 i=p+1 i = p + 1 ( τ − l − 1 ; τ ) p + 1 = ( 1 − τ − l − 1 ) ( τ − l ; τ ) p (\tau^{-l-1};\tau)_{p+1}=(1-\tau^{-l-1})(\tau^{-l};\tau)_p ( τ − l − 1 ; τ ) p + 1 = ( 1 − τ − l − 1 ) ( τ − l ; τ ) p ( τ − l ; τ ) p + 1 = ( 1 − τ − l ) ( τ − l ; τ ) p (\tau^{-l};\tau)_{p+1}=(1-\tau^{-l})(\tau^{-l};\tau)_p ( τ − l ; τ ) p + 1 = ( 1 − τ − l ) ( τ − l ; τ ) p ( τ ; τ ) p + 1 = ( 1 − τ ) ( τ 2 ; τ ) p (\tau;\tau)_{p+1}=(1-\tau)(\tau^2;\tau)_p ( τ ; τ ) p + 1 = ( 1 − τ ) ( τ 2 ; τ ) p
A l ≔ τ 2 l + 1 ( 1 − τ − l − 1 ) ( 1 − τ − l ) 1 − τ − 1 ς l ( ∞ ) = − τ ( 1 − τ l ) ( 1 − τ l + 1 ) 1 − τ ς l ( ∞ ) = − τ l + 1 Λ l ( τ ) 1 − τ ς l ( ∞ ) . A_l\coloneqq\frac{\tau^{2l+1}(1-\tau^{-l-1})(1-\tau^{-l})}{1-\tau^{-1}}\varsigma_l(\infty)
=-\frac{\tau(1-\tau^l)(1-\tau^{l+1})}{1-\tau}\varsigma_l(\infty)
=-\frac{\tau^{l+1}\Lambda_l(\tau)}{1-\tau}\varsigma_l(\infty). A l : = 1 − τ − 1 τ 2 l + 1 ( 1 − τ − l − 1 ) ( 1 − τ − l ) ς l ( ∞ ) = − 1 − τ τ ( 1 − τ l ) ( 1 − τ l + 1 ) ς l ( ∞ ) = − 1 − τ τ l + 1 Λ l ( τ ) ς l ( ∞ ) . Consequently, there exists a constant B l ′ B'_l B l ′ G ( x ) − ς l ( ∞ ) 1 − x − A l ( 1 − τ x ) 2 − B l ′ 1 − τ x G(x)-\frac{\varsigma_l(\infty)}{1-x}-\frac{A_l}{(1-\tau x)^2}-\frac{B'_l}{1-\tau x} G ( x ) − 1 − x ς l ( ∞ ) − ( 1 − τ x ) 2 A l − 1 − τ x B l ′ { ∣ x ∣ < τ − 2 } \{|x|<\tau^{-2}\} { ∣ x ∣ < τ − 2 } { ∣ x ∣ = τ − 3 / 2 } \{|x|=\tau^{-3/2}\} { ∣ x ∣ = τ − 3/2 } O ( τ 3 m / 2 ) O(\tau^{3m/2}) O ( τ 3 m /2 ) 1 ( 1 − τ x ) 2 = ∑ m ≥ 0 ( m + 1 ) τ m x m \frac{1}{(1-\tau x)^2}=\sum_{m\geq0}(m+1)\tau^m x^m ( 1 − τ x ) 2 1 = ∑ m ≥ 0 ( m + 1 ) τ m x m ς l ( l + m ) = ς l ( ∞ ) + A l ( m + 1 ) τ m + B l ′ τ m + O ( τ 3 m / 2 ) \varsigma_l(l+m)=\varsigma_l(\infty)+A_l(m+1)\tau^m+B'_l\tau^m+O(\tau^{3m/2}) ς l ( l + m ) = ς l ( ∞ ) + A l ( m + 1 ) τ m + B l ′ τ m + O ( τ 3 m /2 ) n = l + m n=l+m n = l + m B l = B l ′ τ − l B_l=B'_l\tau^{-l} B l = B l ′ τ − l Λ l ( τ ) ς l ( ∞ ) ≠ 0 \Lambda_l(\tau)\varsigma_l(\infty)\ne0 Λ l ( τ ) ς l ( ∞ ) = 0 τ − n ( ς l ( n ) − ς l ( ∞ ) ) \tau^{-n}(\varsigma_l(n)-\varsigma_l(\infty)) τ − n ( ς l ( n ) − ς l ( ∞ ))
We now describe the Poisson kernel of the diagonal chain, that is the harmonic measures ν v ( j ) ≔ P v ( k ∞ = j ) \nu_v(j)\coloneqq\mathbb{P}_v(k_\infty=j) ν v ( j ) : = P v ( k ∞ = j ) v ∈ D v\in\mathbb{D} v ∈ D j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0 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 ) ) , k ∈ N 0 . (Bf)(k)\coloneqq b_k(f(k)-f(k+1))+b_{k-1}(f(k)-f(k-1)),\qquad k\in\mathbb{N}_0. ( B f ) ( k ) : = b k ( f ( k ) − f ( k + 1 )) + b k − 1 ( f ( k ) − f ( k − 1 )) , k ∈ N 0 . with b − 1 ≔ 0 b_{-1}\coloneqq0 b − 1 : = 0 ⟨ B f , f ⟩ = ∑ k b k ∣ f ( k + 1 ) − f ( k ) ∣ 2 = c τ E τ ( f ) \langle Bf,f\rangle=\sum_k b_k|f(k+1)-f(k)|^2=c_\tau E_\tau(f) ⟨ B f , f ⟩ = ∑ k b k ∣ f ( k + 1 ) − f ( k ) ∣ 2 = c τ E τ ( f ) B B B c τ 2 E τ \frac{c_\tau}{2}E_\tau 2 c τ E τ ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 ℓ 2 ( w ) \ell^2(w) ℓ 2 ( w ) 0 ≤ B ≤ 4 sup k b k 0\leq B\leq4\sup_k b_k 0 ≤ B ≤ 4 sup k b k ϱ d \varrho_d ϱ d d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 ϱ 0 ≔ 1 \varrho_0\coloneqq1 ϱ 0 : = 1
ϱ d + 1 ( λ ) ≔ ϱ d ( λ ) + λ a d ∑ i = 0 d ϱ i ( λ ) , λ ∈ C . (62) \varrho_{d+1}(\lambda)\coloneqq\varrho_d(\lambda)+\frac{\lambda}{a_d}\sum_{i=0}^{d}\varrho_i(\lambda),\qquad\lambda\in\mathbb{C}. \tag*{(62)} ϱ d + 1 ( λ ) : = ϱ d ( λ ) + a d λ i = 0 ∑ d ϱ i ( λ ) , λ ∈ C . (62) Equivalently, a d ( ϱ d + 1 − ϱ d ) + a d − 1 ( ϱ d − 1 − ϱ d ) = λ ϱ d a_d(\varrho_{d+1}-\varrho_d)+a_{d-1}(\varrho_{d-1}-\varrho_d)=\lambda\varrho_d a d ( ϱ d + 1 − ϱ d ) + a d − 1 ( ϱ d − 1 − ϱ d ) = λ ϱ d d ≥ 0 d\geq0 d ≥ 0 a − 1 ≔ 0 a_{-1}\coloneqq0 a − 1 : = 0
Theorem 6.13 1. For any ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0
ν ( n , k ) ( j ) = ( 1 − τ ) τ − j ∑ l = 0 n τ l ς l ( n ) ς l ( ∞ ) ∥ φ l ∥ ℓ N 0 2 ( w ) 2 M l ( n , τ k ) φ l ( τ j ) , (63) \nu_{(n,k)}(j)=(1-\tau)\tau^{-j}\sum_{l=0}^{n}\frac{\tau_l\varsigma_l(n)}{\varsigma_l(\infty)\|\varphi_l\|_{\ell^2_{\mathbb{N}_0}(w)}^2}M_l(n,\tau^k)\varphi_l(\tau^j), \tag*{(63)} ν ( n , k ) ( j ) = ( 1 − τ ) τ − j l = 0 ∑ n ς l ( ∞ ) ∥ φ l ∥ ℓ N 0 2 ( w ) 2 τ l ς l ( n ) M l ( n , τ k ) φ l ( τ j ) , (63) where the term l = 0 l=0 l = 0 ν ( n , k ) \nu_{(n,k)} ν ( n , k ) n n n τ j \tau^j τ j
For any λ ≥ 0 \lambda\ge0 λ ≥ 0 ( ϱ d ( λ ) ) d ≥ 0 (\varrho_d(\lambda))_{d\ge0} ( ϱ d ( λ ) ) d ≥ 0 ϱ ∞ ( λ ) ≥ 1 \varrho_\infty(\lambda) \ge1 ϱ ∞ ( λ ) ≥ 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 . (64) \nu_{(d+k,k)}(j)=\left\langle\frac{\varrho_d}{\varrho_\infty}(\mathrm{B})\delta_j,\delta_k\right\rangle_{\ell^2_{\mathbb{N}_0}}. \tag*{(64)} ν ( d + k , k ) ( j ) = ⟨ ϱ ∞ ϱ d ( B ) δ j , δ k ⟩ ℓ N 0 2 . (64) In particular, ν ( d + k , k ) ( j ) = ν ( d + j , j ) ( k ) \nu_{(d+k,k)}(j)=\nu_{(d+j,j)}(k) ν ( d + k , k ) ( j ) = ν ( d + j , j ) ( k ) ( ν ( k , k ) ( j ) ) k , j ≥ 0 (\nu_{(k,k)}(j))_{k,j\ge0} ( ν ( k , k ) ( j ) ) k , j ≥ 0 ( k , k ) (k,k) ( k , k ) ϱ ∞ ( B ) − 1 \varrho_\infty(\mathrm{B})^{-1} ϱ ∞ ( B ) − 1
Proof : 1. Let v = ( n , k ) v=(n,k) v = ( n , k ) Lemma 6.7 , ν v \nu_v ν v N 0 \mathbb{N}_0 N 0 X τ X_\tau X τ j ↦ τ − j j\mapsto\tau^{-j} j ↦ τ − j H \mathcal{H} H ∫ φ l d ν v = H φ l ( v ) \int\varphi_l\,\mathrm{d}\nu_v=\mathcal{H}\varphi_l(v) ∫ φ l d ν v = H φ l ( v ) l > n l>n l > n Proposition 6.9 . Put e l : = ∥ φ l τ ∥ − 1 φ l τ e_l\mathrel{\mathop:}=\|\varphi_l^\tau\|^{-1}\varphi_l^\tau e l : = ∥ φ l τ ∥ − 1 φ l τ p : = ∑ l = 0 n ( ∫ e l d ν v ) e l p\mathrel{\mathop:}=\sum_{l=0}^n\left(\int e_l\,\mathrm{d}\nu_v\right)e_l p : = ∑ l = 0 n ( ∫ e l d ν v ) e l n n n ν ′ : = ν v − p w \nu'\mathrel{\mathop:}=\nu_v-pw ν ′ : = ν v − pw ∫ e l d ν ′ = 0 \int e_l\,\mathrm{d}\nu'=0 ∫ e l d ν ′ = 0 l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 ( e l ) (e_l) ( e l ) ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) ( e l ) (e_l) ( e l ) C ( X τ ) C(X_\tau) C ( X τ ) ν ′ = 0 \nu'=0 ν ′ = 0 ν v ( j ) = w j p ( τ − j ) \nu_v(j)=w_jp(\tau^{-j}) ν v ( j ) = w j p ( τ − j ) Proposition 6.9 gives ∫ e l d ν v = κ l ς l ( n ) ς l ( ∞ ) ∥ φ l τ ∥ M l , n ( τ − k ) \int e_l\,\mathrm{d}\nu_v=\frac{\kappa_l\varsigma_l(n)}{\varsigma_l(\infty)\|\varphi_l^\tau\|}M_{l,n}(\tau^{-k}) ∫ e l d ν v = ς l ( ∞ ) ∥ φ l τ ∥ κ l ς l ( n ) M l , n ( τ − k ) l ≥ 1 l\ge1 l ≥ 1 ∫ e 0 d ν v = 1 \int e_0\,\mathrm{d}\nu_v=1 ∫ e 0 d ν v = 1 (63) .
Let λ ≥ 0 \lambda\ge0 λ ≥ 0 (62) and induction, we have ϱ d ( λ ) ≥ 1 \varrho_d(\lambda)\ge1 ϱ d ( λ ) ≥ 1 ϱ d + 1 ( λ ) ≥ ϱ d ( λ ) \varrho_{d+1}(\lambda)\ge\varrho_d(\lambda) ϱ d + 1 ( λ ) ≥ ϱ d ( λ ) ϱ d + 1 ( λ ) ≤ ϱ d ( λ ) ( 1 + ( d + 1 ) λ a d ) \varrho_{d+1}(\lambda)\le\varrho_d(\lambda)\left(1+\frac{(d+1)\lambda}{a_d}\right) ϱ d + 1 ( λ ) ≤ ϱ d ( λ ) ( 1 + a d ( d + 1 ) λ ) a d ≥ τ − d 1 + τ a_d\ge\frac{\tau^{-d}}{1+\tau} a d ≥ 1 + τ τ − d ∏ d ≥ 0 ( 1 + ( 1 + τ ) ( d + 1 ) λ τ d ) \prod_{d\ge0}\left(1+(1+\tau)(d+1)\lambda\tau^d\right) ∏ d ≥ 0 ( 1 + ( 1 + τ ) ( d + 1 ) λ τ d ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ϱ d \varrho_d ϱ d d d d (62) again, the series ∑ d ( ϱ d + 1 − ϱ d ) \sum_d(\varrho_{d+1}-\varrho_d) ∑ d ( ϱ d + 1 − ϱ d )
Let D ≥ 1 D\ge1 D ≥ 1 j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0 B ≥ 0 \mathrm{B}\ge0 B ≥ 0 ϱ D ≥ 1 \varrho_D\ge1 ϱ D ≥ 1 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ϱ D ( B ) \varrho_D(\mathrm{B}) ϱ D ( B ) 0 ≤ d ≤ D 0\le d\le D 0 ≤ d ≤ D u D ( d , ⋅ ) : = ϱ d ( B ) ϱ D ( B ) − 1 δ j ∈ ℓ N 0 2 u_D(d,\cdot)\mathrel{\mathop:}=\varrho_d(\mathrm{B})\varrho_D(\mathrm{B})^{-1}\delta_j\in\ell^2_{\mathbb{N}_0} u D ( d , ⋅ ) : = ϱ d ( B ) ϱ D ( B ) − 1 δ j ∈ ℓ N 0 2 u D u_D u D { ( d , k ) : 0 ≤ d ≤ D } \{(d,k):0\le d\le D\} {( d , k ) : 0 ≤ d ≤ D } u D ( D , ⋅ ) = δ j u_D(D,\cdot)=\delta_j u D ( D , ⋅ ) = δ j Proposition 6.4 , a function u u u ( d , k ) (d,k) ( d , k )
a d ( u ( d + 1 , k ) − u ( d , k ) ) + a d − 1 ( u ( d − 1 , k ) − u ( d , k ) ) = ( B u ( d , ⋅ ) ) ( k ) . a_d\bigl(u(d+1,k)-u(d,k)\bigr)+a_{d-1}\bigl(u(d-1,k)-u(d,k)\bigr)=(\mathrm{B}u(d,\cdot))(k). a d ( u ( d + 1 , k ) − u ( d , k ) ) + a d − 1 ( u ( d − 1 , k ) − u ( d , k ) ) = ( B u ( d , ⋅ )) ( k ) . For d < D d<D d < D u = u D u=u_D u = u D k k k ( a d ( ϱ d + 1 − ϱ d ) + a d − 1 ( ϱ d − 1 − ϱ d ) ) ( B ) ϱ D ( B ) − 1 δ j = B ϱ d ( B ) ϱ D ( B ) − 1 δ j \bigl(a_d(\varrho_{d+1}-\varrho_d)+a_{d-1}(\varrho_{d-1}-\varrho_d)\bigr)(\mathrm{B})\varrho_D(\mathrm{B})^{-1}\delta_j=\mathrm{B}\varrho_d(\mathrm{B})\varrho_D(\mathrm{B})^{-1}\delta_j ( a d ( ϱ d + 1 − ϱ d ) + a d − 1 ( ϱ d − 1 − ϱ d ) ) ( B ) ϱ D ( B ) − 1 δ j = B ϱ d ( B ) ϱ D ( B ) − 1 δ j u D u_D u D ( d , k ) (d,k) ( d , k ) d < D d<D d < D v = ( d , k ) v=(d,k) v = ( d , k ) d < D d<D d < D σ D \sigma_D σ D { D } × N 0 \{D\}\times\mathbb{N}_0 { D } × N 0 v v v w b ≤ τ − D 1 − τ w_b\le\frac{\tau^{-D}}{1-\tau} w b ≤ 1 − τ τ − D ( d ′ , k ′ ) (d',k') ( d ′ , k ′ ) ( d ′ + 1 , k ′ ) (d'+1,k') ( d ′ + 1 , k ′ ) a 0 ( 1 − τ ) τ D = a 0 ( 1 − τ ) τ D 1 + τ a_0(1-\tau)\tau^D=\frac{a_0(1-\tau)\tau^D}{1+\tau} a 0 ( 1 − τ ) τ D = 1 + τ a 0 ( 1 − τ ) τ D a d ′ ≥ a 0 a_{d'}\ge a_0 a d ′ ≥ a 0 σ D < ∞ \sigma_D<\infty σ D < ∞ u D u_D u D
P v ( k σ D = j ) = u D ( v ) = ⟨ ϱ d ( B ) ϱ D ( B ) − 1 δ j , δ k ⟩ . \mathbb{P}_v(k_{\sigma_D}=j)=u_D(v)=\left\langle\varrho_d(\mathrm{B})\varrho_D(\mathrm{B})^{-1}\delta_j,\delta_k\right\rangle. P v ( k σ D = j ) = u D ( v ) = ⟨ ϱ d ( B ) ϱ D ( B ) − 1 δ j , δ k ⟩ . As D → ∞ D\to\infty D → ∞ σ D ≥ D − d → ∞ \sigma_D\ge D-d\to\infty σ D ≥ D − d → ∞ k t = k ∞ k_t=k_\infty k t = k ∞ t t t Lemma 6.7 . By dominated convergence, the left-hand side converges to ν v ( j ) \nu_v(j) ν v ( j ) ϱ d ϱ D → ϱ d ϱ ∞ \frac{\varrho_d}{\varrho_D}\to\frac{\varrho_d}{\varrho_\infty} ϱ D ϱ d → ϱ ∞ ϱ d B \mathrm{B} B ϱ D ≥ 1 \varrho_D\ge1 ϱ D ≥ 1 ϱ d ( B ) ϱ D ( B ) − 1 → ϱ d ϱ ∞ ( B ) \varrho_d(\mathrm{B})\varrho_D(\mathrm{B})^{-1}\to\frac{\varrho_d}{\varrho_\infty}(\mathrm{B}) ϱ d ( B ) ϱ D ( B ) − 1 → ϱ ∞ ϱ d ( B ) (64) . Since B \mathrm{B} B ϱ d ϱ ∞ ( B ) \frac{\varrho_d}{\varrho_\infty}(\mathrm{B}) ϱ ∞ ϱ d ( B ) d = 0 d=0 d = 0
Remark 6.14 Theorem 6.13 gives two descriptions of the Poisson kernel. The first one is a finite sum over the spins l ≤ n l\le n l ≤ n q q q ς l ( n ) \varsigma_l(n) ς l ( n ) Proposition 6.12 . The second one separates the variables: the vertical part is the birth–death network ( a d ) (a_d) ( a d ) Remark 6.3 , and the horizontal part is the operator B B B E τ E_\tau E τ ϱ d ϱ ∞ ≤ 1 \frac{\varrho_d}{\varrho_\infty} \le1 ϱ ∞ ϱ d ≤ 1 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) B 1 = 0 B1=0 B 1 = 0 ( ν ( d + k , k ) ( j ) ) k , j (\nu(d+k,k)(j))_{k,j} ( ν ( d + k , k ) ( j ) ) k , j ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 d → ∞ d\to\infty d → ∞ τ = 1 3 \tau=\frac{1}{3} τ = 3 1 d , k , j ≤ 3 d,k,j\le3 d , k , j ≤ 3 ϱ ∞ \varrho_\infty ϱ ∞ B B B ( ν ( k , k ) ( j ) ) k , j (\nu(k,k)(j))_{k,j} ( ν ( k , k ) ( j ) ) k , j Theorem 6.26 ).
The entire function ϱ ∞ \varrho_\infty ϱ ∞
In this subsection, we compute the function ϱ ∞ \varrho_\infty ϱ ∞ Theorem 6.13 , we diagonalize the Jacobi operators of the two networks of Proposition 6.4 , and we describe the consequences for the Poisson kernel. We use the notation ( a ; τ ) n : = ∏ i = 0 n − 1 ( 1 − a τ i ) (a;\tau)_n \mathrel{\mathop:}= \prod_{i=0}^{n-1}(1-a\tau^i) ( a ; τ ) n : = ∏ i = 0 n − 1 ( 1 − a τ i ) n ∈ N 0 ∪ { ∞ } n\in\mathbb{N}_0\cup\{\infty\} n ∈ N 0 ∪ { ∞ } ( a 1 , … , a p ; τ ) n : = ( a 1 ; τ ) n ⋯ ( a p ; τ ) n (a_1,\ldots,a_p;\tau)_n \mathrel{\mathop:}= (a_1;\tau)_n\cdots(a_p;\tau)_n ( a 1 , … , a p ; τ ) n : = ( a 1 ; τ ) n ⋯ ( a p ; τ ) n [27 (17.2.1), (17.2.4)–(17.2.6)] , the Gaussian binomial coefficients ( m k ) τ : = ( τ ; τ ) m ( τ ; τ ) k ( τ ; τ ) m − k \binom{m}{k}_\tau\mathrel{\mathop:}= \frac{(\tau;\tau)_m}{(\tau;\tau)_k(\tau;\tau)_{m-k}} ( k m ) τ : = ( τ ; τ ) k ( τ ; τ ) m − k ( τ ; τ ) m 0 ≤ k ≤ m 0\le k\le m 0 ≤ k ≤ m [27 (17.2.27)] , [45 (0.3.2)] , and the basic hypergeometric series s + 1 ϕ s {}_{s+1}\phi_s s + 1 ϕ s [32 Section 1.2] , see also [27 (17.4.1)] and [45 (0.4.2)] . The recurrence (62) defines polynomials ϱ d \varrho_d ϱ d 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 . (65) z+z^{-1}=2-\frac{(1-\tau^2)\lambda}{\tau}. \tag*{(65)} z + z − 1 = 2 − τ ( 1 − τ 2 ) λ . (65) For any integer j ≥ 0 j\ge0 j ≥ 0 ( 1 − z τ j ) ( 1 − z − 1 τ j ) = 1 − ( z + z − 1 ) τ j + τ 2 j = ( 1 − τ j ) 2 + ( 1 − τ 2 ) λ τ j − 1 (1-z\tau^j)(1-z^{-1}\tau^j)=1-(z+z^{-1})\tau^j+\tau^{2j}=(1-\tau^j)^2+(1-\tau^2)\lambda\tau^{j-1} ( 1 − z τ j ) ( 1 − z − 1 τ j ) = 1 − ( z + z − 1 ) τ j + τ 2 j = ( 1 − τ j ) 2 + ( 1 − τ 2 ) λ τ j − 1 ( z , z − 1 ; τ ) n (z,z^{-1};\tau)_n ( z , z − 1 ; τ ) n λ \lambda λ z z z n ≥ 1 n\ge1 n ≥ 1
μ n : = ( 1 − τ n ) 2 ( 1 − τ 2 ) τ n − 1 = 1 − τ 1 + τ [ n ] τ 2 . (66) \mu_n \mathrel{\mathop:}= \frac{(1-\tau^n)^2}{(1-\tau^2)\tau^{n-1}}=\frac{1-\tau}{1+\tau}[n]_\tau^2. \tag*{(66)} μ n : = ( 1 − τ 2 ) τ n − 1 ( 1 − τ n ) 2 = 1 + τ 1 − τ [ n ] τ 2 . (66) where [ n ] r = r n − r − n r − r − 1 [n]_r=\frac{r^n-r^{-n}}{r-r^{-1}} [ n ] r = r − r − 1 r n − r − n V n − 1 2 V_{\frac{n-1}{2}} V 2 n − 1 μ n + 1 μ n = ( 1 − τ n + 1 ) 2 τ ( 1 − τ n ) 2 > 1 τ \frac{\mu_{n+1}}{\mu_n}=\frac{(1-\tau^{n+1})^2}{\tau(1-\tau^n)^2}>\frac{1}{\tau} μ n μ n + 1 = τ ( 1 − τ n ) 2 ( 1 − τ n + 1 ) 2 > τ 1 ( μ n ) (\mu_n) ( μ n )
Theorem 6.15 Let λ ∈ C \lambda\in\mathbb{C} λ ∈ C
For any y ∈ C y\in\mathbb{C} y ∈ C ∣ y ∣ < 1 |y|<1 ∣ y ∣ < 1
∑ d ≥ 0 ϱ d ( λ ) y d = ∏ j ≥ 0 ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y ( 1 − τ j y ) ( 1 − τ j + 1 y ) = ( τ z y , τ z − 1 y ; τ ) ∞ ( y ; τ ) ∞ ( τ y ; τ ) ∞ . (67) \sum_{d\ge0}\varrho_d(\lambda)y^d=\prod_{j\ge0}\frac{(1-\tau^{j+1}y)^2+(1-\tau^2)\lambda\tau^j y}{(1-\tau^j y)(1-\tau^{j+1}y)}=\frac{(\tau zy,\tau z^{-1}y;\tau)_\infty}{(y;\tau)_\infty(\tau y;\tau)_\infty}. \tag*{(67)} d ≥ 0 ∑ ϱ d ( λ ) y d = j ≥ 0 ∏ ( 1 − τ j y ) ( 1 − τ j + 1 y ) ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y = ( y ; τ ) ∞ ( τ y ; τ ) ∞ ( τ z y , τ z − 1 y ; τ ) ∞ . (67) For any d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0
ϱ d ( λ ) = ∑ k = 0 d ( d k ) τ ( z , z − 1 ; τ ) k ( τ ; τ ) k τ k . (68) \varrho_d(\lambda)=\sum_{k=0}^{d}\binom{d}{k}_\tau\frac{(z,z^{-1};\tau)_k}{(\tau;\tau)_k}\tau^k. \tag*{(68)} ϱ d ( λ ) = k = 0 ∑ d ( k d ) τ ( τ ; τ ) k ( z , z − 1 ; τ ) k τ k . (68) The sequence ( ϱ d ) (\varrho_d) ( ϱ d ) C \mathbb{C} C
ϱ ∞ ( λ ) = ∏ n ≥ 1 ( 1 + λ μ n ) = ( τ z , τ z − 1 ; τ ) ∞ ( τ ; τ ) ∞ = 2 ϕ 1 ( z , z − 1 τ ; τ , τ ) . (69) \varrho_\infty(\lambda)=\prod_{n\ge1}\left(1+\frac{\lambda}{\mu_n}\right)=\frac{(\tau z,\tau z^{-1};\tau)_\infty}{(\tau;\tau)_\infty}={}_2\phi_1\left(\begin{matrix}z,z^{-1}\\ \tau\end{matrix};\tau,\tau\right). \tag*{(69)} ϱ ∞ ( λ ) = n ≥ 1 ∏ ( 1 + μ n λ ) = ( τ ; τ ) ∞ ( τ z , τ z − 1 ; τ ) ∞ = 2 ϕ 1 ( z , z − 1 τ ; τ , τ ) . (69) For any compact subset L L L C \mathbb{C} C C L C_L C L λ ∈ L \lambda\in L λ ∈ L d ∈ N 0 d \in\mathbb{N}_0 d ∈ N 0
∣ ϱ d ( λ ) − ϱ ∞ ( λ ) + ( 1 + τ ) λ ϱ ∞ ( λ ) τ d ∣ ≤ C L τ d . \left|\varrho_d(\lambda)-\varrho_\infty(\lambda)+(1+\tau)\lambda\varrho_\infty(\lambda)\tau^d\right|\le C_L\tau^d. ϱ d ( λ ) − ϱ ∞ ( λ ) + ( 1 + τ ) λ ϱ ∞ ( λ ) τ d ≤ C L τ d . Proof: 1. Put M ≔ ∏ i ≥ 0 ( 1 + ( 1 + τ ) ∣ λ ∣ ( i + 1 ) τ i ) M \coloneqq\prod_{i\ge0}\left(1+(1+\tau)|\lambda|(i+1)\tau^i\right) M : = ∏ i ≥ 0 ( 1 + ( 1 + τ ) ∣ λ ∣ ( i + 1 ) τ i ) a d ≥ τ − d 1 + τ a_d\ge\frac{\tau^{-d}}{1+\tau} a d ≥ 1 + τ τ − d d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 (62) and induction on d d d max i ≤ d ∣ ϱ i ( λ ) ∣ ≤ ∏ i < d ( 1 + ( 1 + τ ) ∣ λ ∣ ( i + 1 ) τ i ) ≤ M \max_{i\le d}|\varrho_i(\lambda)|\le\prod_{i<d}\left(1+(1+\tau)|\lambda|(i+1)\tau^i\right)\le M max i ≤ d ∣ ϱ i ( λ ) ∣ ≤ ∏ i < d ( 1 + ( 1 + τ ) ∣ λ ∣ ( i + 1 ) τ i ) ≤ M
∣ ϱ d + 1 ( λ ) − ϱ d ( λ ) ∣ ≤ ∣ λ ∣ a d ∑ i = 0 d ∣ ϱ i ( λ ) ∣ ≤ ( 1 + τ ) M ∣ λ ∣ ( d + 1 ) τ d . (70) |\varrho_{d+1}(\lambda)-\varrho_d(\lambda)|\le\frac{|\lambda|}{a_d}\sum_{i=0}^{d}|\varrho_i(\lambda)|\le(1+\tau)M|\lambda|(d+1)\tau^d. \tag*{(70)} ∣ ϱ d + 1 ( λ ) − ϱ d ( λ ) ∣ ≤ a d ∣ λ ∣ i = 0 ∑ d ∣ ϱ i ( λ ) ∣ ≤ ( 1 + τ ) M ∣ λ ∣ ( d + 1 ) τ d . (70) In particular, the power series F ( y ) ≔ ∑ d ≥ 0 ϱ d ( λ ) y d F(y)\coloneqq\sum_{d\ge0}\varrho_d(\lambda)y^d F ( y ) : = ∑ d ≥ 0 ϱ d ( λ ) y d ∣ y ∣ < 1 |y|<1 ∣ y ∣ < 1 e 0 ≔ 1 e_0\coloneqq1 e 0 : = 1 e d ≔ ϱ d ( λ ) − ϱ d − 1 ( λ ) e_d\coloneqq\varrho_d(\lambda)-\varrho_{d-1}(\lambda) e d : = ϱ d ( λ ) − ϱ d − 1 ( λ ) d ≥ 1 d\ge1 d ≥ 1 E ( y ) ≔ ∑ d ≥ 0 e d y d = ( 1 − y ) F ( y ) E(y)\coloneqq\sum_{d\ge0}e_dy^d=(1-y)F(y) E ( y ) : = ∑ d ≥ 0 e d y d = ( 1 − y ) F ( y ) ( 1 − τ 2 ) τ d a d = 1 − τ d + 1 (1-\tau^2)\tau^d a_d=1-\tau^{d+1} ( 1 − τ 2 ) τ d a d = 1 − τ d + 1 ( 1 − τ 2 ) τ d a d − 1 = τ ( 1 − τ d ) (1-\tau^2)\tau^d a_{d-1}=\tau(1-\tau^d) ( 1 − τ 2 ) τ d a d − 1 = τ ( 1 − τ d ) (62) , multiplied by ( 1 − τ 2 ) τ d (1-\tau^2)\tau^d ( 1 − τ 2 ) τ d
( 1 − τ d + 1 ) e d + 1 − τ ( 1 − τ d ) e d = ( 1 − τ 2 ) λ τ d ϱ d ( λ ) , d ≥ 0. (1-\tau^{d+1})e_{d+1}-\tau(1-\tau^d)e_d=(1-\tau^2)\lambda\tau^d\varrho_d(\lambda),\qquad d\ge0. ( 1 − τ d + 1 ) e d + 1 − τ ( 1 − τ d ) e d = ( 1 − τ 2 ) λ τ d ϱ d ( λ ) , d ≥ 0. We multiply by y d + 1 y^{d+1} y d + 1 d ≥ 0 d\ge0 d ≥ 0 ∑ d ≥ 0 ( 1 − τ d + 1 ) e d + 1 y d + 1 = ∑ d ≥ 0 ( 1 − τ d ) e d y d = E ( y ) − E ( τ y ) \sum_{d\ge0}(1-\tau^{d+1})e_{d+1}y^{d+1}=\sum_{d\ge0}(1-\tau^d)e_dy^d=E(y)-E(\tau y) ∑ d ≥ 0 ( 1 − τ d + 1 ) e d + 1 y d + 1 = ∑ d ≥ 0 ( 1 − τ d ) e d y d = E ( y ) − E ( τ y ) ( 1 − τ y ) ( E ( y ) − E ( τ y ) ) = ( 1 − τ 2 ) λ y F ( τ y ) (1-\tau y)(E(y)-E(\tau y))=(1-\tau^2)\lambda yF(\tau y) ( 1 − τ y ) ( E ( y ) − E ( τ y )) = ( 1 − τ 2 ) λ y F ( τ y )
( 1 − y ) ( 1 − τ y ) F ( y ) = [ ( 1 − τ y ) 2 + ( 1 − τ 2 ) λ y ] F ( τ y ) , ∣ y ∣ < 1. (71) (1-y)(1-\tau y)F(y)=\left[(1-\tau y)^2+(1-\tau^2)\lambda y\right]F(\tau y),\qquad|y|<1. \tag*{(71)} ( 1 − y ) ( 1 − τ y ) F ( y ) = [ ( 1 − τ y ) 2 + ( 1 − τ 2 ) λ y ] F ( τ y ) , ∣ y ∣ < 1. (71) Let ∣ y ∣ < 1 |y|<1 ∣ y ∣ < 1 j ≥ 0 j\ge0 j ≥ 0 R j ( y ) ≔ ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y ( 1 − τ j y ) ( 1 − τ j + 1 y ) R_j(y)\coloneqq\frac{(1-\tau^{j+1}y)^2+(1-\tau^2)\lambda\tau^j y}{(1-\tau^j y)(1-\tau^{j+1}y)} R j ( y ) : = ( 1 − τ j y ) ( 1 − τ j + 1 y ) ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y R j ( y ) = 1 + O ( τ j ) R_j(y)=1+O(\tau^j) R j ( y ) = 1 + O ( τ j ) (71) at the points y , τ y , … , τ N − 1 y y,\tau y,\ldots,\tau^{N-1}y y , τ y , … , τ N − 1 y F ( y ) = ∏ j = 0 N − 1 R j ( y ) ⋅ F ( τ N y ) F(y)=\prod_{j=0}^{N-1}R_j(y)\cdot F(\tau^N y) F ( y ) = ∏ j = 0 N − 1 R j ( y ) ⋅ F ( τ N y ) N ≥ 1 N\ge1 N ≥ 1 F ( τ N y ) → F ( 0 ) = 1 F(\tau^N y)\to F(0)=1 F ( τ N y ) → F ( 0 ) = 1 N → ∞ N\to\infty N → ∞ (67) . By (65) , we have ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y = ( 1 − τ j + 1 z y ) ( 1 − τ j + 1 z − 1 y ) (1-\tau^{j+1}y)^2+(1-\tau^2)\lambda\tau^j y=(1-\tau^{j+1}zy)(1-\tau^{j+1}z^{-1}y) ( 1 − τ j + 1 y ) 2 + ( 1 − τ 2 ) λ τ j y = ( 1 − τ j + 1 z y ) ( 1 − τ j + 1 z − 1 y )
Let ∣ y ∣ < 1 |y|<1 ∣ y ∣ < 1 ∣ τ y ∣ < 1 = ∣ z ⋅ z − 1 ∣ |\tau y|<1=|z\cdot z^{-1}| ∣ τ y ∣ < 1 = ∣ z ⋅ z − 1 ∣ q q q [27 (17.6.1)] , applied with the parameters ( z , z − 1 , τ y ) (z,z^{-1},\tau y) ( z , z − 1 , τ y )
∑ k ≥ 0 ( z , z − 1 ; τ ) k ( τ ; τ ) k ( τ y ; τ ) k ( τ y ) k = ( τ z y , τ z − 1 y ; τ ) ∞ ( τ y ; τ ) ∞ . \sum_{k\ge0}\frac{(z,z^{-1};\tau)_k}{(\tau;\tau)_k(\tau y;\tau)_k}(\tau y)^k=\frac{(\tau zy,\tau z^{-1}y;\tau)_\infty}{(\tau y;\tau)_\infty}. k ≥ 0 ∑ ( τ ; τ ) k ( τ y ; τ ) k ( z , z − 1 ; τ ) k ( τ y ) k = ( τ y ; τ ) ∞ ( τ z y , τ z − 1 y ; τ ) ∞ . We divide by 1 − y 1-y 1 − y ( 1 − y ) ( τ y ; τ ) k = ( y ; τ ) k + 1 (1-y)(\tau y;\tau)_k=(y;\tau)_{k+1} ( 1 − y ) ( τ y ; τ ) k = ( y ; τ ) k + 1 ( 1 − y ) ( τ y ; τ ) ∞ = ( y ; τ ) ∞ (1-y)(\tau y;\tau)_\infty=(y;\tau)_\infty ( 1 − y ) ( τ y ; τ ) ∞ = ( y ; τ ) ∞
F ( y ) = ∑ k ≥ 0 ( z , z − 1 ; τ ) k ( τ ; τ ) k y k ( y ; τ ) k + 1 . F(y)=\sum_{k\ge0}\frac{(z,z^{-1};\tau)_k}{(\tau;\tau)_k}\frac{y^k}{(y;\tau)_{k+1}}. F ( y ) = k ≥ 0 ∑ ( τ ; τ ) k ( z , z − 1 ; τ ) k ( y ; τ ) k + 1 y k . By the q q q [45 (0.5.2)] , applied with the parameter τ k + 1 \tau^{k+1} τ k + 1 [27 (17.2.38)] ), we have y k ( y ; τ ) k + 1 = ∑ d ≥ k ( d k ) τ y d \frac{y^k}{(y;\tau)_{k+1}}=\sum_{d\ge k}\binom{d}{k}_{\tau}y^d ( y ; τ ) k + 1 y k = ∑ d ≥ k ( k d ) τ y d ∣ ( z , z − 1 ; τ ) k ∣ ≤ ∏ j ≥ 0 ( 1 + ∣ z + z − 1 ∣ τ j + τ 2 j ) |(z,z^{-1};\tau)_k|\le\prod_{j\ge0}(1+|z+z^{-1}|\tau^j+\tau^{2j}) ∣ ( z , z − 1 ; τ ) k ∣ ≤ ∏ j ≥ 0 ( 1 + ∣ z + z − 1 ∣ τ j + τ 2 j ) ∑ d ≥ k ( d k ) τ ∣ y ∣ d = ∣ y ∣ k ( ∣ y ∣ ; τ ) k + 1 ≤ ∣ y ∣ k ( ∣ y ∣ ; τ ) ∞ \sum_{d\ge k}\binom{d}{k}_{\tau}|y|^d=\frac{|y|^k}{(|y|;\tau)_{k+1}}\le\frac{|y|^k}{(|y|;\tau)_\infty} ∑ d ≥ k ( k d ) τ ∣ y ∣ d = ( ∣ y ∣ ; τ ) k + 1 ∣ y ∣ k ≤ ( ∣ y ∣ ; τ ) ∞ ∣ y ∣ k y d y^d y d (68) .
The uniform convergence follows from (70) , and the limit ϱ ∞ \varrho_\infty ϱ ∞ E ( y ) = ∑ d ≥ 0 e d y d E(y)=\sum_{d\ge0}e_dy^d E ( y ) = ∑ d ≥ 0 e d y d ( τ z y , τ z − 1 y ; τ ) ∞ ( τ y ; τ ) ∞ \frac{(\tau zy,\tau z^{-1}y;\tau)_\infty}{(\tau y;\tau)_\infty} ( τ y ; τ ) ∞ ( τ z y , τ z − 1 y ; τ ) ∞ { ∣ y ∣ < τ − 1 } \{|y|<\tau^{-1}\} { ∣ y ∣ < τ − 1 } E E E y = 1 y=1 y = 1 ϱ ∞ ( λ ) = ∑ d ≥ 0 e d = E ( 1 ) = ( τ z , τ z − 1 ; τ ) ∞ ( τ ; τ ) ∞ \varrho_\infty(\lambda)=\sum_{d\ge0}e_d=E(1)=\frac{(\tau z,\tau z^{-1};\tau)_\infty}{(\tau;\tau)_\infty} ϱ ∞ ( λ ) = ∑ d ≥ 0 e d = E ( 1 ) = ( τ ; τ ) ∞ ( τ z , τ z − 1 ; τ ) ∞ ( 1 − τ n z ) ( 1 − τ n z − 1 ) = (1-\tau^n z)(1-\tau^n z^{-1})= ( 1 − τ n z ) ( 1 − τ n z − 1 ) =
( 1 − τ n ) 2 + ( 1 − τ 2 ) λ τ n − 1 (1-\tau^n)^2+(1-\tau^2)\lambda\tau^{n-1} ( 1 − τ n ) 2 + ( 1 − τ 2 ) λ τ n − 1 ( 1 − τ n z ) ( 1 − τ n z − 1 ) ( 1 − τ n ) 2 = (6.19) 1 + λ μ n \frac{(1-\tau^n z)(1-\tau^n z^{-1})}{(1-\tau^n)^2} \overset{\text{(6.19)}}{=} 1+\frac{\lambda}{\mu_n} ( 1 − τ n ) 2 ( 1 − τ n z ) ( 1 − τ n z − 1 ) = (6.19) 1 + μ n λ n n n q q q [27 (17.6.1)] with the parameters ( z , z − 1 , τ ) (z,z^{-1},\tau) ( z , z − 1 , τ )
For ∣ u ∣ < τ − 1 |u|<\tau^{-1} ∣ u ∣ < τ − 1 H ( u ) : = ∏ j ≥ 0 ( 1 − ( z + z − 1 ) τ j u + τ 2 j u 2 ) ( τ u ; τ ) ∞ − 2 H(u)\mathrel{\mathop:}=\prod_{j\geq0}(1-(z+z^{-1})\tau^j u+\tau^{2j}u^2)(\tau u;\tau)_\infty^{-2} H ( u ) : = ∏ j ≥ 0 ( 1 − ( z + z − 1 ) τ j u + τ 2 j u 2 ) ( τ u ; τ ) ∞ − 2 ( λ , u ) ∈ C × { ∣ u ∣ < τ − 1 } (\lambda,u)\in\mathbb{C}\times\{|u|<\tau^{-1}\} ( λ , u ) ∈ C × { ∣ u ∣ < τ − 1 } (65) . By the third part, E ( y ) = H ( τ y ) ( 1 − τ y ) 2 E(y)=\frac{H(\tau y)}{(1-\tau y)^2} E ( y ) = ( 1 − τ y ) 2 H ( τ y ) H ( u ) = H ( 1 ) + ( u − 1 ) H 1 ( u ) H(u)=H(1)+(u-1)H_1(u) H ( u ) = H ( 1 ) + ( u − 1 ) H 1 ( u ) H 1 H_1 H 1
E ( y ) − H ( 1 ) ( 1 − τ y ) 2 + H 1 ( 1 ) 1 − τ y = − H 1 ( τ y ) − H 1 ( 1 ) 1 − τ y E(y)-\frac{H(1)}{(1-\tau y)^2}+\frac{H_1(1)}{1-\tau y}=-\frac{H_1(\tau y)-H_1(1)}{1-\tau y} E ( y ) − ( 1 − τ y ) 2 H ( 1 ) + 1 − τ y H 1 ( 1 ) = − 1 − τ y H 1 ( τ y ) − H 1 ( 1 ) is holomorphic in ( λ , y ) ∈ C × { ∣ y ∣ < τ − 2 } (\lambda,y)\in\mathbb{C}\times\{|y|<\tau^{-2}\} ( λ , y ) ∈ C × { ∣ y ∣ < τ − 2 } { ∣ y ∣ = τ − 3 / 2 } \{|y|=\tau^{-3/2}\} { ∣ y ∣ = τ − 3/2 } C L τ 3 d / 2 C_L\tau^{3d/2} C L τ 3 d /2 λ ∈ L \lambda\in L λ ∈ L 1 ( 1 − τ y ) 2 = ∑ d ≥ 0 ( d + 1 ) τ d y d \frac{1}{(1-\tau y)^2}=\sum_{d\geq0}(d+1)\tau^d y^d ( 1 − τ y ) 2 1 = ∑ d ≥ 0 ( d + 1 ) τ d y d e d = H ( 1 ) ( d + 1 ) τ d − H 1 ( 1 ) τ d + O ( τ 3 d / 2 ) e_d=H(1)(d+1)\tau^d-H_1(1)\tau^d+O(\tau^{3d/2}) e d = H ( 1 ) ( d + 1 ) τ d − H 1 ( 1 ) τ d + O ( τ 3 d /2 ) L L L d d d
ϱ ∞ ( λ ) − ϱ d ( λ ) = H ( 1 ) 1 − τ τ d + 1 + O ( τ d ) \varrho_\infty(\lambda)-\varrho_d(\lambda)=\frac{H(1)}{1-\tau}\tau^{d+1}+\mathcal{O}(\tau^d) ϱ ∞ ( λ ) − ϱ d ( λ ) = 1 − τ H ( 1 ) τ d + 1 + O ( τ d ) uniformly on L L L H ( 1 ) = ( z , z − 1 ; τ ) ∞ ( τ ; τ ) ∞ 2 = ( 1 − z ) ( 1 − z − 1 ) ϱ ∞ ( λ ) = (6.18) ( 1 − τ 2 ) λ τ ϱ ∞ ( λ ) H(1)=\frac{(z,z^{-1};\tau)_\infty}{(\tau;\tau)_\infty^2}=(1-z)(1-z^{-1})\varrho_\infty(\lambda)\overset{\text{(6.18)}}{=}\frac{(1-\tau^2)\lambda}{\tau}\varrho_\infty(\lambda) H ( 1 ) = ( τ ; τ ) ∞ 2 ( z , z − 1 ; τ ) ∞ = ( 1 − z ) ( 1 − z − 1 ) ϱ ∞ ( λ ) = (6.18) τ ( 1 − τ 2 ) λ ϱ ∞ ( λ )
The function ϱ ∞ \varrho_\infty ϱ ∞ m ∈ N 0 m\in\mathbb{N}_0 m ∈ N 0 u m : = 1 { 0 , … , m } ∈ ℓ N 0 2 u_m\mathrel{\mathop:}=1_{\{0,\ldots,m\}}\in\ell^2_{\mathbb{N}_0} u m : = 1 { 0 , … , m } ∈ ℓ N 0 2 K K K ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2
K ( i , j ) : = ∑ m ≥ max ( i , j ) 1 a m , i , j ∈ N 0 . K(i,j)\mathrel{\mathop:}=\sum_{m\geq\max(i,j)}\frac{1}{a_m},\qquad i,j\in\mathbb{N}_0. K ( i , j ) : = m ≥ m a x ( i , j ) ∑ a m 1 , i , j ∈ N 0 . For any N ∈ N 0 N \in\mathbb{N}_0 N ∈ N 0 J N J_N J N ( a d ) (a_d) ( a d ) { 0 , … , N } \{0,\ldots,N\} { 0 , … , N } N + 1 N+1 N + 1
( J N f ) ( d ) ≔ a d ( f ( d ) − f ( d + 1 ) ) + a d − 1 ( f ( d ) − f ( d − 1 ) ) , 0 ≤ d ≤ N , (J_Nf)(d) \coloneqq a_d\bigl(f(d)-f(d+1)\bigr)+a_{d-1}\bigl(f(d)-f(d-1)\bigr), \qquad0\le d\le N, ( J N f ) ( d ) : = a d ( f ( d ) − f ( d + 1 ) ) + a d − 1 ( f ( d ) − f ( d − 1 ) ) , 0 ≤ d ≤ N , with f ( N + 1 ) = 0 f(N+1)=0 f ( N + 1 ) = 0 a − 1 = 0 a_{-1}=0 a − 1 = 0 K N K_N K N K N ( i , j ) ≔ ∑ m = max ( i , j ) N 1 a m K_N(i,j) \coloneqq\sum_{m=\max(i,j)}^N \frac{1}{a_m} K N ( i , j ) : = ∑ m = m a x ( i , j ) N a m 1 0 ≤ i , j ≤ N 0\le i,j\le N 0 ≤ i , j ≤ N T T T κ 1 ≥ κ 2 ≥ ⋯ \kappa_1\ge\kappa_2\ge\cdots κ 1 ≥ κ 2 ≥ ⋯ det ( Id + λ T ) = ∏ i ( 1 + λ κ i ) \det(\operatorname{Id}+\lambda T)=\prod_i(1+\lambda\kappa_i) det ( Id + λ T ) = ∏ i ( 1 + λ κ i )
Theorem 6.17 1. The series ∑ m ≥ 0 1 a m ⟨ ⋅ , u m ⟩ u m \sum_{m\ge0}\frac{1}{a_m}\langle\cdot,u_m\rangle u_m ∑ m ≥ 0 a m 1 ⟨ ⋅ , u m ⟩ u m K K K K K K
Tr K = ∑ m ≥ 0 m + 1 a m = ( 1 − τ 2 ) ∑ m ≥ 1 m τ m − 1 1 − τ m = ∑ n ≥ 1 1 μ n = ϱ ∞ ′ ( 0 ) . \operatorname{Tr}K=\sum_{m\ge0}\frac{m+1}{a_m}=(1-\tau^2)\sum_{m\ge1}\frac{m\tau^{m-1}}{1-\tau^m}=\sum_{n\ge1}\frac{1}{\mu_n}=\varrho_\infty'(0). Tr K = m ≥ 0 ∑ a m m + 1 = ( 1 − τ 2 ) m ≥ 1 ∑ 1 − τ m m τ m − 1 = n ≥ 1 ∑ μ n 1 = ϱ ∞ ′ ( 0 ) . For any N ∈ N 0 N\in\mathbb{N}_0 N ∈ N 0 det J N = ∏ m = 0 N a m \det J_N=\prod_{m=0}^N a_m det J N = ∏ m = 0 N a m J N − 1 = K N J_N^{-1}=K_N J N − 1 = K N λ ∈ C \lambda\in\mathbb{C} λ ∈ C ϱ N + 1 ( λ ) = det ( I + λ K N ) \varrho_{N+1}(\lambda)=\det(I+\lambda K_N) ϱ N + 1 ( λ ) = det ( I + λ K N )
For any λ ∈ C \lambda\in\mathbb{C} λ ∈ C det ( Id + λ K ) = ϱ ∞ ( λ ) \det(\operatorname{Id}+\lambda K)=\varrho_\infty(\lambda) det ( Id + λ K ) = ϱ ∞ ( λ )
The eigenvalues of K K K 1 μ n \frac{1}{\mu_n} μ n 1 n ≥ 1 n\ge1 n ≥ 1 n ≥ 1 n\ge1 n ≥ 1 v n ≔ ( ϱ m ( − μ n ) ) m ≥ 0 v_n\coloneqq\bigl(\varrho_m(-\mu_n)\bigr)_{m\ge0} v n : = ( ϱ m ( − μ n ) ) m ≥ 0 1 μ n \frac{1}{\mu_n} μ n 1
∑ m ≥ 0 ϱ m ( − μ n ) y m = ( τ 1 − n y ; τ ) n − 1 ( τ y ; τ ) n , ∥ v n ∥ ℓ N 0 2 2 = 1 ϖ n , where ϖ n ≔ τ n ( n − 1 ) ( 1 − τ 2 n ) . \sum_{m\ge0}\varrho_m(-\mu_n)y^m=\frac{(\tau^{1-n}y;\tau)_{n-1}}{(\tau y;\tau)_n}, \qquad\lVert v_n\rVert_{\ell^2_{\mathbb{N}_0}}^2=\frac{1}{\varpi_n}, \qquad\text{where }\varpi_n\coloneqq\tau^{n(n-1)}(1-\tau^{2n}). m ≥ 0 ∑ ϱ m ( − μ n ) y m = ( τ y ; τ ) n ( τ 1 − n y ; τ ) n − 1 , ∥ v n ∥ ℓ N 0 2 2 = ϖ n 1 , where ϖ n : = τ n ( n − 1 ) ( 1 − τ 2 n ) . Consequently, ( ϖ n v n ) n ≥ 1 (\sqrt{\varpi_n}v_n)_{n\ge1} ( ϖ n v n ) n ≥ 1 ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 i , j ∈ N 0 i,j\in\mathbb{N}_0 i , j ∈ N 0
∑ n ≥ 1 ϖ n ϱ i ( − μ n ) ϱ j ( − μ n ) = δ i j . (72) \sum_{n\ge1}\varpi_n\varrho_i(-\mu_n)\varrho_j(-\mu_n)=\delta_{ij}. \tag*{(72)} n ≥ 1 ∑ ϖ n ϱ i ( − μ n ) ϱ j ( − μ n ) = δ ij . (72) Proof. 1. The operator ⟨ ⋅ , u m ⟩ u m \langle\cdot,u_m\rangle u_m ⟨ ⋅ , u m ⟩ u m ∥ u m ∥ 2 = m + 1 \lVert u_m\rVert^2=m+1 ∥ u m ∥ 2 = m + 1 m + 1 a m ≤ ( 1 + τ ) ( m + 1 ) τ m \frac{m+1}{a_m}\le(1+\tau)(m+1)\tau^m a m m + 1 ≤ ( 1 + τ ) ( m + 1 ) τ m ∑ m 1 a m u m ( i ) u m ( j ) = K ( i , j ) \sum_m\frac{1}{a_m}u_m(i)u_m(j)=K(i,j) ∑ m a m 1 u m ( i ) u m ( j ) = K ( i , j ) ⟨ K f , f ⟩ = ∑ m 1 a m ∣ ⟨ f , u m ⟩ ∣ 2 = 0 \langle Kf,f\rangle=\sum_m\frac{1}{a_m}|\langle f,u_m\rangle|^2=0 ⟨ K f , f ⟩ = ∑ m a m 1 ∣ ⟨ f , u m ⟩ ∣ 2 = 0 ⟨ f , u m ⟩ = 0 \langle f,u_m\rangle=0 ⟨ f , u m ⟩ = 0 m m m f = 0 f=0 f = 0 δ m = u m − u m − 1 \delta_m=u_m-u_{m-1} δ m = u m − u m − 1 Tr K = ∑ m m + 1 a m = ( 1 − τ 2 ) ∑ m ≥ 1 m τ m − 1 1 − τ m \operatorname{Tr}K=\sum_m\frac{m+1}{a_m}=(1-\tau^2)\sum_{m\ge1}\frac{m\tau^{m-1}}{1-\tau^m} Tr K = ∑ m a m m + 1 = ( 1 − τ 2 ) ∑ m ≥ 1 1 − τ m m τ m − 1 ϱ ∞ ′ ( 0 ) \varrho_\infty'(0) ϱ ∞ ′ ( 0 ) = ( 1 − τ 2 ) ∑ n ≥ 1 1 1 − τ n = ( 1 − τ 2 ) ∑ n ≥ 1 τ n − 1 ( 1 − τ n ) 2 =(1-\tau^2)\sum_{n\ge1}\frac{1}{1-\tau^n}=(1-\tau^2)\sum_{n\ge1}\frac{\tau^{n-1}}{(1-\tau^n)^2} = ( 1 − τ 2 ) ∑ n ≥ 1 1 − τ n 1 = ( 1 − τ 2 ) ∑ n ≥ 1 ( 1 − τ n ) 2 τ n − 1 ∑ m ≥ 1 m τ m 1 − τ m = ∑ m , i ≥ 1 m τ m i = ∑ i ≥ 1 τ i ( 1 − τ i ) 2 \sum_{m\ge1}\frac{m\tau^m}{1-\tau^m}=\sum_{m,i\ge1}m\tau^{mi}=\sum_{i\ge1}\frac{\tau^i}{(1-\tau^i)^2} ∑ m ≥ 1 1 − τ m m τ m = ∑ m , i ≥ 1 m τ mi = ∑ i ≥ 1 ( 1 − τ i ) 2 τ i
For 0 ≤ n ≤ N 0\le n\le N 0 ≤ n ≤ N n + 1 n+1 n + 1 J N + λ J_N+\lambda J N + λ J n + λ J_n+\lambda J n + λ J N J_N J N n n n a n + a n − 1 a_n+a_{n-1} a n + a n − 1 Δ n ≔ det ( J n + λ ) \Delta_n\coloneqq\det(J_n+\lambda) Δ n : = det ( J n + λ ) Δ n = ( a n + a n − 1 + λ ) Δ n − 1 − a n − 1 2 Δ n − 2 \Delta_n=(a_n+a_{n-1}+\lambda)\Delta_{n-1}-a_{n-1}^2\Delta_{n-2} Δ n = ( a n + a n − 1 + λ ) Δ n − 1 − a n − 1 2 Δ n − 2 Δ − 1 = 1 \Delta_{-1}=1 Δ − 1 = 1 Δ − 2 = 0 \Delta_{-2}=0 Δ − 2 = 0 (62) , we have a n ϱ n + 1 = ( a n + a n − 1 + λ ) ϱ n − a n − 1 ϱ n − 1 a_n\varrho_{n+1}=(a_n+a_{n-1}+\lambda)\varrho_n-a_{n-1}\varrho_{n-1} a n ϱ n + 1 = ( a n + a n − 1 + λ ) ϱ n − a n − 1 ϱ n − 1 P n ≔ ( ∏ m = 0 n a m ) ϱ n + 1 ( λ ) P_n\coloneqq\left(\prod_{m=0}^n a_m\right)\varrho_{n+1}(\lambda) P n : = ( ∏ m = 0 n a m ) ϱ n + 1 ( λ ) P − 1 = ϱ 0 = 1 P_{-1}=\varrho_0=1 P − 1 = ϱ 0 = 1 P 0 = a 0 + λ = Δ 0 P_0=a_0+\lambda=\Delta_0 P 0 = a 0 + λ = Δ 0 det ( J N + λ ) = ( ∏ m = 0 N a m ) ϱ N + 1 ( λ ) \det(J_N+\lambda)=\left(\prod_{m=0}^N a_m\right)\varrho_{N+1}(\lambda) det ( J N + λ ) = ( ∏ m = 0 N a m ) ϱ N + 1 ( λ ) λ = 0 \lambda=0 λ = 0 det J N = ∏ m = 0 N a m \det J_N=\prod_{m=0}^N a_m det J N = ∏ m = 0 N a m ϱ N + 1 ( 0 ) = 1 \varrho_{N+1}(0)=1 ϱ N + 1 ( 0 ) = 1 j j j g ( i ) ≔ K N ( i , j ) g(i)\coloneqq K_N(i,j) g ( i ) : = K N ( i , j ) 0 ≤ i ≤ N 0\le i\le N 0 ≤ i ≤ N g ( N + 1 ) = 0 g(N+1)=0 g ( N + 1 ) = 0 g ( i ) − g ( i + 1 ) = 1 a i 1 i ≥ j g(i)-g(i+1)=\frac{1}{a_i}\mathbf{1}_{i\ge j} g ( i ) − g ( i + 1 ) = a i 1 1 i ≥ j ( J N g ) ( i ) = 1 i ≥ j − 1 i − 1 ≥ j = δ i j (J_Ng)(i)=\mathbf{1}_{i\ge j}-\mathbf{1}_{i-1\ge j}=\delta_{ij} ( J N g ) ( i ) = 1 i ≥ j − 1 i − 1 ≥ j = δ ij J N − 1 = K N J_N^{-1}=K_N J N − 1 = K N ϱ N + 1 ( λ ) = det ( J N + λ ) det J N = det ( I + λ K N ) \varrho_{N+1}(\lambda)=\frac{\det(J_N+\lambda)}{\det J_N}=\det(I+\lambda K_N) ϱ N + 1 ( λ ) = d e t J N d e t ( J N + λ ) = det ( I + λ K N )
We identify K N K_N K N ∑ m = 0 N 1 a m ⟨ ⋅ , u m ⟩ u m \sum_{m=0}^{N}\frac{1}{a_m}\langle\cdot,u_m\rangle u_m ∑ m = 0 N a m 1 ⟨ ⋅ , u m ⟩ u m ℓ N 0 2 \ell^2_{N_0} ℓ N 0 2 0 ≤ K N ≤ K N + 1 ≤ K 0\le K_N\le K_{N+1}\le K 0 ≤ K N ≤ K N + 1 ≤ K ∥ K − K N ∥ ≤ Tr ( K − K N ) = ∑ m > N m + 1 a m → 0 \lVert K-K_N\rVert\le\operatorname{Tr}(K-K_N)=\sum_{m>N}\frac{m+1}{a_m}\to0 ∥ K − K N ∥ ≤ Tr ( K − K N ) = ∑ m > N a m m + 1 → 0 [62 Theorem XIII.1] , applied to − K N -K_N − K N − K -K − K i i i κ i ( K N ) \kappa_i(K_N) κ i ( K N ) κ i ( K N ) ≤ κ i ( K ) \kappa_i(K_N)\le\kappa_i(K) κ i ( K N ) ≤ κ i ( K ) ∣ κ i ( K N ) − κ i ( K ) ∣ ≤ ∥ K − K N ∥ \lvert\kappa_i(K_N)-\kappa_i(K)\rvert\le\lVert K-K_N\rVert ∣ κ i ( K N ) − κ i ( K )∣ ≤ ∥ K − K N ∥ λ ≥ 0 \lambda\ge0 λ ≥ 0 0 ≤ log ( 1 + λ κ i ( K N ) ) ≤ λ κ i ( K ) 0\le\log(1+\lambda\kappa_i(K_N))\le\lambda\kappa_i(K) 0 ≤ log ( 1 + λ κ i ( K N )) ≤ λ κ i ( K ) ∑ i κ i ( K ) = Tr K < ∞ \sum_i\kappa_i(K)=\operatorname{Tr}K<\infty ∑ i κ i ( K ) = Tr K < ∞
log det ( Id + λ K ) = lim N → ∞ ∑ i log ( 1 + λ κ i ( K N ) ) = lim N → ∞ log ϱ N + 1 ( λ ) = log ϱ ∞ ( λ ) . \log\det(\operatorname{Id}+\lambda K)=\lim_{N\to\infty}\sum_i\log(1+\lambda\kappa_i(K_N))=\lim_{N\to\infty}\log\varrho_{N+1}(\lambda)=\log\varrho_\infty(\lambda). log det ( Id + λ K ) = N → ∞ lim i ∑ log ( 1 + λ κ i ( K N )) = N → ∞ lim log ϱ N + 1 ( λ ) = log ϱ ∞ ( λ ) . Both λ ↦ ∏ i ( 1 + λ κ i ( K ) ) \lambda\mapsto\prod_i(1+\lambda\kappa_i(K)) λ ↦ ∏ i ( 1 + λ κ i ( K )) ϱ ∞ \varrho_\infty ϱ ∞ C \mathbb{C} C
By the third part and (6.22), we have ∏ i ( 1 + λ κ i ( K ) ) = ∏ n ( 1 + λ μ n ) \prod_i(1+\lambda\kappa_i(K))=\prod_n(1+\frac{\lambda}{\mu_n}) ∏ i ( 1 + λ κ i ( K )) = ∏ n ( 1 + μ n λ ) λ ∈ C \lambda\in\mathbb{C} λ ∈ C μ n \mu_n μ n K K K 1 μ n \frac{1}{\mu_n} μ n 1 f ∈ ℓ N 0 2 f\in\ell^2_{N_0} f ∈ ℓ N 0 2 m ∈ N 0 m\in\mathbb{N}_0 m ∈ N 0 ( K f ) ( m ) = ∑ l ≥ m 1 a l ∑ i ≤ l f ( i ) (Kf)(m)=\sum_{l\ge m}\frac{1}{a_l}\sum_{i\le l}f(i) ( K f ) ( m ) = ∑ l ≥ m a l 1 ∑ i ≤ l f ( i ) ( K f ) ( m + 1 ) − ( K f ) ( m ) = − 1 a m ∑ i ≤ m f ( i ) (Kf)(m+1)-(Kf)(m)=-\frac{1}{a_m}\sum_{i\le m}f(i) ( K f ) ( m + 1 ) − ( K f ) ( m ) = − a m 1 ∑ i ≤ m f ( i ) K f = 1 μ n f Kf=\frac{1}{\mu_n}f K f = μ n 1 f f ≠ 0 f\ne0 f = 0 f ( m + 1 ) − f ( m ) = − μ n a m ∑ i ≤ m f ( i ) f(m+1)-f(m)=-\frac{\mu_n}{a_m}\sum_{i\le m}f(i) f ( m + 1 ) − f ( m ) = − a m μ n ∑ i ≤ m f ( i ) m m m (62) with λ = − μ n \lambda=-\mu_n λ = − μ n f = f ( 0 ) v n f=f(0)v_n f = f ( 0 ) v n v n v_n v n
For λ = − μ n \lambda=-\mu_n λ = − μ n z = τ − n z=\tau^{-n} z = τ − n (65) , since τ − n + τ n = 2 + ( 1 − τ n ) 2 τ n = 2 + ( 1 − τ 2 ) μ n τ \tau^{-n}+\tau^n=2+\frac{(1-\tau^n)^2}{\tau^n}=2+\frac{(1-\tau^2)\mu_n}{\tau} τ − n + τ n = 2 + τ n ( 1 − τ n ) 2 = 2 + τ ( 1 − τ 2 ) μ n ( τ z y ; τ ) ∞ = ( τ 1 − n y ; τ ) n − 1 ( y ; τ ) ∞ (\tau z y;\tau)_\infty=(\tau^{1-n}y;\tau)_{n-1}(y;\tau)_\infty ( τ z y ; τ ) ∞ = ( τ 1 − n y ; τ ) n − 1 ( y ; τ ) ∞ ( τ z − 1 y ; τ ) ∞ = ( τ n + 1 y ; τ ) ∞ (\tau z^{-1}y;\tau)_\infty=(\tau^{n+1}y;\tau)_\infty ( τ z − 1 y ; τ ) ∞ = ( τ n + 1 y ; τ ) ∞ (67) gives the generating function of v n v_n v n τ − 1 , … , τ − n \tau^{-1},\ldots,\tau^{-n} τ − 1 , … , τ − n τ − 1 \tau^{-1} τ − 1 ϱ m ( − μ n ) = c n τ m + O ( τ 2 m ) \varrho_m(-\mu_n)=c_n\tau^m+O(\tau^{2m}) ϱ m ( − μ n ) = c n τ m + O ( τ 2 m )
c n = ( τ − n ; τ ) n − 1 ( τ ; τ ) n − 1 = ( − 1 ) n − 1 τ − n ( n + 1 ) 2 1 − τ n 1 − τ . c_n=\frac{(\tau^{-n};\tau)_{n-1}}{(\tau;\tau)_{n-1}}=(-1)^{n-1}\tau^{-\frac{n(n+1)}{2}}\frac{1-\tau^n}{1-\tau}. c n = ( τ ; τ ) n − 1 ( τ − n ; τ ) n − 1 = ( − 1 ) n − 1 τ − 2 n ( n + 1 ) 1 − τ 1 − τ n . For λ , λ ′ ∈ C \lambda,\lambda'\in\mathbb{C} λ , λ ′ ∈ C T m = d e f a m ( ϱ m + 1 ( λ ) ϱ m ( λ ′ ) − ϱ m ( λ ) ϱ m + 1 ( λ ′ ) ) T_m\mathrel{\overset{\mathrm{def}}{=}}a_m\bigl(\varrho_{m+1}(\lambda)\varrho_m(\lambda')-\varrho_m(\lambda)\varrho_{m+1}(\lambda')\bigr) T m = def a m ( ϱ m + 1 ( λ ) ϱ m ( λ ′ ) − ϱ m ( λ ) ϱ m + 1 ( λ ′ ) ) T − 1 = 0 T_{-1}=0 T − 1 = 0 (62) gives T m − T m − 1 = ( λ − λ ′ ) ϱ m ( λ ) ϱ m ( λ ′ ) T_m-T_{m-1}=(\lambda-\lambda')\varrho_m(\lambda)\varrho_m(\lambda') T m − T m − 1 = ( λ − λ ′ ) ϱ m ( λ ) ϱ m ( λ ′ ) 0 ≤ m ≤ N 0\le m\le N 0 ≤ m ≤ N λ − λ ′ \lambda-\lambda' λ − λ ′ λ ′ → λ \lambda'\to\lambda λ ′ → λ [68 Theorem 3.2.2]
∑ m = 0 N ϱ m ( λ ) 2 = a N ( ϱ N + 1 ′ ( λ ) ϱ N ( λ ) − ϱ N ′ ( λ ) ϱ N + 1 ( λ ) ) . \sum_{m=0}^{N}\varrho_m(\lambda)^2=a_N\bigl(\varrho'_{N+1}(\lambda)\varrho_N(\lambda)-\varrho'_N(\lambda)\varrho_{N+1}(\lambda)\bigr). m = 0 ∑ N ϱ m ( λ ) 2 = a N ( ϱ N + 1 ′ ( λ ) ϱ N ( λ ) − ϱ N ′ ( λ ) ϱ N + 1 ( λ ) ) . Let λ = − μ n \lambda=-\mu_n λ = − μ n N → ∞ N\to\infty N → ∞ Theorem 6.15 , ϱ N ′ → ϱ ∞ ′ \varrho'_N\to\varrho'_\infty ϱ N ′ → ϱ ∞ ′ a N ϱ N ( − μ n ) → c n 1 − τ 2 a_N\varrho_N(-\mu_n)\to\frac{c_n}{1-\tau^2} a N ϱ N ( − μ n ) → 1 − τ 2 c n a N ϱ N + 1 ( − μ n ) → τ c n 1 − τ 2 a_N\varrho_{N+1}(-\mu_n)\to\frac{\tau c_n}{1-\tau^2} a N ϱ N + 1 ( − μ n ) → 1 − τ 2 τ c n ∥ v n ∥ 2 = 1 + τ 1 + τ 2 ϱ ∞ ′ ( − μ n ) \lVert v_n\rVert^2=\frac{1+\tau}{1+\tau^2}\varrho'_\infty(-\mu_n) ∥ v n ∥ 2 = 1 + τ 2 1 + τ ϱ ∞ ′ ( − μ n ) 1 − τ n z 1-\tau^n z 1 − τ n z ( τ z ; τ ) ∞ (\tau z;\tau)_\infty ( τ z ; τ ) ∞ z = τ − n z=\tau^{-n} z = τ − n d z d λ = − τ ( 1 − z − 2 ) 1 − τ 2 \frac{dz}{d\lambda}=-\frac{\tau(1-z^{-2})}{1-\tau^2} d λ d z = − 1 − τ 2 τ ( 1 − z − 2 ) λ \lambda λ − μ n -\mu_n − μ n τ n − 1 ( 1 − τ 2 ) 1 − τ 2 n \frac{\tau^{n-1}(1-\tau^2)}{1-\tau^{2n}} 1 − τ 2 n τ n − 1 ( 1 − τ 2 ) ( − 1 ) n − 1 τ − n ( n − 1 ) 2 ( τ ; τ ) n − 1 ( τ ; τ ) ∞ ( τ n + 1 ; τ ) ∞ ( τ ; τ ) ∞ = ( − 1 ) n − 1 τ − n ( n − 1 ) 2 1 − τ n \frac{(-1)^{n-1}\tau^{-\frac{n(n-1)}{2}}(\tau;\tau)_{n-1}(\tau;\tau)_\infty(\tau^{n+1};\tau)_\infty}{(\tau;\tau)_\infty}= \frac{(-1)^{n-1}\tau^{-\frac{n(n-1)}{2}}}{1-\tau^n} ( τ ; τ ) ∞ ( − 1 ) n − 1 τ − 2 n ( n − 1 ) ( τ ; τ ) n − 1 ( τ ; τ ) ∞ ( τ n + 1 ; τ ) ∞ = 1 − τ n ( − 1 ) n − 1 τ − 2 n ( n − 1 )
ϱ ∞ ′ ( − μ n ) = ( − 1 ) n − 1 τ − ( n − 1 ) ( n − 2 ) 2 1 − τ 2 ( 1 − τ n ) ( 1 − τ 2 n ) . (73) \varrho'_\infty(-\mu_n)=(-1)^{n-1}\tau^{-\frac{(n-1)(n-2)}{2}}\frac{1-\tau^2}{(1-\tau^n)(1-\tau^{2n})}. \tag*{(73)} ϱ ∞ ′ ( − μ n ) = ( − 1 ) n − 1 τ − 2 ( n − 1 ) ( n − 2 ) ( 1 − τ n ) ( 1 − τ 2 n ) 1 − τ 2 . (73) and ∥ v n ∥ 2 = τ − n ( n − 1 ) 1 − τ 2 n = 1 ϖ n \lVert v_n\rVert^2=\frac{\tau^{-n(n-1)}}{1-\tau^{2n}}=\frac{1}{\varpi_n} ∥ v n ∥ 2 = 1 − τ 2 n τ − n ( n − 1 ) = ϖ n 1 K K K C v n \mathbb{C}v_n C v n ( ϖ n v n ) (\sqrt{\varpi_n}v_n) ( ϖ n v n ) ⟨ δ i , δ j ⟩ = ∑ n ϖ n ⟨ δ i , v n ⟩ ⟨ v n , δ j ⟩ \langle\delta_i,\delta_j\rangle=\sum_n\varpi_n\langle\delta_i,v_n\rangle\langle v_n,\delta_j\rangle ⟨ δ i , δ j ⟩ = ∑ n ϖ n ⟨ δ i , v n ⟩ ⟨ v n , δ j ⟩
Corollary 6.19
The zeros of ϱ ∞ \varrho_\infty ϱ ∞ − μ n -\mu_n − μ n n ≥ 1 n\ge1 n ≥ 1 ϱ ∞ \varrho_\infty ϱ ∞ 0 0 0 max ∣ λ ∣ = R ∣ ϱ ∞ ( λ ) ∣ = ϱ ∞ ( R ) \max_{|\lambda|=R}|\varrho_\infty(\lambda)|=\varrho_\infty(R) max ∣ λ ∣ = R ∣ ϱ ∞ ( λ ) ∣ = ϱ ∞ ( R ) log ϱ ∞ ( R ) = ( log R ) 2 2 log ( 1 / τ ) + O ( log R ) \log\varrho_\infty(R)=\frac{(\log R)^2}{2\log(1/\tau)}+O(\log R) log ϱ ∞ ( R ) = 2 l o g ( 1/ τ ) ( l o g R ) 2 + O ( log R ) R → ∞ R\to\infty R → ∞
Let W n W_n W n W n ( cos θ ) = sin ( ( n + 1 2 ) θ ) sin θ 2 W_n(\cos\theta)=\frac{\sin((n+\frac12)\theta)}{\sin\frac{\theta}{2}} W n ( cos θ ) = s i n 2 θ s i n (( n + 2 1 ) θ ) [27 (18.5.4)] . For any λ ∈ C \lambda\in\mathbb{C} λ ∈ C x = 1 − ( 1 − τ 2 ) λ 2 τ x=1-\frac{(1-\tau^2)\lambda}{2\tau} x = 1 − 2 τ ( 1 − τ 2 ) λ
ϱ ∞ ( λ ) = 1 ( τ ; τ ) ∞ 3 ∑ n ≥ 0 ( − 1 ) n τ n ( n + 1 ) 2 W n ( x ) = θ 1 ( u , τ 1 / 2 ) θ 1 ′ ( 0 , τ 1 / 2 ) sin u , \varrho_\infty(\lambda)=\frac{1}{(\tau;\tau)_\infty^3}\sum_{n\ge0}(-1)^n\tau^{\frac{n(n+1)}{2}}W_n(x)=\frac{\theta_1(u,\tau^{1/2})}{\theta_1'(0,\tau^{1/2})\sin u}, ϱ ∞ ( λ ) = ( τ ; τ ) ∞ 3 1 n ≥ 0 ∑ ( − 1 ) n τ 2 n ( n + 1 ) W n ( x ) = θ 1 ′ ( 0 , τ 1/2 ) sin u θ 1 ( u , τ 1/2 ) , where u ∈ C u\in\mathbb{C} u ∈ C sin 2 u = ( 1 − τ 2 ) λ 4 τ \sin^2u=\frac{(1-\tau^2)\lambda}{4\tau} sin 2 u = 4 τ ( 1 − τ 2 ) λ θ 1 \theta_1 θ 1 [27 (20.2.1)] , the last expression being 1 1 1 u = 0 u=0 u = 0
The function f ( z ) = ( τ z , τ z − 1 ; τ ) ∞ ( τ ; τ ) ∞ f(z)=\frac{(\tau z,\tau z^{-1};\tau)_\infty}{(\tau;\tau)_\infty} f ( z ) = ( τ ; τ ) ∞ ( τ z , τ z − 1 ; τ ) ∞ ( 1 − τ z ) f ( τ z ) = ( 1 − z − 1 ) f ( z ) (1-\tau z)f(\tau z)=(1-z^{-1})f(z) ( 1 − τ z ) f ( τ z ) = ( 1 − z − 1 ) f ( z ) z ∈ C ∖ { 0 } z\in\mathbb{C}\setminus\{0\} z ∈ C ∖ { 0 } C ∖ { 0 } \mathbb{C}\setminus\{0\} C ∖ { 0 } f f f
Write ϱ ∞ ( λ ) = ∑ n ≥ 0 c n λ n \varrho_\infty(\lambda)=\sum_{n\ge0}c_n\lambda^n ϱ ∞ ( λ ) = ∑ n ≥ 0 c n λ n n ≥ 1 n\ge1 n ≥ 1 c n > 0 c_n>0 c n > 0
c n = ( 1 − τ 2 ) n ∑ 1 ≤ i 1 < ⋯ < i n ∏ l = 1 n τ i l − 1 ( 1 − τ i l ) 2 c_n=(1-\tau^2)^n\sum_{1\le i_1<\cdots<i_n}\prod_{l=1}^n\frac{\tau^{i_l-1}}{(1-\tau^{i_l})^2} c n = ( 1 − τ 2 ) n 1 ≤ i 1 < ⋯ < i n ∑ l = 1 ∏ n ( 1 − τ i l ) 2 τ i l − 1 = 1 ( τ ; τ ) ∞ 3 ( τ 2 − 1 τ ) n ∑ m ≥ n ( − 1 ) m 2 m + 1 2 n + 1 ( m + n 2 n ) τ m ( m + 1 ) 2 . =\frac{1}{(\tau;\tau)_\infty^3}\left(\frac{\tau^2-1}{\tau}\right)^n\sum_{m\ge n}(-1)^m\frac{2m+1}{2n+1}\binom{m+n}{2n}\tau^{\frac{m(m+1)}{2}}. = ( τ ; τ ) ∞ 3 1 ( τ τ 2 − 1 ) n m ≥ n ∑ ( − 1 ) m 2 n + 1 2 m + 1 ( 2 n m + n ) τ 2 m ( m + 1 ) . For any integer k ≥ 1 k\ge1 k ≥ 1 E 2 k ( τ ) ≔ 1 − 4 k B 2 k ∑ N ≥ 1 σ 2 k − 1 ( N ) τ N E_{2k}(\tau)\coloneqq1-\frac{4k}{B_{2k}}\sum_{N\ge1}\sigma_{2k-1}(N)\tau^N E 2 k ( τ ) : = 1 − B 2 k 4 k ∑ N ≥ 1 σ 2 k − 1 ( N ) τ N 2 k 2k 2 k τ \tau τ B 2 k B_{2k} B 2 k [27 Section 24.2] and σ 2 k − 1 ( N ) = ∑ e ∣ N e 2 k − 1 \sigma_{2k-1}(N)=\sum_{e\mid N}e^{2k-1} σ 2 k − 1 ( N ) = ∑ e ∣ N e 2 k − 1 u ∈ C u\in\mathbb{C} u ∈ C ∣ u ∣ < min ( π , 1 2 log 1 τ ) |u|<\min(\pi,\frac12\log\frac1\tau) ∣ u ∣ < min ( π , 2 1 log τ 1 )
sin u u ϱ ∞ ( 4 τ sin 2 u 1 − τ 2 ) = exp ( ∑ k ≥ 1 ( − 1 ) k 2 2 k − 1 B 2 k k ( 2 k ) ! E 2 k ( τ ) u 2 k ) . (74) \frac{\sin u}{u}\varrho_\infty\left(\frac{4\tau\sin^2u}{1-\tau^2}\right)=\exp\left(\sum_{k\ge1}\frac{(-1)^k2^{2k-1}B_{2k}}{k(2k)!}E_{2k}(\tau)u^{2k}\right). \tag*{(74)} u sin u ϱ ∞ ( 1 − τ 2 4 τ sin 2 u ) = exp ( k ≥ 1 ∑ k ( 2 k )! ( − 1 ) k 2 2 k − 1 B 2 k E 2 k ( τ ) u 2 k ) . (74) Consequently, for any n ≥ 1 n \ge1 n ≥ 1 ( 4 τ 1 − τ 2 ) n c n \left(\frac{4\tau}{1-\tau^2}\right)^n c_n ( 1 − τ 2 4 τ ) n c n E 2 ( τ ) E_2(\tau) E 2 ( τ ) E 4 ( τ ) E_4(\tau) E 4 ( τ ) E 6 ( τ ) E_6(\tau) E 6 ( τ ) 0 , 2 , … , 2 n 0,2,\ldots,2n 0 , 2 , … , 2 n E 2 k E_{2k} E 2 k 2 k 2k 2 k
c 1 = 1 − τ 2 24 τ ( 1 − E 2 ) , c 2 = ( 1 − τ 2 4 τ ) 2 ( E 2 2 72 − E 2 12 − E 4 480 + 3 40 ) . c_1=\frac{1-\tau^2}{24\tau}(1-E_2),\qquad c_2=\left(\frac{1-\tau^2}{4\tau}\right)^2\left(\frac{E_2^2}{72}-\frac{E_2}{12}-\frac{E_4}{480}+\frac{3}{40}\right). c 1 = 24 τ 1 − τ 2 ( 1 − E 2 ) , c 2 = ( 4 τ 1 − τ 2 ) 2 ( 72 E 2 2 − 12 E 2 − 480 E 4 + 40 3 ) . Proof: 1. The first assertion follows from (6.22), since the μ n \mu_n μ n c n > 0 c_n>0 c n > 0 n n n max ∣ λ ∣ = R ∣ ϱ ∞ ( λ ) ∣ = ϱ ∞ ( R ) \max_{|\lambda|=R}|\varrho_\infty(\lambda)|=\varrho_\infty(R) max ∣ λ ∣ = R ∣ ϱ ∞ ( λ ) ∣ = ϱ ∞ ( R ) ℓ = log 1 τ \ell=\log\frac{1}{\tau} ℓ = log τ 1 (66) , there exists C ≥ 1 C\ge1 C ≥ 1 C − 1 τ − n ≤ μ n ≤ C τ − n C^{-1}\tau^{-n}\le\mu_n\le C\tau^{-n} C − 1 τ − n ≤ μ n ≤ C τ − n n ≥ 1 n\ge1 n ≥ 1 R ≥ 1 R\ge1 R ≥ 1 N N N log R ℓ \frac{\log R}{\ell} ℓ l o g R n ≤ N n\le N n ≤ N log ( 1 + R μ n ) = log R − n ℓ + O ( 1 ) \log(1+\frac{R}{\mu_n})=\log R-n\ell+O(1) log ( 1 + μ n R ) = log R − n ℓ + O ( 1 ) n > N n>N n > N log ( 1 + R μ n ) ≤ R μ n ≤ C R τ n \log(1+\frac{R}{\mu_n})\le\frac{R}{\mu_n}\le CR\tau^n log ( 1 + μ n R ) ≤ μ n R ≤ C R τ n n > N n>N n > N O ( 1 ) O(1) O ( 1 )
log ϱ ∞ ( R ) = N log R − N ( N + 1 ) 2 ℓ + O ( N ) = ( log R ) 2 2 ℓ + O ( log R ) . \log\varrho_\infty(R)=N\log R-\frac{N(N+1)}{2}\ell+O(N)=\frac{(\log R)^2}{2\ell}+O(\log R). log ϱ ∞ ( R ) = N log R − 2 N ( N + 1 ) ℓ + O ( N ) = 2 ℓ ( log R ) 2 + O ( log R ) . In particular, ϱ ∞ \varrho_\infty ϱ ∞ 0 0 0
For n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 w ∈ C ∖ { 0 } w\in\mathbb{C}\setminus\{0\} w ∈ C ∖ { 0 } D n ( w ) ≔ ∑ ∣ i ∣ ≤ n w i D_n(w)\coloneqq\sum_{|i|\le n}w^i D n ( w ) : = ∑ ∣ i ∣ ≤ n w i D n ( e i θ ) = W n ( cos θ ) D_n(e^{i\theta})=W_n(\cos\theta) D n ( e i θ ) = W n ( cos θ ) D n ( w ) = W n ( w + w − 1 2 ) D_n(w)=W_n\left(\frac{w+w^{-1}}{2}\right) D n ( w ) = W n ( 2 w + w − 1 ) z ≠ 1 z\ne1 z = 1 [27 (17.8.1)] ,
∑ m ∈ Z ( − 1 ) m τ m ( m − 1 ) 2 z m = ( τ , z , τ z − 1 ; τ ) ∞ . \sum_{m\in\mathbb{Z}}(-1)^m\tau^{\frac{m(m-1)}{2}}z^m=(\tau,z,\tau z^{-1};\tau)_\infty. m ∈ Z ∑ ( − 1 ) m τ 2 m ( m − 1 ) z m = ( τ , z , τ z − 1 ; τ ) ∞ . Since ( z ; τ ) ∞ = ( 1 − z ) ( τ z ; τ ) ∞ (z;\tau)_\infty=(1-z)(\tau z;\tau)_\infty ( z ; τ ) ∞ = ( 1 − z ) ( τ z ; τ ) ∞
ϱ ∞ ( λ ) = 1 ( 1 − z ) ( τ ; τ ) ∞ 3 ∑ m ∈ Z ( − 1 ) m τ m ( m − 1 ) 2 z m . \varrho_\infty(\lambda)=\frac{1}{(1-z)(\tau;\tau)_\infty^3}\sum_{m\in\mathbb{Z}}(-1)^m\tau^{\frac{m(m-1)}{2}}z^m. ϱ ∞ ( λ ) = ( 1 − z ) ( τ ; τ ) ∞ 3 1 m ∈ Z ∑ ( − 1 ) m τ 2 m ( m − 1 ) z m . We group the terms m m m 1 − m 1-m 1 − m m ≥ 1 m\ge1 m ≥ 1 z m − z 1 − m 1 − z = − D m − 1 ( z ) \frac{z^m-z^{1-m}}{1-z}=-D_{m-1}(z) 1 − z z m − z 1 − m = − D m − 1 ( z ) n = m − 1 n=m-1 n = m − 1 z = 1 z=1 z = 1 z = e 2 i u z=e^{2iu} z = e 2 i u D n ( z ) = sin ( ( 2 n + 1 ) u ) sin u D_n(z)=\frac{\sin((2n+1)u)}{\sin u} D n ( z ) = s i n u s i n (( 2 n + 1 ) u ) sin 2 u = 2 − z − z − 1 4 = − ( 1 − τ 2 ) λ 4 τ \sin^2u=\frac{2-z-z^{-1}}{4}=-\frac{(1-\tau^2)\lambda}{4\tau} sin 2 u = 4 2 − z − z − 1 = − 4 τ ( 1 − τ 2 ) λ (65) . By [27 (20.2.1)] , we have
θ 1 ( u , τ 1 2 ) = 2 τ 1 8 ∑ n ≥ 0 ( − 1 ) n τ n ( n + 1 ) 2 sin ( ( 2 n + 1 ) u ) . \theta_1\left(u,\tau^{\frac12}\right)=2\tau^{\frac18}\sum_{n\ge0}(-1)^n\tau^{\frac{n(n+1)}{2}}\sin((2n+1)u). θ 1 ( u , τ 2 1 ) = 2 τ 8 1 n ≥ 0 ∑ ( − 1 ) n τ 2 n ( n + 1 ) sin (( 2 n + 1 ) u ) . Hence ϱ ∞ ( λ ) = θ 1 ( u , τ 1 2 ) 2 τ 1 8 ( τ ; τ ) ∞ 3 sin u \varrho_\infty(\lambda)=\frac{\theta_1(u,\tau^{\frac12})}{2\tau^{\frac18}(\tau;\tau)_\infty^3\sin u} ϱ ∞ ( λ ) = 2 τ 8 1 ( τ ; τ ) ∞ 3 s i n u θ 1 ( u , τ 2 1 ) u → 0 u\to0 u → 0 ϱ ∞ ( 0 ) = 1 \varrho_\infty(0)=1 ϱ ∞ ( 0 ) = 1 θ 1 ′ ( 0 , τ 1 2 ) = 2 τ 1 8 ( τ ; τ ) ∞ 3 \theta_1'\left(0,\tau^{\frac12}\right)=2\tau^{\frac18}(\tau;\tau)_\infty^3 θ 1 ′ ( 0 , τ 2 1 ) = 2 τ 8 1 ( τ ; τ ) ∞ 3 [27 (20.4.2)] , which gives the second formula.
We have f ( τ z ) = ( τ 2 z , z − 1 ; τ ) ∞ ( τ ; τ ) ∞ 2 f(\tau z)=\frac{(\tau^2z,z^{-1};\tau)_\infty}{(\tau;\tau)_\infty^2} f ( τ z ) = ( τ ; τ ) ∞ 2 ( τ 2 z , z − 1 ; τ ) ∞ ( τ z ; τ ) ∞ = ( 1 − τ z ) ( τ 2 z ; τ ) ∞ (\tau z;\tau)_\infty=(1-\tau z)(\tau^2z;\tau)_\infty ( τ z ; τ ) ∞ = ( 1 − τ z ) ( τ 2 z ; τ ) ∞ ( z − 1 ; τ ) ∞ = ( 1 − z − 1 ) ( τ z − 1 ; τ ) ∞ (z^{-1};\tau)_\infty=(1-z^{-1})(\tau z^{-1};\tau)_\infty ( z − 1 ; τ ) ∞ = ( 1 − z − 1 ) ( τ z − 1 ; τ ) ∞ g g g C ∖ { 0 } \mathbb{C}\setminus\{0\} C ∖ { 0 } z = τ − 1 z=\tau^{-1} z = τ − 1 g ( τ − 1 ) = 0 g(\tau^{-1})=0 g ( τ − 1 ) = 0 z = τ − k z=\tau^{-k} z = τ − k k ≥ 2 k\ge2 k ≥ 2 ( 1 − τ 1 − k ) g ( τ 1 − k ) = ( 1 − τ k ) g ( τ − k ) (1-\tau^{1-k})g(\tau^{1-k})=(1-\tau^k)g(\tau^{-k}) ( 1 − τ 1 − k ) g ( τ 1 − k ) = ( 1 − τ k ) g ( τ − k ) g ( τ − k ) = 0 g(\tau^{-k})=0 g ( τ − k ) = 0 k ≥ 1 k\ge1 k ≥ 1 z = τ k z=\tau^k z = τ k k ≥ 1 k\ge1 k ≥ 1 ( 1 − τ k + 1 ) g ( τ k + 1 ) = ( 1 − τ − k ) g ( τ k ) (1-\tau^{k+1})g(\tau^{k+1})=(1-\tau^{-k})g(\tau^k) ( 1 − τ k + 1 ) g ( τ k + 1 ) = ( 1 − τ − k ) g ( τ k ) z = 1 z=1 z = 1 g ( τ ) = 0 g(\tau)=0 g ( τ ) = 0 g ( τ k ) = 0 g(\tau^k)=0 g ( τ k ) = 0 k ≥ 1 k\ge1 k ≥ 1 f f f τ ± k \tau^{\pm k} τ ± k k ≥ 1 k\ge1 k ≥ 1 h = g f h=\frac{g}{f} h = f g C ∖ { 0 } \mathbb{C}\setminus\{0\} C ∖ { 0 } h ( τ z ) = h ( z ) h(\tau z)=h(z) h ( τ z ) = h ( z ) h h h ( C ∖ { 0 } ) / τ Z (\mathbb{C}\setminus\{0\})/\tau^{\mathbb{Z}} ( C ∖ { 0 }) / τ Z h h h
The first formula follows from the expansion of the absolutely convergent product (6.22), and it shows that c n > 0 c_n>0 c n > 0
W m ( x ) = ∑ j = 0 m 2 m + 1 2 j + 1 ( m + j 2 j ) ( 2 x − 2 ) j , where 2 x − 2 = − ( 1 − τ 2 ) λ τ . W_m(x)=\sum_{j=0}^{m}\frac{2m+1}{2j+1}\binom{m+j}{2j}(2x-2)^j,\qquad\text{where }2x-2=-\frac{(1-\tau^2)\lambda}{\tau}. W m ( x ) = j = 0 ∑ m 2 j + 1 2 m + 1 ( 2 j m + j ) ( 2 x − 2 ) j , where 2 x − 2 = − τ ( 1 − τ 2 ) λ . It holds for m = 0 m=0 m = 0 m = 1 m=1 m = 1 W 1 ( x ) = 2 x + 1 W_1(x)=2x+1 W 1 ( x ) = 2 x + 1 D m + 1 ( w ) = ( w + w − 1 ) D m ( w ) − D m − 1 ( w ) D_{m+1}(w)=(w+w^{-1})D_m(w)-D_{m-1}(w) D m + 1 ( w ) = ( w + w − 1 ) D m ( w ) − D m − 1 ( w ) ( 2 m + 3 ) ( m + (2m+3)(m+ ( 2 m + 3 ) ( m + j + 1 ) ( m + j ) = 2 ( 2 m + 1 ) ( m + j ) ( m + 1 − j ) + 2 j ( 2 m + 1 ) ( 2 j + 1 ) − ( 2 m − 1 ) ( m + 1 − j ) ( m − j ) j+1)(m+j)=2(2m+1)(m+j)(m+1-j)+2j(2m+1)(2j+1)-(2m-1)(m+1-j)(m-j) j + 1 ) ( m + j ) = 2 ( 2 m + 1 ) ( m + j ) ( m + 1 − j ) + 2 j ( 2 m + 1 ) ( 2 j + 1 ) − ( 2 m − 1 ) ( m + 1 − j ) ( m − j ) m m m j j j ( 2 m + 1 2 j + 1 ) ( m + j 2 j ) ≤ ( 2 m + 1 ) 4 m \binom{2m+1}{2j+1}\binom{m+j}{2j}\le(2m+1)4^m ( 2 j + 1 2 m + 1 ) ( 2 j m + j ) ≤ ( 2 m + 1 ) 4 m τ m ( m + 1 ) 2 \frac{\tau^{m(m+1)}}{2} 2 τ m ( m + 1 )
Let ∣ u ∣ < min ( π , 1 2 log 1 τ ) |u|<\min(\pi,\frac{1}{2}\log\frac{1}{\tau}) ∣ u ∣ < min ( π , 2 1 log τ 1 ) z = e 2 i u z=e^{2iu} z = e 2 i u ∣ τ n z ± 1 ∣ < 1 |\tau^n z^{\pm1}|<1 ∣ τ n z ± 1 ∣ < 1 n ≥ 1 n\ge1 n ≥ 1
log ϱ ∞ ( λ ) = − ∑ n ≥ 1 ∑ m ≥ 1 τ n m m ( z m + z − m − 2 ) = ∑ m ≥ 1 4 sin 2 ( m u ) m ⋅ τ m 1 − τ m . \log\varrho_\infty(\lambda)=-\sum_{n\ge1}\sum_{m\ge1}\frac{\tau^{nm}}{m}(z^m+z^{-m}-2)=\sum_{m\ge1}\frac{4\sin^2(mu)}{m}\cdot\frac{\tau^m}{1-\tau^m}. log ϱ ∞ ( λ ) = − n ≥ 1 ∑ m ≥ 1 ∑ m τ nm ( z m + z − m − 2 ) = m ≥ 1 ∑ m 4 sin 2 ( m u ) ⋅ 1 − τ m τ m . the double series being absolutely convergent. We expand 4 sin 2 ( m u ) = − 2 ∑ k ≥ 1 ( − 1 ) k ( 2 m u ) 2 k ( 2 k ) ! 4\sin^2(mu)=-2\sum_{k\ge1}\frac{(-1)^k(2mu)^{2k}}{(2k)!} 4 sin 2 ( m u ) = − 2 ∑ k ≥ 1 ( 2 k )! ( − 1 ) k ( 2 m u ) 2 k e 2 ∣ u ∣ τ < 1 e^{2|u|}\tau<1 e 2∣ u ∣ τ < 1 ∑ m cosh ( 2 m ∣ u ∣ ) τ m m ( 1 − τ m ) \sum_m\cosh(2m|u|)\frac{\tau^m}{m(1-\tau^m)} ∑ m cosh ( 2 m ∣ u ∣ ) m ( 1 − τ m ) τ m k ≥ 1 k\ge1 k ≥ 1
S k ≔ ∑ m ≥ 1 m 2 k − 1 τ m 1 − τ m = ∑ N ≥ 1 σ 2 k − 1 ( N ) τ N = B 2 k 4 k ( 1 − E 2 k ( τ ) ) . S_k\coloneqq\sum_{m\ge1}\frac{m^{2k-1}\tau^m}{1-\tau^m}=\sum_{N\ge1}\sigma_{2k-1}(N)\tau^N=\frac{B_{2k}}{4k}(1-E_{2k}(\tau)). S k : = m ≥ 1 ∑ 1 − τ m m 2 k − 1 τ m = N ≥ 1 ∑ σ 2 k − 1 ( N ) τ N = 4 k B 2 k ( 1 − E 2 k ( τ )) . We obtain log ϱ ∞ ( λ ) = − ∑ k ≥ 1 ( − 1 ) k 2 2 k − 1 B 2 k k ( 2 k ) ! ( 1 − E 2 k ( τ ) ) u 2 k \log\varrho_\infty(\lambda)=-\sum_{k\ge1}\frac{(-1)^k2^{2k-1}B_{2k}}{k(2k)!}(1-E_{2k}(\tau))u^{2k} log ϱ ∞ ( λ ) = − ∑ k ≥ 1 k ( 2 k )! ( − 1 ) k 2 2 k − 1 B 2 k ( 1 − E 2 k ( τ )) u 2 k [27 (4.22.1)] and the values ζ ( 2 k ) = ( − 1 ) k + 1 B 2 k ( 2 π ) 2 k 2 ( 2 k ) ! \zeta(2k)=\frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!} ζ ( 2 k ) = 2 ( 2 k )! ( − 1 ) k + 1 B 2 k ( 2 π ) 2 k [27 (25.6.2)] give log sin u u = − ∑ k ≥ 1 ζ ( 2 k ) k π 2 k u 2 k = ∑ k ≥ 1 ( − 1 ) k 2 2 k − 1 B 2 k k ( 2 k ) ! u 2 k \log\frac{\sin u}{u}=-\sum_{k\ge1}\frac{\zeta(2k)}{k\pi^{2k}}u^{2k}=\sum_{k\ge1}\frac{(-1)^k2^{2k-1}B_{2k}}{k(2k)!}u^{2k} log u s i n u = − ∑ k ≥ 1 k π 2 k ζ ( 2 k ) u 2 k = ∑ k ≥ 1 k ( 2 k )! ( − 1 ) k 2 2 k − 1 B 2 k u 2 k ∣ u ∣ < π |u|<\pi ∣ u ∣ < π [27 (4.19.7)] . Adding the two expansions gives (74) . Since u 2 = arcsin 2 s 4 τ u^2=\arcsin^2\sqrt{\frac{s}{4\tau}} u 2 = arcsin 2 4 τ s s = ( 1 − τ ) 2 λ 2 4 τ s=\frac{(1-\tau)^2\lambda^2}{4\tau} s = 4 τ ( 1 − τ ) 2 λ 2 s n s^n s n (74) is a polynomial with rational coefficients in the E 2 k E_{2k} E 2 k k ≤ n k\le n k ≤ n 2 n 2n 2 n k ≥ 2 k\ge2 k ≥ 2 E 2 k E_{2k} E 2 k E 4 E_4 E 4 E 6 E_6 E 6 c 1 c_1 c 1 c 2 c_2 c 2 2 2 2
The second-order recurrence of the vertical network is also a q q q x = τ d x=\tau^d x = τ d
Proposition 6.20 Let λ ∈ C \lambda\in\mathbb{C} λ ∈ C q q q
( 1 − τ x ) f ( τ x ) − [ 1 + τ − τ ( z + z − 1 ) x ] f ( x ) + τ ( 1 − x ) f ( τ − 1 x ) = 0. (75) (1-\tau x)f(\tau x)-[1+\tau-\tau(z+z^{-1})x]f(x)+\tau(1-x)f(\tau^{-1}x)=0. \tag*{(75)} ( 1 − τ x ) f ( τ x ) − [ 1 + τ − τ ( z + z − 1 ) x ] f ( x ) + τ ( 1 − x ) f ( τ − 1 x ) = 0. (75) The function τ d ↦ ϱ d ( λ ) \tau^d\mapsto\varrho_d(\lambda) τ d ↦ ϱ d ( λ ) (75) at the points x = τ d x=\tau^d x = τ d d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 d = 0 d=0 d = 0
The function ψ ( x ) ≔ x 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ x ) \psi(x)\coloneqq x\,{}_2\phi_1\left(\begin{matrix}\tau z,\tau z^{-1}\\ \tau^2\end{matrix};\tau,\tau x\right) ψ ( x ) : = x 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ x ) { ∣ x ∣ < τ − 1 } \{|x|<\tau^{-1}\} { ∣ x ∣ < τ − 1 } (75) for any ∣ x ∣ < 1 |x|<1 ∣ x ∣ < 1 (75) at x = 0 x=0 x = 0 0 0 0 1 1 1 λ ≠ 0 \lambda\ne0 λ = 0 f f f 0 0 0 f ( 0 ) ≠ 0 f(0)\ne0 f ( 0 ) = 0 (75) on a smaller disc.
For any d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 ψ d ≔ ψ ( τ d ) \psi_d\coloneqq\psi(\tau^d) ψ d : = ψ ( τ d ) a d ( ϱ d + 1 ( λ ) ψ d − ϱ d ( λ ) ψ d + 1 ) = ϱ ∞ ( λ ) 1 + τ \mathfrak{a}_d(\varrho_{d+1}(\lambda)\psi_d-\varrho_d(\lambda)\psi_{d+1})=\frac{\varrho_\infty(\lambda)}{1+\tau} a d ( ϱ d + 1 ( λ ) ψ d − ϱ d ( λ ) ψ d + 1 ) = 1 + τ ϱ ∞ ( λ )
ϱ ∞ ( λ ) = ( 1 + ( 1 + τ ) λ ) 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ ) − τ 2 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ 2 ) . \varrho_\infty(\lambda)=(1+(1+\tau)\lambda){}_2\phi_1\left(\begin{matrix}\tau z,\tau z^{-1}\\ \tau^2\end{matrix};\tau,\tau\right)-\tau^2{}_2\phi_1\left(\begin{matrix}\tau z,\tau z^{-1}\\ \tau^2\end{matrix};\tau,\tau^2\right). ϱ ∞ ( λ ) = ( 1 + ( 1 + τ ) λ ) 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ ) − τ 2 2 ϕ 1 ( τ z , τ z − 1 τ 2 ; τ , τ 2 ) . Proof. 1. This is the relation ( 1 − τ d + 1 ) e d + 1 − τ ( 1 − τ d ) e d = ( 1 − τ 2 ) λ τ d ϱ d (1-\tau^{d+1})e_{d+1}-\tau(1-\tau^d)e_d=(1-\tau^2)\lambda\tau^d\varrho_d ( 1 − τ d + 1 ) e d + 1 − τ ( 1 − τ d ) e d = ( 1 − τ 2 ) λ τ d ϱ d Theorem 6.15 , rewritten with x = τ d x=\tau^d x = τ d (65) .
Let a , b , c ∈ C a,b,c \in\mathbb{C} a , b , c ∈ C c ≠ τ − N 0 c \ne\tau^{-N_0} c = τ − N 0 v ( x ) = ∑ n ≥ 0 v n x n = 2 ϕ 1 ( a , b ; c ; τ , x ) v(x)=\sum_{n\geq0}v_nx^n={}_2\phi_1(a,b;c;\tau,x) v ( x ) = ∑ n ≥ 0 v n x n = 2 ϕ 1 ( a , b ; c ; τ , x )
( 1 − τ n + 1 ) ( 1 − c τ n ) v n + 1 = ( 1 − a τ n ) ( 1 − b τ n ) v n . (1-\tau^{n+1})(1-c\tau^n)v_{n+1}=(1-a\tau^n)(1-b\tau^n)v_n. ( 1 − τ n + 1 ) ( 1 − c τ n ) v n + 1 = ( 1 − a τ n ) ( 1 − b τ n ) v n . Comparing the coefficients, we obtain Heine’s q q q [27 (17.6.27)] , where it is written with the q q q
( c − a b τ x ) v ( τ 2 x ) − [ c + τ − ( a + b ) τ x ] v ( τ x ) + τ ( 1 − x ) v ( x ) = 0 , ∣ x ∣ < 1. (c-ab\tau x)v(\tau^2x)-[c+\tau-(a+b)\tau x]v(\tau x)+\tau(1-x)v(x)=0,\qquad|x|<1. ( c − ab τ x ) v ( τ 2 x ) − [ c + τ − ( a + b ) τ x ] v ( τ x ) + τ ( 1 − x ) v ( x ) = 0 , ∣ x ∣ < 1. For ( a , b , c ) = ( τ z , τ z − 1 , τ 2 ) (a,b,c)=(\tau z,\tau z^{-1},\tau^2) ( a , b , c ) = ( τ z , τ z − 1 , τ 2 ) a b = τ 2 ab=\tau^2 ab = τ 2 a + b = τ ( z + z − 1 ) a+b=\tau(z+z^{-1}) a + b = τ ( z + z − 1 ) (75) for f ( x ) = x v ( τ x ) f(x)=xv(\tau x) f ( x ) = xv ( τ x ) x x x x x x f ( x ) = x s ( 1 + O ( x ) ) f(x)=x^s(1+O(x)) f ( x ) = x s ( 1 + O ( x )) x s x^s x s (75) is τ s − ( 1 + τ ) + τ 1 − s = τ − s ( τ s − 1 ) ( τ s − τ ) \tau^s-(1+\tau)+\tau^{1-s}=\tau^{-s}(\tau^s-1)(\tau^s-\tau) τ s − ( 1 + τ ) + τ 1 − s = τ − s ( τ s − 1 ) ( τ s − τ ) 0 0 0 1 1 1 f ( x ) = ∑ n ≥ 0 f n x n f(x)=\sum_{n\geq0}f_nx^n f ( x ) = ∑ n ≥ 0 f n x n 0 0 0 x x x (75) is f 1 ( τ − 1 − τ + 1 ) + f 0 ( τ ( z + z − 1 ) − 2 τ ) = − ( 1 − τ 2 ) λ f 0 f_1(\tau-1-\tau+1)+f_0(\tau(z+z^{-1})-2\tau)=-(1-\tau^2)\lambda f_0 f 1 ( τ − 1 − τ + 1 ) + f 0 ( τ ( z + z − 1 ) − 2 τ ) = − ( 1 − τ 2 ) λ f 0 f 0 = 0 f_0=0 f 0 = 0 λ ≠ 0 \lambda\ne0 λ = 0
Put W d : = a d ( ϱ d + 1 ( λ ) ψ d − ϱ d ( λ ) ψ d + 1 ) W_d\mathrel{\mathop:}=a_d(\varrho_{d+1}(\lambda)\psi_d-\varrho_d(\lambda)\psi_{d+1}) W d : = a d ( ϱ d + 1 ( λ ) ψ d − ϱ d ( λ ) ψ d + 1 ) d ≥ 1 d\geq1 d ≥ 1 a d g d + 1 = ( a d + a d − 1 + λ ) g d − a d − 1 g d − 1 a_dg_{d+1}=(a_d+a_{d-1}+\lambda)g_d-a_{d-1}g_{d-1} a d g d + 1 = ( a d + a d − 1 + λ ) g d − a d − 1 g d − 1 W d = W d − 1 W_d=W_{d-1} W d = W d − 1 ψ d = τ d ( 1 + O ( τ d ) ) \psi_d=\tau^d(1+O(\tau^d)) ψ d = τ d ( 1 + O ( τ d )) a d ( ψ d − ψ d + 1 ) → 1 − τ 1 − τ 2 = 1 1 + τ a_d(\psi_d-\psi_{d+1})\to\frac{1-\tau}{1-\tau^2}=\frac{1}{1+\tau} a d ( ψ d − ψ d + 1 ) → 1 − τ 2 1 − τ = 1 + τ 1 a d ( ϱ d + 1 − ϱ d ) ψ d = λ ψ d ∑ i ≤ d ϱ i → 0 a_d(\varrho_{d+1}-\varrho_d)\psi_d=\lambda\psi_d\sum_{i\leq d}\varrho_i\to0 a d ( ϱ d + 1 − ϱ d ) ψ d = λ ψ d ∑ i ≤ d ϱ i → 0 (62) and the boundedness of ( ϱ i ( λ ) ) (\varrho_i(\lambda)) ( ϱ i ( λ )) W d = a d ( ϱ d + 1 − ϱ d ) ψ d + ϱ d a d ( ψ d − ψ d + 1 ) W_d=a_d(\varrho_{d+1}-\varrho_d)\psi_d+\varrho_da_d(\psi_d-\psi_{d+1}) W d = a d ( ϱ d + 1 − ϱ d ) ψ d + ϱ d a d ( ψ d − ψ d + 1 ) W d = lim d ′ → ∞ W d ′ = ϱ ∞ ( λ ) 1 + τ W_d=\lim_{d'\to\infty}W_{d'}=\frac{\varrho_\infty(\lambda)}{1+\tau} W d = lim d ′ → ∞ W d ′ = 1 + τ ϱ ∞ ( λ ) W 0 = ϱ ∞ ( λ ) 1 + τ W_0=\frac{\varrho_\infty(\lambda)}{1+\tau} W 0 = 1 + τ ϱ ∞ ( λ ) a 0 = 1 1 + τ a_0=\frac{1}{1+\tau} a 0 = 1 + τ 1 ϱ 1 ( λ ) = 1 + ( 1 + τ ) λ \varrho_1(\lambda)=1+(1+\tau)\lambda ϱ 1 ( λ ) = 1 + ( 1 + τ ) λ
Remark 6.21 Proposition 6.20 describes a resonant case. The parameter c = τ 2 c=\tau^2 c = τ 2 q q q c c c (75) has a solution which is holomorphic at 0 0 0 ψ \psi ψ ϱ d \varrho_d ϱ d − ( 1 + τ ) λ ∞ ( λ ) d τ d -(1+\tau)\lambda_\infty(\lambda)d\tau^d − ( 1 + τ ) λ ∞ ( λ ) d τ d Theorem 6.15 , where d = log x log τ d=\frac{\log x}{\log\tau} d = l o g τ l o g x 2 ϕ 1 {}_2\phi_1 2 ϕ 1 x = τ d x=\tau^d x = τ d ϱ d \varrho_d ϱ d (68) is a terminating series in the base τ − 1 \tau^{-1} τ − 1 Remark 6.16 ), which diverges for generic x x x ( 1 + τ ) − 1 ϱ ∞ (1+\tau)^{-1}\varrho_\infty ( 1 + τ ) − 1 ϱ ∞ ( ϱ d ) (\varrho_d) ( ϱ d ) d = 0 d=0 d = 0 ( ψ d ) (\psi_d) ( ψ d ) ϱ ∞ \varrho_\infty ϱ ∞ λ \lambda λ 0 0 0 v n v_n v n Theorem 6.17 . The same computation as in the proof of the third part gives the Weyl function
⟨ ( J + λ ) − 1 δ 0 , δ 0 ⟩ = ( 1 + τ ) ψ 0 ϱ ∞ ( λ ) \langle(J+\lambda)^{-1}\delta_0,\delta_0\rangle=\frac{(1+\tau)\psi_0}{\varrho_\infty(\lambda)} ⟨( J + λ ) − 1 δ 0 , δ 0 ⟩ = ϱ ∞ ( λ ) ( 1 + τ ) ψ 0 for λ ≥ 0 \lambda\geq0 λ ≥ 0 ϖ n \varpi_n ϖ n
We now give a probabilistic meaning to ϱ ∞ \varrho_\infty ϱ ∞ X = ( X t ) 0 ≤ t < ζ X=(X_t)_{0\leq t<\zeta} X = ( X t ) 0 ≤ t < ζ N 0 \mathbb{N}_0 N 0 d d d d + 1 d+1 d + 1 a d a_d a d d d d d − 1 d-1 d − 1 a d − 1 a_{d-1} a d − 1 ζ \zeta ζ [55 Chapter 2] . We denote by P d \mathbb{P}_d P d E d \mathbb{E}_d E d d d d T D : = inf { t : X t = D } T_D\mathrel{\mathop:}=\inf\{t:X_t=D\} T D : = inf { t : X t = D } D ∈ N 0 D\in\mathbb{N}_0 D ∈ N 0 Y = ( Y t ) t ≥ 0 Y=(Y_t)_{t\geq0} Y = ( Y t ) t ≥ 0 N 0 \mathbb{N}_0 N 0 X X X k k k k + 1 k+1 k + 1 b k b_k b k k k k k − 1 k-1 k − 1 b k − 1 b_{k-1} b k − 1 Y Y Y − B -B − B P ( Y t = j ∣ Y 0 = k ) = ⟨ e − t B δ j , δ k ⟩ \mathbb{P}(Y_t=j\mid Y_0=k)=\langle e^{-tB}\delta_j,\delta_k\rangle P ( Y t = j ∣ Y 0 = k ) = ⟨ e − tB δ j , δ k ⟩ t ≥ 0 t\geq0 t ≥ 0 j , k ∈ N 0 j,k\in\mathbb{N}_0 j , k ∈ N 0
Theorem 6.22 1. For any d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 λ ≥ 0 \lambda\geq0 λ ≥ 0
E d [ e − λ ζ ] = ϱ d ( λ ) ϱ ∞ ( λ ) . \mathbb{E}_d[e^{-\lambda\zeta}]=\frac{\varrho_d(\lambda)}{\varrho_\infty(\lambda)}. E d [ e − λ ζ ] = ϱ ∞ ( λ ) ϱ d ( λ ) . {#eq:6.29}
In particular P d ( ζ < ∞ ) = 1 \mathbb{P}_d(\zeta<\infty)=1 P d ( ζ < ∞ ) = 1 E d [ ζ ] = ∑ m ≥ d m + 1 a m \mathbb{E}_d[\zeta]=\sum_{m\geq d}\frac{m+1}{a_m} E d [ ζ ] = ∑ m ≥ d a m m + 1 P 0 \mathbb{P}_0 P 0 ζ \zeta ζ ∑ n ≥ 1 ϵ n μ n \sum_{n\geq1}\frac{\epsilon_n}{\mu_n} ∑ n ≥ 1 μ n ϵ n ( ϵ n ) n ≥ 1 (\epsilon_n)_{n\geq1} ( ϵ n ) n ≥ 1 ∑ n ≥ 1 R n e − μ n t \sum_{n\ge1}R_n e^{-\mu_n t} ∑ n ≥ 1 R n e − μ n t t > 0 t>0 t > 0 R n = ( − 1 ) n − 1 τ ( n − 1 ) ( n − 2 ) / 2 ( 1 − τ n ) 1 − τ 2 R_n=(-1)^{n-1}\frac{\tau^{(n-1)(n-2)/2}(1-\tau^n)}{1-\tau^2} R n = ( − 1 ) n − 1 1 − τ 2 τ ( n − 1 ) ( n − 2 ) /2 ( 1 − τ n )
Suppose that ( X 0 , Y 0 ) = ( d , k ) (X_0,Y_0)=(d,k) ( X 0 , Y 0 ) = ( d , k ) J 0 = 0 < J 1 < J 2 < ⋯ J_0=0<J_1<J_2<\cdots J 0 = 0 < J 1 < J 2 < ⋯ ( X , Y ) (X,Y) ( X , Y ) [ 0 , ζ ) [0,\zeta) [ 0 , ζ ) J l → ζ J_l\to\zeta J l → ζ ( X J l + Y J l , Y J l ) l ≥ 0 (X_{J_l}+Y_{J_l},Y_{J_l})_{l\ge0} ( X J l + Y J l , Y J l ) l ≥ 0 ( d + k , k ) (d+k,k) ( d + k , k ) k ∞ k_\infty k ∞ Y ζ Y_\zeta Y ζ j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0
ν ( d + k , k ) ( j ) = E d [ e − ξ B δ j , δ k ] = ⟨ ϱ d ϱ ∞ ( B ) δ j , δ k ⟩ , \nu_{(d+k,k)}(j)=\mathbb{E}_d\left[e^{-\xi B}\delta_j,\delta_k\right]=\left\langle\frac{\varrho_d}{\varrho_\infty}(B)\delta_j,\delta_k\right\rangle, ν ( d + k , k ) ( j ) = E d [ e − ξ B δ j , δ k ] = ⟨ ϱ ∞ ϱ d ( B ) δ j , δ k ⟩ , and ϱ ∞ ( B ) − 1 = ∏ n ≥ 1 μ n ( μ n + B ) − 1 \varrho_\infty(B)^{-1}=\prod_{n\ge1}\mu_n(\mu_n+B)^{-1} ϱ ∞ ( B ) − 1 = ∏ n ≥ 1 μ n ( μ n + B ) − 1
For any d , k ∈ N 0 d,k\in\mathbb{N}_0 d , k ∈ N 0
∑ j ≥ 0 ∣ ν ( d + k , k ) ( j ) − δ j k ∣ ≤ 2 ( b k + b k − 1 ) E d [ ζ ] ≤ 4 τ ( 1 − τ ) 3 ( d + 1 ) τ d , \sum_{j\ge0}\left|\nu_{(d+k,k)}(j)-\delta_{jk}\right|\le2(b_k+b_{k-1})\mathbb{E}_d[\zeta]\le\frac{4\tau}{(1-\tau)^3}(d+1)\tau^d, j ≥ 0 ∑ ν ( d + k , k ) ( j ) − δ j k ≤ 2 ( b k + b k − 1 ) E d [ ζ ] ≤ ( 1 − τ ) 3 4 τ ( d + 1 ) τ d , and lim d → ∞ 1 d τ d ∑ j ≥ 0 ∣ ν ( d + k , k ) ( j ) − δ j k ∣ = 2 ( 1 + τ ) ( b k + b k − 1 ) \lim_{d\to\infty}\frac{1}{d\tau^d}\sum_{j\ge0}\left|\nu_{(d+k,k)}(j)-\delta_{jk}\right|=2(1+\tau)(b_k+b_{k-1}) lim d → ∞ d τ d 1 ∑ j ≥ 0 ν ( d + k , k ) ( j ) − δ j k = 2 ( 1 + τ ) ( b k + b k − 1 )
Proof: 1. Let λ ≥ 0 \lambda\ge0 λ ≥ 0 D > 0 D>0 D > 0 0 ≤ d ≤ D 0\le d\le D 0 ≤ d ≤ D g ( d ) : = E d [ e − λ T D ] g(d)\mathrel{\mathop:}=\mathbb{E}_d[e^{-\lambda T_D}] g ( d ) : = E d [ e − λ T D ] g ( D ) = 1 g(D)=1 g ( D ) = 1 0 ≤ d < D 0\le d<D 0 ≤ d < D P d \mathbb{P}_d P d a d + a d − 1 a_d+a_{d-1} a d + a d − 1 d + 1 d+1 d + 1 a d a d + a d − 1 \frac{a_d}{a_d+a_{d-1}} a d + a d − 1 a d ( a d + a d − 1 + λ ) g ( d ) = a d g ( d + 1 ) + a d − 1 g ( d − 1 ) (a_d+a_{d-1}+\lambda)g(d)=a_dg(d+1)+a_{d-1}g(d-1) ( a d + a d − 1 + λ ) g ( d ) = a d g ( d + 1 ) + a d − 1 g ( d − 1 ) a d ( g ( d + 1 ) − g ( d ) ) + a d − 1 ( g ( d − 1 ) − g ( d ) ) = λ g ( d ) a_d(g(d+1)-g(d))+a_{d-1}(g(d-1)-g(d))=\lambda g(d) a d ( g ( d + 1 ) − g ( d )) + a d − 1 ( g ( d − 1 ) − g ( d )) = λ g ( d ) d d d (62) , we obtain g ( d ) = g ( 0 ) ϱ d ( λ ) g(d)=g(0)\varrho_d(\lambda) g ( d ) = g ( 0 ) ϱ d ( λ ) 0 ≤ d ≤ D 0\le d\le D 0 ≤ d ≤ D g ( d ) = ϱ d ( λ ) ϱ D ( λ ) g(d)=\frac{\varrho_d(\lambda)}{\varrho_D(\lambda)} g ( d ) = ϱ D ( λ ) ϱ d ( λ ) ϱ D ( λ ) ≥ 1 \varrho_D(\lambda)\ge1 ϱ D ( λ ) ≥ 1 T D = ∞ T_D=\infty T D = ∞ X X X D D D e − λ ∞ = 0 e^{-\lambda\infty}=0 e − λ ∞ = 0 λ > 0 \lambda>0 λ > 0 P d \mathbb{P}_d P d T D T_D T D D ≥ d D\ge d D ≥ d D D D ζ = lim D → ∞ T D \zeta=\lim_{D\to\infty}T_D ζ = lim D → ∞ T D D l D_l D l l + 1 l+1 l + 1 X X X l l l T D l T_{D_l} T D l { 0 , … , D } \{0,\ldots,D\} { 0 , … , D } { 0 , … , D } \{0,\ldots,D\} { 0 , … , D } T D ≤ ζ T_D\le\zeta T D ≤ ζ { ζ < ∞ } \{\zeta<\infty\} { ζ < ∞ } [33 ] . By monotone convergence, we obtain (6.29) for λ > 0 \lambda>0 λ > 0 λ → 0 + \lambda\to0^+ λ → 0 + P d ( ζ < ∞ ) = 1 \mathbb{P}_d(\zeta<\infty)=1 P d ( ζ < ∞ ) = 1 0 0 0 ζ \zeta ζ E d [ ζ ] = ϱ ∞ ′ ( 0 ) − ϱ d ′ ( 0 ) \mathbb{E}_d[\zeta]=\varrho_\infty'(0)-\varrho_d'(0) E d [ ζ ] = ϱ ∞ ′ ( 0 ) − ϱ d ′ ( 0 ) (62) , ϱ m ′ ( 0 ) = ∑ i ≤ m i + 1 a i \varrho_m'(0)=\sum_{i\le m}\frac{i+1}{a_i} ϱ m ′ ( 0 ) = ∑ i ≤ m a i i + 1 m ∈ N 0 ∪ { ∞ } m\in\mathbb{N}_0\cup\{\infty\} m ∈ N 0 ∪ { ∞ } E d [ ζ ] \mathbb{E}_d[\zeta] E d [ ζ ]
By (6.22), E 0 [ e − λ ζ ] = ∏ n μ n μ n + λ = E [ e − λ ∑ n e n / μ n ] \mathbb{E}_0[e^{-\lambda\zeta}]=\prod_n\frac{\mu_n}{\mu_n+\lambda}=\mathbb{E}\left[e^{-\lambda\sum_n e_n/\mu_n}\right] E 0 [ e − λ ζ ] = ∏ n μ n + λ μ n = E [ e − λ ∑ n e n / μ n ] ∑ n e n / μ n \sum_n e_n/\mu_n ∑ n e n / μ n ∑ n 1 μ n < ∞ \sum_n\frac{1}{\mu_n}<\infty ∑ n μ n 1 < ∞ ζ \zeta ζ (73) , R n = 1 ϱ ∞ ′ ( − μ n ) R_n=\frac{1}{\varrho_\infty'(-\mu_n)} R n = ϱ ∞ ′ ( − μ n ) 1 ∑ n ∣ R n ∣ < ∞ \sum_n|R_n|<\infty ∑ n ∣ R n ∣ < ∞ ρ N : = μ N μ N + 1 \rho_N\mathrel{\mathop:}=\sqrt{\mu_N\mu_{N+1}} ρ N : = μ N μ N + 1 ∣ λ ∣ = ρ N |\lambda|=\rho_N ∣ λ ∣ = ρ N μ n + 1 ≥ τ − 1 μ n \mu_{n+1}\ge\tau^{-1}\mu_n μ n + 1 ≥ τ − 1 μ n n n n ∣ 1 + λ μ n ∣ ≥ ρ N μ n − 1 ≥ τ − ( N − n ) − 1 / 2 − 1 |1+\frac{\lambda}{\mu_n}|\ge\frac{\rho_N}{\mu_n}-1\ge\tau^{-(N-n)-1/2}-1 ∣1 + μ n λ ∣ ≥ μ n ρ N − 1 ≥ τ − ( N − n ) − 1/2 − 1 n ≤ N n\le N n ≤ N ∣ 1 + λ μ n ∣ ≥ 1 − ρ N μ n ≥ 1 − τ ( n − N − 1 ) + 1 / 2 |1+\frac{\lambda}{\mu_n}|\ge1-\frac{\rho_N}{\mu_n}\ge1-\tau^{(n-N-1)+1/2} ∣1 + μ n λ ∣ ≥ 1 − μ n ρ N ≥ 1 − τ ( n − N − 1 ) + 1/2 n > N n>N n > N
min ∣ λ ∣ = ρ N ∣ ϱ ∞ ( λ ) ∣ ≥ ∏ i = 0 N − 1 ( τ − i − 1 / 2 − 1 ) ⋅ ( τ 1 / 2 ; τ ) ∞ , \min_{|\lambda|=\rho_N}|\varrho_\infty(\lambda)|\ge\prod_{i=0}^{N-1}(\tau^{-i-1/2}-1)\cdot(\tau^{1/2};\tau)_\infty, ∣ λ ∣ = ρ N min ∣ ϱ ∞ ( λ ) ∣ ≥ i = 0 ∏ N − 1 ( τ − i − 1/2 − 1 ) ⋅ ( τ 1/2 ; τ ) ∞ , which tends to ∞ \infty ∞ N → ∞ N\to\infty N → ∞ 1 ϱ ∞ ( λ ) − ∑ n R n λ + μ n \frac{1}{\varrho_\infty(\lambda)}-\sum_n\frac{R_n}{\lambda+\mu_n} ϱ ∞ ( λ ) 1 − ∑ n λ + μ n R n { ∣ λ ∣ = ρ N } \{|\lambda|=\rho_N\} { ∣ λ ∣ = ρ N } N → ∞ , s i n c e N\to\infty, since N → ∞ , s in ce ∣ λ ∣ = ρ N |\lambda|=\rho_N ∣ λ ∣ = ρ N n n n λ ≥ 0 \lambda\ge0 λ ≥ 0
1 ϱ ∞ ( λ ) = ∑ n ≥ 1 R n λ + μ n = ∫ 0 ∞ e − λ t ∑ n ≥ 1 R n e − μ n t d t . \frac{1}{\varrho_\infty(\lambda)}=\sum_{n\ge1}\frac{R_n}{\lambda+\mu_n}=\int_0^\infty e^{-\lambda t}\sum_{n\ge1}R_ne^{-\mu_n t}\,dt. ϱ ∞ ( λ ) 1 = n ≥ 1 ∑ λ + μ n R n = ∫ 0 ∞ e − λ t n ≥ 1 ∑ R n e − μ n t d t . by Fubini’s theorem, since ∑ n ∣ R n ∣ μ n < ∞ \sum_n \frac{|R_n|}{\mu_n}<\infty ∑ n μ n ∣ R n ∣ < ∞
By the memoryless property of the exponential holding times and the independence of X X X Y Y Y ( X , Y ) (X,Y) ( X , Y ) ζ \zeta ζ N 0 2 \mathbb{N}_0^2 N 0 2 ( d ′ , k ′ ) (d',k') ( d ′ , k ′ ) ( d ′ ± 1 , k ′ ) (d'\pm1,k') ( d ′ ± 1 , k ′ ) ( d ′ , k ′ ± 1 ) (d',k'\pm1) ( d ′ , k ′ ± 1 ) a d ′ , a d ′ − 1 , b k ′ a_{d'},a_{d'-1},b_{k'} a d ′ , a d ′ − 1 , b k ′ b k ′ − 1 b_{k'-1} b k ′ − 1 cond ( v , v ′ ) ∑ v ′ ′ cond ( v , v ′ ′ ) \frac{\operatorname{cond}(v,v')}{\sum_{v''}\operatorname{cond}(v,v'')} ∑ v ′′ cond ( v , v ′′ ) cond ( v , v ′ ) cond ( v , v ′ ) = w ^ ( v ) P ( v , v ′ ) \operatorname{cond}(v,v')=\widehat{w}(v)P(v,v') cond ( v , v ′ ) = w ( v ) P ( v , v ′ ) Proposition 6.4 , written in the coordinates ( n − k , k ) (n-k,k) ( n − k , k ) [55 Section 2.6] . Since ∑ v ′ ′ cond ( v , v ′ ′ ) = w ^ ( v ) ∑ v ′ ′ P ( v , v ′ ′ ) = w ^ ( v ) \sum_{v''}\operatorname{cond}(v,v'')=\widehat{w}(v)\sum_{v''}P(v,v'')=\widehat{w}(v) ∑ v ′′ cond ( v , v ′′ ) = w ( v ) ∑ v ′′ P ( v , v ′′ ) = w ( v ) ( n − k , k ) (n-k,k) ( n − k , k ) Lemma 6.7 , X J l → ∞ X_{J_l}\to\infty X J l → ∞ l → ∞ l\to\infty l → ∞ X X X J l J_l J l T D T_D T D J l J_l J l J l J_l J l T D T_D T D D D D sup l J l = lim D T D = ζ \sup_l J_l=\lim_D T_D=\zeta sup l J l = lim D T D = ζ Y Y Y [ 0 , ζ ] [0,\zeta] [ 0 , ζ ] ζ < ∞ \zeta<\infty ζ < ∞ ζ \zeta ζ ζ \zeta ζ Y Y Y X X X k ∞ = lim l Y J l = Y ζ k_\infty=\lim_l Y_{J_l}=Y_\zeta k ∞ = lim l Y J l = Y ζ ν ( d + k , k ) ( j ) = E d [ p ζ ( k , j ) ] \nu_{(d+k,k)}(j)=\mathbb{E}_d[p_\zeta(k,j)] ν ( d + k , k ) ( j ) = E d [ p ζ ( k , j )] p t ( k , j ) = ⟨ e − t B δ j , δ k ⟩ p_t(k,j)=\langle e^{-tB}\delta_j,\delta_k\rangle p t ( k , j ) = ⟨ e − tB δ j , δ k ⟩ s j , k s_{j,k} s j , k B B B δ j \delta_j δ j δ k \delta_k δ k σ ( B ) ⊂ [ 0 , ∞ ) \sigma(B)\subset[0,\infty) σ ( B ) ⊂ [ 0 , ∞ ) E d [ p ζ ( k , j ) ] = ∫ E d [ e − λ ζ ] d s j , k ( λ ) = ⟨ ϱ d ϱ ∞ ( B ) δ j , δ k ⟩ \mathbb{E}_d[p_\zeta(k,j)]=\int\mathbb{E}_d[e^{-\lambda\zeta}]\,ds_{j,k}(\lambda)=\left\langle\frac{\varrho_d}{\varrho_\infty}(B)\delta_j,\delta_k\right\rangle E d [ p ζ ( k , j )] = ∫ E d [ e − λ ζ ] d s j , k ( λ ) = ⟨ ϱ ∞ ϱ d ( B ) δ j , δ k ⟩ ∥ Id − μ n ( μ n + B ) − 1 ∥ ≤ ∥ B ∥ μ n \lVert\operatorname{Id}-\mu_n(\mu_n+B)^{-1}\rVert\le\frac{\lVert B\rVert}{\mu_n} ∥ Id − μ n ( μ n + B ) − 1 ∥ ≤ μ n ∥ B ∥ ϱ ∞ ( B ) − 1 \varrho_\infty(B)^{-1} ϱ ∞ ( B ) − 1
Put q k : = b k + b k − 1 q_k\mathrel{\mathop:}=b_k+b_{k-1} q k : = b k + b k − 1 Y Y Y k k k q ‾ : = sup i q i ≤ 2 τ 1 − τ 2 \overline{q}\mathrel{\mathop:}=\sup_i q_i\le\frac{2\tau}{1-\tau^2} q : = sup i q i ≤ 1 − τ 2 2 τ ∑ j ∣ ν ( d + k , k ) ( j ) − δ j k ∣ = 2 P ( Y ζ ≠ k ) \sum_j|\nu_{(d+k,k)}(j)-\delta_{jk}|=2\mathbb{P}(Y_\zeta\ne k) ∑ j ∣ ν ( d + k , k ) ( j ) − δ j k ∣ = 2 P ( Y ζ = k ) Y ζ ≠ k Y_\zeta\ne k Y ζ = k Y Y Y ζ \zeta ζ E d [ 1 − e − q k ζ ] ≤ q k E d [ ζ ] \mathbb{E}_d[1-e^{-q_k\zeta}]\le q_k\mathbb{E}_d[\zeta] E d [ 1 − e − q k ζ ] ≤ q k E d [ ζ ] m + 1 a m ≤ ( 1 + τ ) ( m + 1 ) τ m \frac{m+1}{a_m}\le(1+\tau)(m+1)\tau^m a m m + 1 ≤ ( 1 + τ ) ( m + 1 ) τ m ∑ m ≥ d ( m + 1 ) τ m ≤ ( d + 1 ) τ d ( 1 − τ ) 2 \sum_{m\ge d}(m+1)\tau^m\le\frac{(d+1)\tau^d}{(1-\tau)^2} ∑ m ≥ d ( m + 1 ) τ m ≤ ( 1 − τ ) 2 ( d + 1 ) τ d Y Y Y [ 0 , ζ ) [0,\zeta) [ 0 , ζ ) Y ζ ≠ k Y_\zeta\ne k Y ζ = k t ≥ 0 t\ge0 t ≥ 0 Y Y Y k k k [ 0 , t ] [0,t] [ 0 , t ] q k t e − q ‾ t ≥ q k ( t − q ‾ t 2 ) q_kte^{-\overline{q}t}\ge q_k(t-\overline{q}t^2) q k t e − q t ≥ q k ( t − q t 2 ) P ( Y ζ ≠ k ) ≥ q k ( E d [ ζ ] − q ‾ E d [ ζ 2 ] ) \mathbb{P}(Y_\zeta\ne k)\ge q_k(\mathbb{E}_d[\zeta]-\overline{q}\mathbb{E}_d[\zeta^2]) P ( Y ζ = k ) ≥ q k ( E d [ ζ ] − q E d [ ζ 2 ])
E d [ ζ ] = ( 1 − τ 2 ) ∑ m ≥ d ( m + 1 ) τ m 1 − τ m + 1 = ( 1 + τ ) d τ d + O ( τ 2 d ) . \mathbb{E}_d[\zeta]=(1-\tau^2)\sum_{m\ge d}\frac{(m+1)\tau^m}{1-\tau^{m+1}}=(1+\tau)d\tau^d+O(\tau^{2d}). E d [ ζ ] = ( 1 − τ 2 ) m ≥ d ∑ 1 − τ m + 1 ( m + 1 ) τ m = ( 1 + τ ) d τ d + O ( τ 2 d ) . On the other hand, (62) gives ϱ ∞ − ϱ d = λ G d \varrho_\infty-\varrho_d=\lambda G_d ϱ ∞ − ϱ d = λ G d G d ( λ ) = ∑ m ≥ d 1 a m ∑ i ≤ m ϱ i ( λ ) G_d(\lambda)=\sum_{m\ge d}\frac{1}{a_m}\sum_{i\le m}\varrho_i(\lambda) G d ( λ ) = ∑ m ≥ d a m 1 ∑ i ≤ m ϱ i ( λ ) E d [ e − λ ζ ] = 1 − λ G d ( λ ) ϱ ∞ ( λ ) \mathbb{E}_d[e^{-\lambda\zeta}]=1-\frac{\lambda G_d(\lambda)}{\varrho_\infty(\lambda)} E d [ e − λ ζ ] = 1 − ϱ ∞ ( λ ) λ G d ( λ )
E d [ ζ 2 ] = − 2 ( G d ϱ ∞ ) ′ ( 0 ) = 2 ∑ m ≥ d 1 a m ∑ i ≤ m ( ϱ ∞ ′ ( 0 ) − ϱ i ′ ( 0 ) ) = 2 ∑ m ≥ d 1 a m ∑ i ≤ m E i [ ζ ] ≤ 2 ( ∑ i ≥ 0 E i [ ζ ] ) ∑ m ≥ d 1 a m , \begin{aligned}
\mathbb{E}_d[\zeta^2]=-2\left(\frac{G_d}{\varrho_\infty}\right)'(0)=2\sum_{m\ge d}\frac{1}{a_m}\sum_{i\le m}\left(\varrho_\infty'(0)-\varrho_i'(0)\right) \\
&=2\sum_{m\ge d}\frac{1}{a_m}\sum_{i\le m}\mathbb{E}_i[\zeta]\le2\left(\sum_{i\ge0}\mathbb{E}_i[\zeta]\right)\sum_{m\ge d}\frac{1}{a_m},
\end{aligned} E d [ ζ 2 ] = − 2 ( ϱ ∞ G d ) ′ ( 0 ) = 2 m ≥ d ∑ a m 1 i ≤ m ∑ ( ϱ ∞ ′ ( 0 ) − ϱ i ′ ( 0 ) ) = 2 m ≥ d ∑ a m 1 i ≤ m ∑ E i [ ζ ] ≤ 2 ( i ≥ 0 ∑ E i [ ζ ] ) m ≥ d ∑ a m 1 , which is O ( τ d ) O(\tau^d) O ( τ d ) ∑ i E i [ ζ ] = ∑ m ( m + 1 ) 2 a m < ∞ \sum_i\mathbb{E}_i[\zeta]=\sum_m\frac{(m+1)^2}{a_m}<\infty ∑ i E i [ ζ ] = ∑ m a m ( m + 1 ) 2 < ∞ P ( Y ζ ≠ k ) = q k ( 1 + τ ) d τ d + O ( τ 2 d ) \mathbb{P}(Y_\zeta\ne k)=q_k(1+\tau)d\tau^d+O(\tau^{2d}) P ( Y ζ = k ) = q k ( 1 + τ ) d τ d + O ( τ 2 d )
of π 2 2 T \frac{\pi^2}{2}T 2 π 2 T T T T E [ e − λ T ] = 2 λ sinh 2 λ \mathbb{E}[e^{-\lambda T}]=\frac{\sqrt{2\lambda}}{\sinh\sqrt{2\lambda}} E [ e − λ T ] = s i n h 2 λ 2 λ Y Y Y ∑ n ≥ 1 2 ( − 1 ) n − 1 n 2 e − n 2 t \sum_{n\geq1}2(-1)^{n-1}n^2e^{-n^2t} ∑ n ≥ 1 2 ( − 1 ) n − 1 n 2 e − n 2 t [18 ] and [14 ] . Accordingly, 1 + τ 1 − τ R n = ( − 1 ) n − 1 n ( n − 1 ) ( n − 2 ) ( 1 − τ ) 2 ( 1 − τ n ) ( 1 − τ 2 n ) 1 − τ → 2 ( − 1 ) n − 1 n 2 \frac{1+\tau}{1-\tau}R_n=(-1)^{n-1}\frac{n(n-1)(n-2)}{(1-\tau)^2}\frac{(1-\tau^n)(1-\tau^{2n})}{1-\tau}\to2(-1)^{n-1}n^2 1 − τ 1 + τ R n = ( − 1 ) n − 1 ( 1 − τ ) 2 n ( n − 1 ) ( n − 2 ) 1 − τ ( 1 − τ n ) ( 1 − τ 2 n ) → 2 ( − 1 ) n − 1 n 2 ϱ ∞ ( 1 + τ 1 + λ ) = ∏ n ≥ 1 ( 1 + λ [ n ] r 2 ) \varrho_\infty\left(\frac{1+\tau}{1+\lambda}\right)=\prod_{n\geq1}\left(1+\frac{\lambda}{[n]_r^2}\right) ϱ ∞ ( 1 + λ 1 + τ ) = ∏ n ≥ 1 ( 1 + [ n ] r 2 λ ) sinh ( π λ ) π λ \frac{\sinh(\pi\sqrt{\lambda})}{\pi\sqrt{\lambda}} π λ s i n h ( π λ ) C \mathbb{C} C Corollary 6.19 is a q q q s ∈ C s\in\mathbb{C} s ∈ C Re s > − 1 \operatorname{Re}s>-1 Re s > − 1 ∑ n ∣ R n ∣ μ n − Re s − 1 < ∞ \sum_n|R_n|\mu_n^{-\operatorname{Re}s-1}<\infty ∑ n ∣ R n ∣ μ n − Re s − 1 < ∞
E 0 [ ζ s ] = Γ ( s + 1 ) ∑ n ≥ 1 R n μ n − s − 1 . \mathbb{E}_0[\zeta^s]=\Gamma(s+1)\sum_{n\geq1}R_n\mu_n^{-s-1}. E 0 [ ζ s ] = Γ ( s + 1 ) n ≥ 1 ∑ R n μ n − s − 1 . a Dirichlet series in the squares of the quantum dimensions [ n ] r [n]_r [ n ] r τ → 1 − \tau\to1^- τ → 1 − ζ τ \zeta_\tau ζ τ 2 Γ ( s + 1 ) ( 1 − 2 1 − 2 s ) ζ ( 2 s ) = E [ Y s ] 2\Gamma(s+1)(1-2^{1-2s})\zeta(2s)=\mathbb{E}[Y^s] 2Γ ( s + 1 ) ( 1 − 2 1 − 2 s ) ζ ( 2 s ) = E [ Y s ] [14 ] . We do not know whether this q q q
Corollary 6.24 For any bounded function f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D
∣ H f ( n , k ) − f ( k ) ∣ ≤ 4 τ ( 1 − τ ) 3 ∥ f ∥ ∞ ( n − k + 1 ) τ n − k . |\mathcal{H}f(n,k)-f(k)|\leq\frac{4\tau}{(1-\tau)^3}\|f\|_\infty(n-k+1)\tau^{n-k}. ∣ H f ( n , k ) − f ( k ) ∣ ≤ ( 1 − τ ) 3 4 τ ∥ f ∥ ∞ ( n − k + 1 ) τ n − k . For f = 1 { k } f=\mathbf{1}_{\{k\}} f = 1 { k } ( 1 + τ ) ( b k + b k − 1 ) ( n − k ) τ n − k (1+\tau)(b_k+b_{k-1})(n-k)\tau^{n-k} ( 1 + τ ) ( b k + b k − 1 ) ( n − k ) τ n − k n → ∞ n\to\infty n → ∞
Proof. We have H f ( n , k ) − f ( k ) = ∑ j ν ( n , k ) ( j ) ( f ( j ) − f ( k ) ) \mathcal{H}f(n,k)-f(k)=\sum_j\nu_{(n,k)}(j)(f(j)-f(k)) H f ( n , k ) − f ( k ) = ∑ j ν ( n , k ) ( j ) ( f ( j ) − f ( k )) ∥ f ∥ ∞ ∑ j ∣ ν ( n , k ) ( j ) − δ j k ∣ \|f\|_\infty\sum_j|\nu_{(n,k)}(j)-\delta_{jk}| ∥ f ∥ ∞ ∑ j ∣ ν ( n , k ) ( j ) − δ j k ∣ d = n − k d=n-k d = n − k f = 1 { k } f=\mathbf{1}_{\{k\}} f = 1 { k } 1 − ν ( n , k ) ( k ) = 1 2 ∑ j ∣ ν ( n , k ) ( j ) − δ j k ∣ 1-\nu_{(n,k)}(k)=\frac{1}{2}\sum_j|\nu_{(n,k)}(j)-\delta_{jk}| 1 − ν ( n , k ) ( k ) = 2 1 ∑ j ∣ ν ( n , k ) ( j ) − δ j k ∣
Thus the rate τ n − k = τ n − k 2 \tau^{n-k}=\tau^{\frac{n-k}{2}} τ n − k = τ 2 n − k (56) can be replaced by ( n − k + 1 ) τ n − k (n-k+1)\tau^{n-k} ( n − k + 1 ) τ n − k Theorem 6.13 : the discrete chain is reversible for w ^ \widehat{w} w Y Y Y ζ \zeta ζ
Lemma 6.25 For any k , n ∈ N 0 k,n\in\mathbb{N}_0 k , n ∈ N 0
Q ~ k ≔ 3 ϕ 2 ( τ − k , z , z − 1 τ , 0 ; τ , τ ) , H n ≔ 3 ϕ 2 ( τ − n , z , z − 1 τ , τ ; τ , τ ) , \widetilde{Q}_k\coloneqq{}_3\phi_2\left(\begin{matrix}\tau^{-k},z,z^{-1}\\ \tau,0\end{matrix};\tau,\tau\right),\qquad H_n\coloneqq{}_3\phi_2\left(\begin{matrix}\tau^{-n},z,z^{-1}\\ \tau,\tau\end{matrix};\tau,\tau\right), Q k : = 3 ϕ 2 ( τ − k , z , z − 1 τ , 0 ; τ , τ ) , H n : = 3 ϕ 2 ( τ − n , z , z − 1 τ , τ ; τ , τ ) , which are polynomials in λ \lambda λ k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0
Q ~ k = ∑ n = 0 k ( τ ; τ ) k ( τ ; τ ) k − n τ k − n H n . \widetilde{Q}_k=\sum_{n=0}^{k}\frac{(\tau;\tau)_k}{(\tau;\tau)_{k-n}}\tau^{k-n}H_n. Q k = n = 0 ∑ k ( τ ; τ ) k − n ( τ ; τ ) k τ k − n H n . Proof. Put φ i ≔ ( z , z − 1 ; τ ) i \varphi_i\coloneqq(z,z^{-1};\tau)_i φ i : = ( z , z − 1 ; τ ) i Q ~ k = ∑ i A k , i φ i \widetilde{Q}_k=\sum_iA_{k,i}\varphi_i Q k = ∑ i A k , i φ i H n = ∑ i A n , i φ i ( τ ; τ ) i H_n=\sum_iA_{n,i}\frac{\varphi_i}{(\tau;\tau)_i} H n = ∑ i A n , i ( τ ; τ ) i φ i A k , i = ( τ − k ; τ ) i ( τ ; τ ) i 2 τ i A_{k,i}=\frac{(\tau^{-k};\tau)_i}{(\tau;\tau)_i^2}\tau^i A k , i = ( τ ; τ ) i 2 ( τ − k ; τ ) i τ i ( φ i ) (\varphi_i) ( φ i ) λ \lambda λ A k , i = ∑ n = i k ( τ ; τ ) k ( τ ; τ ) k − n τ k − n A n , i ( τ ; τ ) i A_{k,i}=\sum_{n=i}^{k}\frac{(\tau;\tau)_k}{(\tau;\tau)_{k-n}}\tau^{k-n}\frac{A_{n,i}}{(\tau;\tau)_i} A k , i = ∑ n = i k ( τ ; τ ) k − n ( τ ; τ ) k τ k − n ( τ ; τ ) i A n , i 0 ≤ i ≤ k 0\leq i\leq k 0 ≤ i ≤ k ( τ − n ; τ ) i = ( − 1 ) i τ ( i 2 ) − n i ( τ ; τ ) n ( τ ; τ ) n − i (\tau^{-n};\tau)_i=(-1)^i\tau^{\binom{i}{2}-ni}\frac{(\tau;\tau)_n}{(\tau;\tau)_{n-i}} ( τ − n ; τ ) i = ( − 1 ) i τ ( 2 i ) − ni ( τ ; τ ) n − i ( τ ; τ ) n n = i + l n=i+l n = i + l N = k − i N=k-i N = k − i
∑ l = 0 N [ N l ] τ ∏ p = 0 l − 1 ( ξ − τ p ) = ξ N , where ξ = τ − ( i + 1 ) . (76) \sum_{l=0}^{N}\begin{bmatrix}N\\l\end{bmatrix}_{\tau}\prod_{p=0}^{l-1}(\xi-\tau^p)=\xi^N,\qquad\text{where }\xi=\tau^{-(i+1)}. \tag*{(76)} l = 0 ∑ N [ N l ] τ p = 0 ∏ l − 1 ( ξ − τ p ) = ξ N , where ξ = τ − ( i + 1 ) . (76) This holds for any ξ ∈ C \xi\in\mathbb{C} ξ ∈ C N N N [ N + 1 l ] τ = τ l [ N l ] τ + [ N l − 1 ] τ \left[\begin{matrix}N+1\\l\end{matrix}\right]_{\tau}=\tau^l\left[\begin{matrix}N\\l\end{matrix}\right]_{\tau}+\left[\begin{matrix}N\\l-1\end{matrix}\right]_{\tau} [ N + 1 l ] τ = τ l [ N l ] τ + [ N l − 1 ] τ τ l + ( ξ − τ l ) = ξ \tau^l+(\xi-\tau^l)=\xi τ l + ( ξ − τ l ) = ξ
Theorem 6.26 Let ϱ ~ k \widetilde{\varrho}_k ϱ k Lemma 6.25 , that is ϱ ~ k ( λ ) = Q k ( x ; 1 , τ ∣ τ ) ( τ ; τ ) k \widetilde{\varrho}_k(\lambda)=\frac{Q_k(x;1,\tau\mid\tau)}{(\tau;\tau)_k} ϱ k ( λ ) = ( τ ; τ ) k Q k ( x ; 1 , τ ∣ τ ) x = 1 − ( 1 − τ 2 ) λ 2 τ x=1-\frac{(1-\tau^2)\lambda}{2\tau} x = 1 − 2 τ ( 1 − τ 2 ) λ θ ∈ [ 0 , π ] \theta\in[0,\pi] θ ∈ [ 0 , π ] 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 )
The sequence ( ϱ ~ k ( λ ) ) k ≥ 0 (\widetilde{\varrho}_k(\lambda))_{k\geq0} ( ϱ k ( λ ) ) k ≥ 0 B ϱ ~ ( λ ) = λ ϱ ~ ( λ ) B\widetilde{\varrho}(\lambda)=\lambda\widetilde{\varrho}(\lambda) B ϱ ( λ ) = λ ϱ ( λ ) B B B [ 0 , 4 τ 1 − τ 2 ] \left[0,\frac{4\tau}{1-\tau^2}\right] [ 0 , 1 − τ 2 4 τ ] s s s B B B δ 0 \delta_0 δ 0 θ ↦ λ θ \theta\mapsto\lambda_\theta θ ↦ λ θ
( τ ; τ ) ∞ 2 ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 ∣ ( e i θ , τ e i θ ; τ ) ∞ ∣ 2 d θ 2 π , θ ∈ [ 0 , π ] . (\tau;\tau)_\infty^2\frac{|(e^{2i\theta};\tau)_\infty|^2}{|(e^{i\theta},\tau e^{i\theta};\tau)_\infty|^2}\frac{d\theta}{2\pi},\qquad\theta\in[0,\pi]. ( τ ; τ ) ∞ 2 ∣ ( e i θ , τ e i θ ; τ ) ∞ ∣ 2 ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 2 π d θ , θ ∈ [ 0 , π ] . Moreover, ⟨ g ( B ) δ j , δ k ⟩ = ∫ ϱ ~ j ϱ ~ k d s \langle g(B)\delta_j,\delta_k\rangle=\int\widetilde{\varrho}_j\widetilde{\varrho}_k\,ds ⟨ g ( B ) δ j , δ k ⟩ = ∫ ϱ j ϱ k d s g g g [ 0 , 4 τ 1 − τ 2 ] \left[0,\frac{4\tau}{1-\tau^2}\right] [ 0 , 1 − τ 2 4 τ ] j , k ∈ N 0 j,k\in\mathbb{N}_0 j , k ∈ N 0
For any d , k , j ∈ N 0 d,k,j\in\mathbb{N}_0 d , k , j ∈ N 0
ν ( d + k , k ) ( j ) = ( τ ; τ ) ∞ 4 2 π ∫ 0 π ϱ ~ d ( λ θ ) ϱ ~ j ( λ θ ) ϱ ~ k ( λ θ ) ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 ∣ ( e i θ ; τ ) ∞ ∣ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 4 d θ . \nu_{(d+k,k)}(j)=\frac{(\tau;\tau)_\infty^4}{2\pi}\int_0^\pi\widetilde{\varrho}_d(\lambda_\theta)\widetilde{\varrho}_j(\lambda_\theta)\widetilde{\varrho}_k(\lambda_\theta)\frac{|(e^{2i\theta};\tau)_\infty|^2}{|(e^{i\theta};\tau)_\infty|^2 |(\tau e^{i\theta};\tau)_\infty|^4}\,d\theta. ν ( d + k , k ) ( j ) = 2 π ( τ ; τ ) ∞ 4 ∫ 0 π ϱ d ( λ θ ) ϱ j ( λ θ ) ϱ k ( λ θ ) ∣ ( e i θ ; τ ) ∞ ∣ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 4 ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 d θ . For any j , k ∈ N 0 j,k\in\mathbb{N}_0 j , k ∈ N 0
ν ( k , k ) ( j ) = ∑ n = 0 min ( j , k ) [ j n ] τ [ k n ] τ ( τ ; τ ) n ( τ ; τ ) n + 1 τ j + k − 2 n . (77) \nu_{(k,k)}(j)=\sum_{n=0}^{\min(j,k)}\begin{bmatrix}j\\n\end{bmatrix}_{\tau}\begin{bmatrix}k\\n\end{bmatrix}_{\tau}(\tau;\tau)_n(\tau;\tau)_{n+1}\tau^{j+k-2n}. \tag*{(77)} ν ( k , k ) ( j ) = n = 0 ∑ m i n ( j , k ) [ j n ] τ [ k n ] τ ( τ ; τ ) n ( τ ; τ ) n + 1 τ j + k − 2 n . (77) and, for any d ∈ N 0 d\in\mathbb{N}_0 d ∈ N 0 ν ( d + k , k ) ( j ) = ∑ i ≥ 0 ⟨ ϱ ~ d ( B ) δ i , δ k ⟩ ν ( i , i ) ( j ) \nu_{(d+k,k)}(j)=\sum_{i\geq0}\langle\widetilde{\varrho}_d(B)\delta_i,\delta_k\rangle\nu_{(i,i)}(j) ν ( d + k , k ) ( j ) = ∑ i ≥ 0 ⟨ ϱ d ( B ) δ i , δ k ⟩ ν ( i , i ) ( j ) ⟨ ϱ ~ d ( B ) δ i , δ k ⟩ = 0 \langle\widetilde{\varrho}_d(B)\delta_i,\delta_k\rangle=0 ⟨ ϱ d ( B ) δ i , δ k ⟩ = 0 ∣ i − k ∣ > d |i-k|>d ∣ i − k ∣ > d
Proof. 1. By [45 (3.8.1)] with ( a , b ) = ( 1 , τ ) (a,b)=(1,\tau) ( a , b ) = ( 1 , τ ) ϱ ~ k = Q k ( x ; 1 , τ ∣ τ ) ( τ ; τ ) k \widetilde{\varrho}_k=\frac{Q_k(x;1,\tau\mid\tau)}{(\tau;\tau)_k} ϱ k = ( τ ; τ ) k Q k ( x ; 1 , τ ∣ τ ) [45 (3.8.4)] reads ( 1 − τ k + 1 ) ϱ ~ k + 1 + ( 1 − τ k ) ϱ ~ k − 1 = ( 2 x − τ k − τ k + 1 ) ϱ ~ k (1-\tau^{k+1})\widetilde{\varrho}_{k+1}+(1-\tau^k)\widetilde{\varrho}_{k-1}=(2x-\tau^k-\tau^{k+1})\widetilde{\varrho}_k ( 1 − τ k + 1 ) ϱ k + 1 + ( 1 − τ k ) ϱ k − 1 = ( 2 x − τ k − τ k + 1 ) ϱ k τ 1 − τ 2 \frac{\tau}{1-\tau^2} 1 − τ 2 τ τ ( 1 − τ k + 1 ) 1 − τ 2 = b k \frac{\tau(1-\tau^{k+1})}{1-\tau^2}=b_k 1 − τ 2 τ ( 1 − τ k + 1 ) = b k τ ( 2 x − τ k − τ k + 1 ) 1 − τ 2 = b k + b k − 1 − λ \frac{\tau(2x-\tau^k-\tau^{k+1})}{1-\tau^2}=b_k+b_{k-1}-\lambda 1 − τ 2 τ ( 2 x − τ k − τ k + 1 ) = b k + b k − 1 − λ b k ( ϱ ~ k − ϱ ~ k + 1 ) + b k − 1 ( ϱ ~ k − ϱ ~ k − 1 ) = λ ϱ ~ k b_k(\widetilde{\varrho}_k-\widetilde{\varrho}_{k+1})+b_{k-1}(\widetilde{\varrho}_k-\widetilde{\varrho}_{k-1})=\lambda\widetilde{\varrho}_k b k ( ϱ k − ϱ k + 1 ) + b k − 1 ( ϱ k − ϱ k − 1 ) = λ ϱ k B ϱ ~ = λ ϱ ~ B\widetilde{\varrho}=\lambda\widetilde{\varrho} B ϱ = λ ϱ B B B ϱ ~ 0 = 1 \widetilde{\varrho}_0=1 ϱ 0 = 1 δ k ↦ ϱ ~ k \delta_k\mapsto\widetilde{\varrho}_k δ k ↦ ϱ k ℓ N 0 2 \ell^2_{\mathbb{N}_0} ℓ N 0 2 L 2 ( s ) L^2(s) L 2 ( s ) B B B λ \lambda λ [71 ] . This gives the last assertion. To identify s s s 1 2 ≤ a < 1 \frac{1}{2}\leq a<1 2 1 ≤ a < 1 w a ( θ ) ≔ ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 ∣ ( a e i θ , τ e i θ ; τ ) ∞ ∣ 2 w_a(\theta)\coloneqq\frac{|(e^{2i\theta};\tau)_\infty|^2}{|(ae^{i\theta},\tau e^{i\theta};\tau)_\infty|^2} w a ( θ ) : = ∣ ( a e i θ , τ e i θ ; τ ) ∞ ∣ 2 ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 [45 (3.8.2)] , the polynomials Q k ( x ; a , τ ∣ τ ) ( ( τ ; τ ) k ( a τ ; τ ) k ) 1 / 2 \frac{Q_k(x;a,\tau\mid\tau)}{\bigl((\tau;\tau)_k(a\tau;\tau)_k\bigr)^{1/2}} ( ( τ ; τ ) k ( a τ ; τ ) k ) 1/2 Q k ( x ; a , τ ∣ τ ) ( τ , a τ ; τ ) ∞ w a ( θ ) d θ 2 π (\tau,a\tau;\tau)_\infty w_a(\theta)\frac{d\theta}{2\pi} ( τ , a τ ; τ ) ∞ w a ( θ ) 2 π d θ [ 0 , π ] [0,\pi] [ 0 , π ] x = cos θ x=\cos\theta x = cos θ [45 (3.8.4)] , converge to those of ( ϱ ~ k ) (\widetilde{\varrho}_k) ( ϱ k ) a → 1 a\to1 a → 1 λ θ \lambda_\theta λ θ ⟨ B n δ 0 , δ 0 ⟩ \langle B^n\delta_0,\delta_0\rangle ⟨ B n δ 0 , δ 0 ⟩ s s s ∣ 1 − e i θ ∣ 2 ≤ 2 ∣ 1 − a e i θ ∣ 2 |1-e^{i\theta}|^2\leq2|1-ae^{i\theta}|^2 ∣1 − e i θ ∣ 2 ≤ 2∣1 − a e i θ ∣ 2 a ≥ 1 2 a\geq\frac{1}{2} a ≥ 2 1 w a w_a w a a a a w a → w 1 w_a\to w_1 w a → w 1 w 1 w_1 w 1 [ 0 , π ] [0,\pi] [ 0 , π ] w a = w 1 ∣ ( e i θ ; τ ) ∞ ∣ 2 ∣ ( a e i θ ; τ ) ∞ ∣ 2 w_a=w_1\frac{|(e^{i\theta};\tau)_\infty|^2}{|(ae^{i\theta};\tau)_\infty|^2} w a = w 1 ∣ ( a e i θ ; τ ) ∞ ∣ 2 ∣ ( e i θ ; τ ) ∞ ∣ 2 2 ∏ j ≥ 1 ( 1 + τ j 1 − τ j ) 2 2\prod_{j\geq1}\left(\frac{1+\tau^j}{1-\tau^j}\right)^2 2 ∏ j ≥ 1 ( 1 − τ j 1 + τ j ) 2 s s s { λ θ : θ ∈ [ 0 , π ] } = [ 0 , 4 τ 1 − τ 2 ] \{\lambda_\theta:\theta\in[0,\pi]\}=\left[0,\frac{4\tau}{1-\tau^2}\right] { λ θ : θ ∈ [ 0 , π ]} = [ 0 , 1 − τ 2 4 τ ] B B B
For θ ∈ [ 0 , π ] \theta\in[0,\pi] θ ∈ [ 0 , π ] z = e i θ z=e^{i\theta} z = e i θ (65) with λ = λ θ \lambda=\lambda_\theta λ = λ θ
ϱ ~ ∞ ( λ θ ) = ∣ ( τ e i θ ; τ ) ∞ ∣ 2 ( τ ; τ ) ∞ 2 . (78) \widetilde{\varrho}_\infty(\lambda_\theta)=\frac{|(\tau e^{i\theta};\tau)_\infty|^2}{(\tau;\tau)_\infty^2}. \tag*{(78)} ϱ ∞ ( λ θ ) = ( τ ; τ ) ∞ 2 ∣ ( τ e i θ ; τ ) ∞ ∣ 2 . (78) The second part follows from (64) and from the first part with g = ϱ d ϱ ∞ g=\frac{\varrho_d}{\varrho_\infty} g = ϱ ∞ ϱ d
Let d = 0 d=0 d = 0 ν ( k , k ) ( j ) = ( τ ; τ ) ∞ 4 ∫ 0 π ϱ j ϱ k w 1 , τ , τ d θ 2 π \nu_{(k,k)}(j)=(\tau;\tau)_{\infty}^{4}\int_{0}^{\pi}\varrho_j\varrho_k w_{1,\tau,\tau}\,\frac{\mathrm{d}\theta}{2\pi} ν ( k , k ) ( j ) = ( τ ; τ ) ∞ 4 ∫ 0 π ϱ j ϱ k w 1 , τ , τ 2 π d θ w a , b , c ( θ ) = ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 ∣ ( a e i θ , b e i θ , c e i θ ; τ ) ∞ ∣ 2 w_{a,b,c}(\theta)=\frac{|(e^{2\mathrm{i}\theta};\tau)_{\infty}|^{2}}{|(ae^{\mathrm{i}\theta},be^{\mathrm{i}\theta},ce^{\mathrm{i}\theta};\tau)_{\infty}|^{2}} w a , b , c ( θ ) = ∣ ( a e i θ , b e i θ , c e i θ ; τ ) ∞ ∣ 2 ∣ ( e 2 i θ ; τ ) ∞ ∣ 2 τ \tau τ [45 (3.3.1) and (3.3.2)] with ( a , b , c ) = ( a , τ , τ ) (a,b,c)=(a,\tau,\tau) ( a , b , c ) = ( a , τ , τ ) 0 < a < 1 0<a<1 0 < a < 1 a → 1 a\to1 a → 1 H n H_n H n Lemma 6.25 satisfy
∫ 0 π H m H n w 1 , τ , τ d θ 2 π = δ m n ( τ ; τ ) n 4 ( τ n + 1 ; τ ) ∞ 3 ( τ n + 2 ; τ ) ∞ = 1 − τ n + 1 ( τ ; τ ) ∞ 4 δ m n . \int_{0}^{\pi} H_mH_n w_{1,\tau,\tau}\,\frac{\mathrm{d}\theta}{2\pi}
=\frac{\delta_{mn}}{(\tau;\tau)_n^4(\tau^{n+1};\tau)_{\infty}^3(\tau^{n+2};\tau)_{\infty}}
=\frac{1-\tau^{n+1}}{(\tau;\tau)_{\infty}^4}\delta_{mn}. ∫ 0 π H m H n w 1 , τ , τ 2 π d θ = ( τ ; τ ) n 4 ( τ n + 1 ; τ ) ∞ 3 ( τ n + 2 ; τ ) ∞ δ mn = ( τ ; τ ) ∞ 4 1 − τ n + 1 δ mn . By Lemma 6.25 , we deduce that
ν ( k , k ) ( j ) = ∑ n ≥ 0 ( τ ; τ ) j ( τ ; τ ) k ( τ ; τ ) j − n ( τ ; τ ) k − n τ j + k − 2 n ( 1 − τ n + 1 ) , \nu_{(k,k)}(j)=\sum_{n\ge0}\frac{(\tau;\tau)_j(\tau;\tau)_k}{(\tau;\tau)_{j-n}(\tau;\tau)_{k-n}}\tau^{j+k-2n}(1-\tau^{n+1}), ν ( k , k ) ( j ) = n ≥ 0 ∑ ( τ ; τ ) j − n ( τ ; τ ) k − n ( τ ; τ ) j ( τ ; τ ) k τ j + k − 2 n ( 1 − τ n + 1 ) , the sum being over n ≤ min ( j , k ) n\le\min(j,k) n ≤ min ( j , k ) (77) , since ( τ ; τ ) j ( τ ; τ ) j − n = ( j n ) τ ( τ ; τ ) n \frac{(\tau;\tau)_j}{(\tau;\tau)_{j-n}}=\binom{j}{n}_{\tau}(\tau;\tau)_n ( τ ; τ ) j − n ( τ ; τ ) j = ( n j ) τ ( τ ; τ ) n ϱ d ϱ ∞ ( B ) = ϱ d ( B ) ϱ ∞ ( B ) − 1 \frac{\varrho_d}{\varrho_{\infty}}(B)=\varrho_d(B)\varrho_{\infty}(B)^{-1} ϱ ∞ ϱ d ( B ) = ϱ d ( B ) ϱ ∞ ( B ) − 1 ϱ ∞ ( B ) − 1 \varrho_{\infty}(B)^{-1} ϱ ∞ ( B ) − 1 ( ν ( i , i ) ( j ) ) i , j (\nu_{(i,i)}(j))_{i,j} ( ν ( i , i ) ( j ) ) i , j Theorem 6.13 , and ϱ d ( B ) \varrho_d(B) ϱ d ( B ) d d d B B B
Remark 6.27 For instance, (77) gives ν ( 1 , 1 ) ( 1 ) = ( 1 − τ ) ( 1 − τ + τ 3 ) \nu_{(1,1)}(1)=(1-\tau)(1-\tau+\tau^3) ν ( 1 , 1 ) ( 1 ) = ( 1 − τ ) ( 1 − τ + τ 3 ) Theorem 6.26 gives ν ( 1 , 0 ) ( 0 ) = ( 1 + τ ) ν ( 0 , 0 ) ( 0 ) − τ ν ( 1 , 1 ) ( 0 ) = ( 1 − τ ) ( 1 + τ − τ 2 ) \nu_{(1,0)}(0)=(1+\tau)\nu_{(0,0)}(0)-\tau\nu_{(1,1)}(0)=(1-\tau)(1+\tau-\tau^2) ν ( 1 , 0 ) ( 0 ) = ( 1 + τ ) ν ( 0 , 0 ) ( 0 ) − τ ν ( 1 , 1 ) ( 0 ) = ( 1 − τ ) ( 1 + τ − τ 2 ) ϱ 1 ( B ) = I d + ( 1 + τ ) B \varrho_1(B)=\mathrm{Id}+(1+\tau)B ϱ 1 ( B ) = Id + ( 1 + τ ) B (77) are nonnegative, which makes the positivity of the Poisson kernel visible, and the matrix ( ν ( k , k ) ( j ) ) (\nu_{(k,k)}(j)) ( ν ( k , k ) ( j )) C Λ C T C\Lambda C^{\mathsf{T}} C Λ C T C C C C ( k , n ) = ( τ ; τ ) k ( τ ; τ ) k − n τ k − n C(k,n)=\frac{(\tau;\tau)_k}{(\tau;\tau)_{k-n}}\tau^{k-n} C ( k , n ) = ( τ ; τ ) k − n ( τ ; τ ) k τ k − n Λ = diag ( 1 − τ n + 1 ) \Lambda=\operatorname{diag}(1-\tau^{n+1}) Λ = diag ( 1 − τ n + 1 )
Concerning the literature: the evaluation (6.22) is a special case of [22 Lemma 4.17] , the discrete orthogonality of Theorem 6.17 is a special case of [4 ] , see [22 Theorem 4.16] , the eigenvectors of Remark 6.18 are given by the duality of [6 ] , and the law of the explosion time of a birth–death process with an exit boundary as a sum of independent exponential variables is due to [33 ] , which extends a theorem of Keilson for finite chains, see [28 ] . The transition probabilities of the diagonal chain are due to Biane [15 Section 4] , see the paragraph after Proposition 6.2 , and its restrictions to the centre and to the torus are those considered in [11 ] for S U ( n ) \mathrm{SU}(n) SU ( n ) [23 ] . The identification of the objects above with the Poisson kernel of the diagonal chain, Theorem 6.22, Corollary 6.24 , (77) and parts 3 and 4 of Proposition 6.11 are new, as far as we know. The Martin boundary theory of quantum random walks and their restrictions to commutative subalgebras were studied in [13 ] , [53 ] , [23 ] and [41 ] . The formulas of this subsection were also checked numerically, with 20 to 40 digits, for τ ∈ { 1 2 , 1 3 , 1 4 } \tau\in\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\} τ ∈ { 2 1 , 3 1 , 4 1 } (77) were compared with Monte Carlo simulations of the chain X X X Proposition 6.2 .
Energy of Poisson integrals
Let cond ( v , v ′ ) = d e f w ^ ( v ) P ( v , v ′ ) = cond ( v ′ , v ) \operatorname{cond}(v,v')\overset{\mathrm{def}}{=}\widehat{w}(v)P(v,v')=\operatorname{cond}(v',v) cond ( v , v ′ ) = def w ( v ) P ( v , v ′ ) = cond ( v ′ , v ) u u u D \mathbb{D} D Γ ( u ) ( v ) = ∑ v ′ P ( v , v ′ ) ∣ u ( v ′ ) ∣ 2 − ∣ u ( v ) ∣ 2 = ∑ v ′ P ( v , v ′ ) ∣ u ( v ′ ) − u ( v ) ∣ 2 \Gamma(u)(v)=\sum_{v'}P(v,v')|u(v')|^2-|u(v)|^2=\sum_{v'}P(v,v')|u(v')-u(v)|^2 Γ ( u ) ( v ) = ∑ v ′ P ( v , v ′ ) ∣ u ( v ′ ) ∣ 2 − ∣ u ( v ) ∣ 2 = ∑ v ′ P ( v , v ′ ) ∣ u ( v ′ ) − u ( v ) ∣ 2
E ( n ) ( u ) = ∑ k = 0 n ∑ v ′ cond ( ( n , k ) , v ′ ) ∣ u ( v ′ ) − u ( n , k ) ∣ 2 . (79) \mathcal{E}^{(n)}(u)=\sum_{k=0}^{n}\sum_{v'}\operatorname{cond}((n,k),v')|u(v')-u(n,k)|^2. \tag*{(79)} E ( n ) ( u ) = k = 0 ∑ n v ′ ∑ cond (( n , k ) , v ′ ) ∣ u ( v ′ ) − u ( n , k ) ∣ 2 . (79) where v ′ v' v ′ ( n , k ) (n,k) ( n , k ) E ( n ) ( u ) \mathcal{E}^{(n)}(u) E ( n ) ( u ) n n n n + 1 n+1 n + 1 n − 1 n-1 n − 1 n n n ( n , k ) (n,k) ( n , k ) ( n + 1 , k + 1 ) (n+1,k+1) ( n + 1 , k + 1 ) c τ 2 w k β k \frac{c_\tau}{2}w_k\beta_k 2 c τ w k β k n ≥ k n\ge k n ≥ k
cond ( ( n , k ) , ( n ± 1 , k ) ) ≤ 1 ( 1 − τ ) τ n − k . (80) \operatorname{cond}((n,k),(n\pm1,k))\le\frac{1}{(1-\tau)\tau^{n-k}}. \tag*{(80)} cond (( n , k ) , ( n ± 1 , k )) ≤ ( 1 − τ ) τ n − k 1 . (80) Proposition 6.28 For any bounded function f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C
lim inf n → ∞ E ( n ) ( H f ) ≥ c τ E τ ( f ) ∈ [ 0 , ∞ ] . \liminf_{n\to\infty}\mathcal{E}^{(n)}(\mathcal{H}f)\ge c_\tau\mathcal{E}_\tau(f)\in[0,\infty]. n → ∞ lim inf E ( n ) ( H f ) ≥ c τ E τ ( f ) ∈ [ 0 , ∞ ] . In particular, if f f f E G ( H f ) = ∞ \mathcal{E}_{G}(\mathcal{H}f)=\infty E G ( H f ) = ∞
Proof: Keeping only the horizontal edges in (79) , we obtain for any integers $0\le J<n
E ( n ) ( H f ) ≥ c τ 2 ∑ k = 0 J w k β k ( ∣ H f ( n + 1 , k + 1 ) − H f ( n , k ) ∣ 2 + ∣ H f ( n , k + 1 ) − H f ( n − 1 , k ) ∣ 2 ) . \mathcal{E}^{(n)}(\mathcal{H}f)\ge\frac{c_\tau}{2}\sum_{k=0}^{J}w_k\beta_k\left(\left|\mathcal{H}f(n+1,k+1)-\mathcal{H}f(n,k)\right|^2+\left|\mathcal{H}f(n,k+1)-\mathcal{H}f(n-1,k)\right|^2\right). E ( n ) ( H f ) ≥ 2 c τ k = 0 ∑ J w k β k ( ∣ H f ( n + 1 , k + 1 ) − H f ( n , k ) ∣ 2 + ∣ H f ( n , k + 1 ) − H f ( n − 1 , k ) ∣ 2 ) . By (56) , each term of the sum converges to 2 w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 2w_k\beta_k|f(k+1)-f(k)|^2 2 w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 n → ∞ n\to\infty n → ∞ lim inf n E ( n ) ( H f ) ≥ c τ ∑ k = 0 J w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 \liminf_{n}\mathcal{E}^{(n)}(\mathcal{H}f)\ge c_\tau\sum_{k=0}^{J}w_k\beta_k|f(k+1)-f(k)|^2 lim inf n E ( n ) ( H f ) ≥ c τ ∑ k = 0 J w k β k ∣ f ( k + 1 ) − f ( k ) ∣ 2 J → ∞ J\to\infty J → ∞ f f f E τ ( f ) > 0 \mathcal{E}_\tau(f)>0 E τ ( f ) > 0 w k β k w_k\beta_k w k β k ∑ n E ( n ) ( H f ) \sum_n\mathcal{E}^{(n)}(\mathcal{H}f) ∑ n E ( n ) ( H f )
Theorem 6.29 Let F F F k ↦ F ( τ k ) k\mapsto F(\tau^k) k ↦ F ( τ k )
E ( n ) ( H F ) = c τ E τ ( F ) + O ( ( n + 1 ) 2 τ n ) . \mathcal{E}^{(n)}(\mathcal{H}F)=c_\tau\mathcal{E}_\tau(F)+O\left((n+1)^2\tau^n\right). E ( n ) ( H F ) = c τ E τ ( F ) + O ( ( n + 1 ) 2 τ n ) . In particular, lim n → ∞ E ( n ) ( H F ) = c τ E τ ( F ) = c τ ⟨ F , L τ F ⟩ ℓ N 0 2 ( ω ) \lim_{n\to\infty}\mathcal{E}^{(n)}(\mathcal{H}F)=c_\tau\mathcal{E}_\tau(F)=c_\tau\langle F,L_\tau F\rangle_{\ell^2_{\mathbb{N}_0}(\omega)} lim n → ∞ E ( n ) ( H F ) = c τ E τ ( F ) = c τ ⟨ F , L τ F ⟩ ℓ N 0 2 ( ω )
Proof: Write F = ∑ l = 0 L a l φ l τ F=\sum_{l=0}^{L}a_l\varphi_l^\tau F = ∑ l = 0 L a l φ l τ (60) and H φ 0 τ = 1 \mathcal{H}\varphi_0^\tau=1 H φ 0 τ = 1 C F C_F C F ∣ H F ( n , k ) − F ( τ k ) ∣ ≤ C F ( n + 1 ) τ n |\mathcal{H}F(n,k)-F(\tau^k)|\le C_F(n+1)\tau^n ∣ H F ( n , k ) − F ( τ k ) ∣ ≤ C F ( n + 1 ) τ n ( n , k ) ∈ D (n,k)\in\mathbb{D} ( n , k ) ∈ D ∣ F ( τ k + 1 ) − F ( τ k ) ∣ ≤ ∥ F ′ ∥ L ∞ ( [ 0 , 1 ] ) τ k |F(\tau^{k+1})-F(\tau^k)|\le\|F'\|_{L^\infty([0,1])}\tau^k ∣ F ( τ k + 1 ) − F ( τ k ) ∣ ≤ ∥ F ′ ∥ L ∞ ([ 0 , 1 ]) τ k
We first consider the vertical edges at the level n ≥ 1 n\ge1 n ≥ 1 k k k ∣ H F ( n ± 1 , k ) − H F ( n , k ) ∣ ≤ 2 C F ( n + 2 ) τ n − 1 |\mathcal{H}F(n\pm1,k)-\mathcal{H}F(n,k)|\le2C_F(n+2)\tau^{n-1} ∣ H F ( n ± 1 , k ) − H F ( n , k ) ∣ ≤ 2 C F ( n + 2 ) τ n − 1 ( n ± 1 , k ) ∈ D (n\pm1,k)\in\mathbb{D} ( n ± 1 , k ) ∈ D ∑ k = 0 n τ − ( n − k ) ≤ τ − n 1 − τ \sum_{k=0}^{n}\tau^{-(n-k)}\le\frac{\tau^{-n}}{1-\tau} ∑ k = 0 n τ − ( n − k ) ≤ 1 − τ τ − n (80) shows that the contribution of the vertical edges to (79) is at most
2 ∑ k = 0 n τ − ( n − k ) 1 − τ ⋅ 4 C F 2 ( n + 2 ) 2 τ 2 n − 2 ≤ 8 C F 2 τ 2 ( 1 − τ ) 2 ( n + 2 ) 2 τ n . 2\sum_{k=0}^{n}\frac{\tau^{-(n-k)}}{1-\tau}\cdot4C_F^2(n+2)^2\tau^{2n-2}\le\frac{8C_F^2}{\tau^2(1-\tau)^2}(n+2)^2\tau^n. 2 k = 0 ∑ n 1 − τ τ − ( n − k ) ⋅ 4 C F 2 ( n + 2 ) 2 τ 2 n − 2 ≤ τ 2 ( 1 − τ ) 2 8 C F 2 ( n + 2 ) 2 τ n . Now, we turn to the horizontal edges between the levels n n n n + 1 n+1 n + 1 H F ( n + 1 , k + 1 ) − H F ( n , k ) = F ( τ k + 1 ) − F ( τ k ) + ε n ( k ) \mathcal{H}F(n+1,k+1)-\mathcal{H}F(n,k)=F(\tau^{k+1})-F(\tau^k)+\varepsilon_n(k) H F ( n + 1 , k + 1 ) − H F ( n , k ) = F ( τ k + 1 ) − F ( τ k ) + ε n ( k ) ∣ ε n ( k ) ∣ ≤ 2 C F ( n + 2 ) τ n |\varepsilon_n(k)|\le2C_F(n+2)\tau^n ∣ ε n ( k ) ∣ ≤ 2 C F ( n + 2 ) τ n c τ 2 w k β k ≤ c τ ( 1 − τ ) 2 τ k \frac{c_\tau}{2}w_k\beta_k\le\frac{c_\tau(1-\tau)}{2}\tau^k 2 c τ w k β k ≤ 2 c τ ( 1 − τ ) τ k
∣ c τ 2 ∑ k = 0 n w k β k ( ∣ H F ( n + 1 , k + 1 ) − H F ( n , k ) ∣ 2 − ∣ F ( τ k + 1 ) − F ( τ k ) ∣ 2 ) ∣ ≤ C ∑ k = 0 n ( τ k ( n + 2 ) τ n + ( n + 2 ) 2 τ 2 n ) . \left|\frac{c_\tau}{2}\sum_{k=0}^{n}w_k\beta_k\left(\left|\mathcal{H}F(n+1,k+1)-\mathcal{H}F(n,k)\right|^2-\left|F(\tau^{k+1})-F(\tau^k)\right|^2\right)\right|
\le C\sum_{k=0}^{n}\left(\tau^k(n+2)\tau^n+(n+2)^2\tau^{2n}\right). 2 c τ k = 0 ∑ n w k β k ( ∣ H F ( n + 1 , k + 1 ) − H F ( n , k ) ∣ 2 − F ( τ k + 1 ) − F ( τ k ) 2 ) ≤ C k = 0 ∑ n ( τ k ( n + 2 ) τ n + ( n + 2 ) 2 τ 2 n ) . for some constant C C C O ( ( n + 1 ) τ n ) O((n+1)\tau^n) O (( n + 1 ) τ n ) ∑ k > n w k β k ∣ F ( τ k + 1 ) − F ( τ k ) ∣ 2 = O ( τ 2 n ) \sum_{k>n}w_k\beta_k|F(\tau^{k+1})-F(\tau^k)|^2=O(\tau^{2n}) ∑ k > n w k β k ∣ F ( τ k + 1 ) − F ( τ k ) ∣ 2 = O ( τ 2 n ) n n n n + 1 n+1 n + 1 c τ 2 E τ ( F ) + O ( ( n + 1 ) τ n ) \frac{c_\tau}{2}\mathcal{E}_\tau(F)+O((n+1)\tau^n) 2 c τ E τ ( F ) + O (( n + 1 ) τ n ) n − 1 n-1 n − 1 n n n c τ 2 E τ ( F ) + O ( ( n + 1 ) τ n ) \frac{c_\tau}{2}\mathcal{E}_\tau(F)+O((n+1)\tau^n) 2 c τ E τ ( F ) + O (( n + 1 ) τ n ) (24) .
For integers 0 ≤ l ≤ m 0\le l\le m 0 ≤ l ≤ m
h l , m ≔ ∑ k = 0 m τ k M l , m ( τ k ) 2 , h l , ∞ ≔ ∑ k ≥ 0 τ k p ^ l ( τ k ) 2 . h_{l,m}\coloneqq\sum_{k=0}^{m}\tau^k M_{l,m}(\tau^k)^2,
\qquad
h_{l,\infty}\coloneqq\sum_{k\geq0}\tau^k\widehat p_l(\tau^k)^2. h l , m : = k = 0 ∑ m τ k M l , m ( τ k ) 2 , h l , ∞ : = k ≥ 0 ∑ τ k p l ( τ k ) 2 . For any function u u u D \mathbb{D} D m ≥ 0 m\geq0 m ≥ 0
ε m ( u ) ≔ ∑ k = 0 m ∑ k ′ = 0 m + 1 cond ( ( m , k ) , ( m + 1 , k ′ ) ) ∣ u ( m + 1 , k ′ ) − u ( m , k ) ∣ 2 \varepsilon_m(u)\coloneqq\sum_{k=0}^{m}\sum_{k'=0}^{m+1}\operatorname{cond}((m,k),(m+1,k'))\lvert u(m+1,k')-u(m,k)\rvert^2 ε m ( u ) : = k = 0 ∑ m k ′ = 0 ∑ m + 1 cond (( m , k ) , ( m + 1 , k ′ )) ∣ u ( m + 1 , k ′ ) − u ( m , k ) ∣ 2 the energy of the edges between the levels m m m m + 1 m+1 m + 1 ε − 1 ( u ) ≔ 0 \varepsilon_{-1}(u)\coloneqq0 ε − 1 ( u ) : = 0 (79) , we have E ( n ) ( u ) = ε n − 1 ( u ) + ε n ( u ) \mathcal{E}^{(n)}(u)=\varepsilon_{n-1}(u)+\varepsilon_n(u) E ( n ) ( u ) = ε n − 1 ( u ) + ε n ( u ) n ≥ 0 n\geq0 n ≥ 0
Proposition 6.30 Let l ≥ 1 l\geq1 l ≥ 1 u l ≔ H φ l τ u_l\coloneqq H\varphi_l^\tau u l : = H φ l τ
For any integer m ≥ l m\geq l m ≥ l ( 1 − τ m + 1 − l ) h l , m + 1 = ( 1 − τ m + l + 2 ) h l , m (1-\tau^{m+1-l})h_{l,m+1}=(1-\tau^{m+l+2})h_{l,m} ( 1 − τ m + 1 − l ) h l , m + 1 = ( 1 − τ m + l + 2 ) h l , m
h l , m = ( τ 2 l + 2 ; τ ) m − l ( τ ; τ ) ∞ ( τ ; τ ) m − l ( τ 2 l + 2 ; τ ) ∞ h l , ∞ ≤ h l , ∞ ( τ ; τ ) ∞ , m ≥ l . h_{l,m}=\frac{(\tau^{2l+2};\tau)_{m-l}(\tau;\tau)_\infty}{(\tau;\tau)_{m-l}(\tau^{2l+2};\tau)_\infty}h_{l,\infty}\leq\frac{h_{l,\infty}}{(\tau;\tau)_\infty},\qquad m\geq l. h l , m = ( τ ; τ ) m − l ( τ 2 l + 2 ; τ ) ∞ ( τ 2 l + 2 ; τ ) m − l ( τ ; τ ) ∞ h l , ∞ ≤ ( τ ; τ ) ∞ h l , ∞ , m ≥ l . We have ε m ( u l ) = 0 \varepsilon_m(u_l)=0 ε m ( u l ) = 0 m ≤ l − 2 m\leq l-2 m ≤ l − 2 ε l − 1 ( u l ) = ( 1 − τ l ) 2 h l , l ( 1 − τ 2 ) τ l − 1 ς l ( ∞ ) 2 \varepsilon_{l-1}(u_l)=\frac{(1-\tau^l)^2h_{l,l}}{(1-\tau^2)\tau^{l-1}\varsigma_l(\infty)^2} ε l − 1 ( u l ) = ( 1 − τ 2 ) τ l − 1 ς l ( ∞ ) 2 ( 1 − τ l ) 2 h l , l m ≥ l m\geq l m ≥ l
ε m ( u l ) = ϰ l 2 h l , m ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 [ ( 1 − τ m + 2 ) ς l ( m ) 2 − 2 ( 1 − τ m + l + 2 ) ς l ( m ) ς l ( m + 1 ) + ( 1 − τ m + 1 ) ( 1 − τ m + l + 2 ) 1 − τ m + 1 − l ς l ( m + 1 ) 2 ] . (81) \varepsilon_m(u_l)=\frac{\varkappa_l^2h_{l,m}}{(1-\tau^2)\tau^m\varsigma_l(\infty)^2}\left[(1-\tau^{m+2})\varsigma_l(m)^2-2(1-\tau^{m+l+2})\varsigma_l(m)\varsigma_l(m+1)+\frac{(1-\tau^{m+1})(1-\tau^{m+l+2})}{1-\tau^{m+1-l}}\varsigma_l(m+1)^2\right]. \tag*{(81)} ε m ( u l ) = ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 ϰ l 2 h l , m [ ( 1 − τ m + 2 ) ς l ( m ) 2 − 2 ( 1 − τ m + l + 2 ) ς l ( m ) ς l ( m + 1 ) + 1 − τ m + 1 − l ( 1 − τ m + 1 ) ( 1 − τ m + l + 2 ) ς l ( m + 1 ) 2 ] . (81) There exists a constant C τ C_\tau C τ τ \tau τ E ( n ) ( u l ) ≤ C τ Λ l ( τ ) ∥ φ l τ ∥ ℓ 0 2 ( w ) 2 \mathcal{E}^{(n)}(u_l)\leq C_\tau\Lambda_l(\tau)\|\varphi_l^\tau\|_{\ell^2_0(\mathbf{w})}^2 E ( n ) ( u l ) ≤ C τ Λ l ( τ ) ∥ φ l τ ∥ ℓ 0 2 ( w ) 2 l ≥ 1 l\geq1 l ≥ 1 n ≥ 0 n\geq0 n ≥ 0
Proof: By Proposition 6.2, from any vertex ( m , k ) (m,k) ( m , k ) m + 1 m+1 m + 1 m − 1 m-1 m − 1
p ↑ ( m ) = 1 − τ m + 2 ( 1 + τ ) ( 1 − τ m + 1 ) , p ↓ ( m ) = τ ( 1 − τ m ) ( 1 + τ ) ( 1 − τ m + 1 ) , p_\uparrow(m)=\frac{1-\tau^{m+2}}{(1+\tau)(1-\tau^{m+1})},\qquad
p_\downarrow(m)=\frac{\tau(1-\tau^m)}{(1+\tau)(1-\tau^{m+1})}, p ↑ ( m ) = ( 1 + τ ) ( 1 − τ m + 1 ) 1 − τ m + 2 , p ↓ ( m ) = ( 1 + τ ) ( 1 − τ m + 1 ) τ ( 1 − τ m ) , which do not depend on k k k (52) , we have w ^ ( m , k ) = 1 − τ m + 1 ( 1 − τ ) τ m τ k \widehat w(m,k)=\frac{1-\tau^{m+1}}{(1-\tau)\tau^m}\tau^k w ( m , k ) = ( 1 − τ ) τ m 1 − τ m + 1 τ k
Let m ≥ l m\geq l m ≥ l m ≥ 1 m\geq1 m ≥ 1 j ≥ 0 j\geq0 j ≥ 0 m m m ( n ′ , k ′ ) ↦ 1 { m + 1 } ( n ′ ) τ j k ′ (n',k')\mapsto\mathbf{1}_{\{m+1\}}(n')\tau^{j k'} ( n ′ , k ′ ) ↦ 1 { m + 1 } ( n ′ ) τ j k ′ k ↦ ∑ k ′ P ( ( m , k ) , ( m + 1 , k ′ ) ) τ j k ′ k\mapsto\sum_{k'}P((m,k),(m+1,k'))\tau^{j k'} k ↦ ∑ k ′ P (( m , k ) , ( m + 1 , k ′ )) τ j k ′ 1 − τ m + j + 2 ( 1 + τ ) ( 1 − τ m + 1 ) τ j k \frac{1-\tau^{m+j+2}}{(1+\tau)(1-\tau^{m+1})}\tau^{jk} ( 1 + τ ) ( 1 − τ m + 1 ) 1 − τ m + j + 2 τ j k τ ( j − 1 ) k \tau^{(j-1)k} τ ( j − 1 ) k m + 1 m+1 m + 1 ( n ′ , k ′ ) ↦ 1 { m } ( n ′ ) τ j k ′ (n',k')\mapsto\mathbf{1}_{\{m\}}(n')\tau^{j k'} ( n ′ , k ′ ) ↦ 1 { m } ( n ′ ) τ j k ′ k ′ ↦ ∑ k P ( ( m + 1 , k ′ ) , ( m , k ) ) τ j k k'\mapsto\sum_kP((m+1,k'),(m,k))\tau^{j k} k ′ ↦ ∑ k P (( m + 1 , k ′ ) , ( m , k )) τ j k τ − τ m + 2 − j ( 1 + τ ) ( 1 − τ m + 2 ) τ j k ′ \frac{\tau-\tau^{m+2-j}}{(1+\tau)(1-\tau^{m+2})}\tau^{j k'} ( 1 + τ ) ( 1 − τ m + 2 ) τ − τ m + 2 − j τ j k ′ τ ( j − 1 ) k ′ \tau^{(j-1)k'} τ ( j − 1 ) k ′
b m ≔ ∑ k , k ′ cond ( ( m , k ) , ( m + 1 , k ′ ) ) M l , m ( τ k ) M l , m + 1 ( τ k ′ ) . b_m\coloneqq\sum_{k,k'}\operatorname{cond}((m,k),(m+1,k'))M_{l,m}(\tau^k)M_{l,m+1}(\tau^{k'}). b m : = k , k ′ ∑ cond (( m , k ) , ( m + 1 , k ′ )) M l , m ( τ k ) M l , m + 1 ( τ k ′ ) . Since M l , m M_{l,m} M l , m M l , m + 1 M_{l,m+1} M l , m + 1 l l l l − 1 l-1 l − 1 L 2 ( m m ) L^2(\mathbf{m}_m) L 2 ( m m ) L 2 ( m m + 1 ) L^2(\mathbf{m}_{m+1}) L 2 ( m m + 1 )
b m = ∑ k w ^ ( m , k ) M l , m ( τ k ) ∑ k ′ P ( ( m , k ) , ( m + 1 , k ′ ) ) M l , m + 1 ( τ k ′ ) = ( 1 − τ m + l + 2 ) h l , m ( 1 − τ 2 ) τ m , b_m=\sum_k\widehat w(m,k)M_{l,m}(\tau^k)\sum_{k'}P((m,k),(m+1,k'))M_{l,m+1}(\tau^{k'})=\frac{(1-\tau^{m+l+2})h_{l,m}}{(1-\tau^2)\tau^m}, b m = k ∑ w ( m , k ) M l , m ( τ k ) k ′ ∑ P (( m , k ) , ( m + 1 , k ′ )) M l , m + 1 ( τ k ′ ) = ( 1 − τ 2 ) τ m ( 1 − τ m + l + 2 ) h l , m , and, by reversibility,
b m = ∑ k ′ w ^ ( m + 1 , k ′ ) M l , m + 1 ( τ k ′ ) ∑ k P ( ( m + 1 , k ′ ) , ( m , k ) ) M l , m ( τ k ) = ( 1 − τ m + 1 − l ) h l , m + 1 ( 1 − τ 2 ) τ m . b_m=\sum_{k'}\widehat{w}(m+1,k')M_{l,m+1}(\tau^{k'})\sum_k\mathbb{P}((m+1,k'),(m,k))M_{l,m}(\tau^k)=\frac{(1-\tau^{m+1-l})h_{l,m+1}}{(1-\tau^2)\tau^m}. b m = k ′ ∑ w ( m + 1 , k ′ ) M l , m + 1 ( τ k ′ ) k ∑ P (( m + 1 , k ′ ) , ( m , k )) M l , m ( τ k ) = ( 1 − τ 2 ) τ m ( 1 − τ m + 1 − l ) h l , m + 1 . This proves the first identity. By the proof of part 2 of Proposition 6.9 , the coefficients of M l , m M_{l,m} M l , m p ^ l \widehat{p}_l p l m m \mathfrak{m}_m m m m ∞ \mathfrak{m}_\infty m ∞ h l , m → h l , ∞ h_{l,m}\to h_{l,\infty} h l , m → h l , ∞ h l , m = h l , ∞ ∏ i ≥ m 1 − τ i + 1 − l 1 − τ i + l + 2 h_{l,m}=h_{l,\infty}\prod_{i\geq m}\frac{1-\tau^{i+1-l}}{1-\tau^{i+l+2}} h l , m = h l , ∞ ∏ i ≥ m 1 − τ i + l + 2 1 − τ i + 1 − l ( τ 2 l + 2 ; τ ) m − l ≤ 1 (\tau^{2l+2};\tau)_{m-l}\leq1 ( τ 2 l + 2 ; τ ) m − l ≤ 1 ( τ ; τ ) ∞ ≤ ( τ ; τ ) m − l (\tau;\tau)_\infty\leq(\tau;\tau)_{m-l} ( τ ; τ ) ∞ ≤ ( τ ; τ ) m − l ( τ 2 l + 2 ; τ ) ∞ ≥ ( τ ; τ ) ∞ (\tau^{2l+2};\tau)_\infty\geq(\tau;\tau)_\infty ( τ 2 l + 2 ; τ ) ∞ ≥ ( τ ; τ ) ∞
By part 3 of Proposition 6.9 , we have u l = ς l ( ∞ ) ς l p l u_l=\frac{\varsigma_l(\infty)}{\varsigma_l}p_l u l = ς l ς l ( ∞ ) p l n < l n<l n < l E m ( u l ) = 0 \mathcal{E}_m(u_l)=0 E m ( u l ) = 0 m ≤ l − 2 m\leq l-2 m ≤ l − 2 m ≥ l − 1 m\geq l-1 m ≥ l − 1 E m ( u l ) \mathcal{E}_m(u_l) E m ( u l ) ∑ k ′ cond ( ( m , k ) , ( m + 1 , k ′ ) ) = w ^ ( m , k ) p ↑ ( m ) \sum_{k'}\operatorname{cond}((m,k),(m+1,k'))=\widehat{w}(m,k)p_\uparrow(m) ∑ k ′ cond (( m , k ) , ( m + 1 , k ′ )) = w ( m , k ) p ↑ ( m ) ∑ k cond ( ( m , k ) , ( m + 1 , k ′ ) ) = w ^ ( m + 1 , k ′ ) p ↓ ( m + 1 ) \sum_k\operatorname{cond}((m,k),(m+1,k'))=\widehat{w}(m+1,k')p_\downarrow(m+1) ∑ k cond (( m , k ) , ( m + 1 , k ′ )) = w ( m + 1 , k ′ ) p ↓ ( m + 1 ) E m ( u l ) = A m − 2 B m + C m \mathcal{E}_m(u_l)=A_m-2B_m+C_m E m ( u l ) = A m − 2 B m + C m
A m : = p ↑ ( m ) ∑ k w ^ ( m , k ) u l ( m , k ) 2 , C m : = p ↓ ( m + 1 ) ∑ k ′ w ^ ( m + 1 , k ′ ) u l ( m + 1 , k ′ ) 2 , A_m\mathrel{\mathop:}=p_\uparrow(m)\sum_k\widehat{w}(m,k)u_l(m,k)^2,\qquad C_m\mathrel{\mathop:}=p_\downarrow(m+1)\sum_{k'}\widehat{w}(m+1,k')u_l(m+1,k')^2, A m : = p ↑ ( m ) k ∑ w ( m , k ) u l ( m , k ) 2 , C m : = p ↓ ( m + 1 ) k ′ ∑ w ( m + 1 , k ′ ) u l ( m + 1 , k ′ ) 2 , B m : = ∑ k , k ′ cond ( ( m , k ) , ( m + 1 , k ′ ) ) u l ( m , k ) u l ( m + 1 , k ′ ) . B_m\mathrel{\mathop:}=\sum_{k,k'}\operatorname{cond}((m,k),(m+1,k'))u_l(m,k)u_l(m+1,k'). B m : = k , k ′ ∑ cond (( m , k ) , ( m + 1 , k ′ )) u l ( m , k ) u l ( m + 1 , k ′ ) . For m = l − 1 m=l-1 m = l − 1 A m = B m = 0 A_m=B_m=0 A m = B m = 0 ς l ( l ) = 1 \varsigma_l(l)=1 ς l ( l ) = 1
C l − 1 = τ ( 1 − τ l ) ( 1 + τ ) ( 1 − τ l + 1 ) ⋅ 1 − τ l + 1 ( 1 − τ ) τ l ⋅ χ l 2 ς l ( ∞ ) 2 h l , l = ( 1 − τ l ) χ l 2 h l , l ( 1 − τ 2 ) τ l − 1 ς l ( ∞ ) 2 . C_{l-1}=\frac{\tau(1-\tau^l)}{(1+\tau)(1-\tau^{l+1})}\cdot\frac{1-\tau^{l+1}}{(1-\tau)\tau^l}\cdot\frac{\chi_l^2}{\varsigma_l(\infty)^2}h_{l,l}=\frac{(1-\tau^l)\chi_l^2h_{l,l}}{(1-\tau^2)\tau^{l-1}\varsigma_l(\infty)^2}. C l − 1 = ( 1 + τ ) ( 1 − τ l + 1 ) τ ( 1 − τ l ) ⋅ ( 1 − τ ) τ l 1 − τ l + 1 ⋅ ς l ( ∞ ) 2 χ l 2 h l , l = ( 1 − τ 2 ) τ l − 1 ς l ( ∞ ) 2 ( 1 − τ l ) χ l 2 h l , l . For m ≥ l m\geq l m ≥ l
A m = ( 1 − τ m + 2 ) χ l 2 ς l ( m ) 2 h l , m ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 , C m = ( 1 − τ m + 1 ) χ l 2 ς l ( m + 1 ) 2 h l , m + 1 ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 , A_m=\frac{(1-\tau^{m+2})\chi_l^2\varsigma_l(m)^2h_{l,m}}{(1-\tau^2)\tau^m\varsigma_l(\infty)^2},\qquad C_m=\frac{(1-\tau^{m+1})\chi_l^2\varsigma_l(m+1)^2h_{l,m+1}}{(1-\tau^2)\tau^m\varsigma_l(\infty)^2}, A m = ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 ( 1 − τ m + 2 ) χ l 2 ς l ( m ) 2 h l , m , C m = ( 1 − τ 2 ) τ m ς l ( ∞ ) 2 ( 1 − τ m + 1 ) χ l 2 ς l ( m + 1 ) 2 h l , m + 1 , and B m = χ l 2 ς l ( ∞ ) 2 ς l ( m ) ς l ( m + 1 ) b m B_m=\frac{\chi_l^2}{\varsigma_l(\infty)^2}\varsigma_l(m)\varsigma_l(m+1)b_m B m = ς l ( ∞ ) 2 χ l 2 ς l ( m ) ς l ( m + 1 ) b m h l , m + 1 h_{l,m+1} h l , m + 1 b m b_m b m h l , m h_{l,m} h l , m (81) .
Recall that φ l T = χ l p ^ l \varphi_l^T=\chi_l\widehat{p}_l φ l T = χ l p l w k = ( 1 − τ ) τ k w_k=(1-\tau)\tau^k w k = ( 1 − τ ) τ k ∥ φ l T ∥ 2 = ( 1 − τ ) χ l 2 h l , ∞ \lVert\varphi_l^T\rVert^2=(1-\tau)\chi_l^2h_{l,\infty} ∥ φ l T ∥ 2 = ( 1 − τ ) χ l 2 h l , ∞ m ≥ l m\geq l m ≥ l y : = τ m + 1 − l ≤ τ y\mathrel{\mathop:}=\tau^{m+1-l}\leq\tau y : = τ m + 1 − l ≤ τ a : = ς l ( m ) a\mathrel{\mathop:}=\varsigma_l(m) a : = ς l ( m ) δ : = ς l ( m + 1 ) − ς l ( m ) \delta\mathrel{\mathop:}=\varsigma_l(m+1)-\varsigma_l(m) δ : = ς l ( m + 1 ) − ς l ( m ) Q Q Q (81) . Since τ m + 1 = y τ l \tau^{m+1}=y\tau^l τ m + 1 = y τ l τ m + 2 = y τ l + 1 \tau^{m+2}=y\tau^{l+1} τ m + 2 = y τ l + 1 τ m + l + 2 = y τ 2 l + 1 \tau^{m+l+2}=y\tau^{2l+1} τ m + l + 2 = y τ 2 l + 1 Q = F 1 a 2 + 2 F 2 a δ + R δ 2 Q=F_1a^2+2F_2a\delta+R\delta^2 Q = F 1 a 2 + 2 F 2 a δ + R δ 2
F 1 = y ( 1 − τ l ) [ 1 − τ l + 1 + y τ l + 1 ( 1 − τ l ) ] 1 − y , F_1=\frac{y(1-\tau^l)[1-\tau^{l+1}+y\tau^{l+1}(1-\tau^l)]}{1-y}, F 1 = 1 − y y ( 1 − τ l ) [ 1 − τ l + 1 + y τ l + 1 ( 1 − τ l )] , F 2 = y ( 1 − τ l ) ( 1 − y τ 2 l + 1 ) 1 − y , R = ( 1 − y τ l ) ( 1 − y τ 2 l + 1 ) 1 − y . F_2=\frac{y(1-\tau^l)(1-y\tau^{2l+1})}{1-y},\qquad R=\frac{(1-y\tau^l)(1-y\tau^{2l+1})}{1-y}. F 2 = 1 − y y ( 1 − τ l ) ( 1 − y τ 2 l + 1 ) , R = 1 − y ( 1 − y τ l ) ( 1 − y τ 2 l + 1 ) . We have y ( 1 − τ l ) ( 1 − τ l + 1 ) = τ m + 1 Λ l ( τ ) y(1-\tau^l)(1-\tau^{l+1})=\tau^{m+1}\Lambda_l(\tau) y ( 1 − τ l ) ( 1 − τ l + 1 ) = τ m + 1 Λ l ( τ ) τ l + 1 ( 1 − τ l ) ≤ 1 − τ l + 1 \tau^{l+1}(1-\tau^l)\leq1-\tau^{l+1} τ l + 1 ( 1 − τ l ) ≤ 1 − τ l + 1 1 + τ 2 l + 1 − 2 τ l + 1 ≥ ( 1 − τ l + 1 ) 2 ≥ 0 1+\tau^{2l+1}-2\tau^{l+1}\geq(1-\tau^{l+1})^2\geq0 1 + τ 2 l + 1 − 2 τ l + 1 ≥ ( 1 − τ l + 1 ) 2 ≥ 0 y ≤ τ y\leq\tau y ≤ τ
0 ≤ F 1 ≤ 1 + τ 1 − τ τ m + 1 Λ l ( τ ) , 0 ≤ F 2 ≤ τ m + 1 Λ l ( τ ) ( 1 − τ ) 2 , 0 < R ≤ 1 1 − τ . 0\leq F_1\leq\frac{1+\tau}{1-\tau}\tau^{m+1}\Lambda_l(\tau),\qquad0\leq F_2\leq\frac{\tau^{m+1}\Lambda_l(\tau)}{(1-\tau)^2},\qquad0<R\leq\frac{1}{1-\tau}. 0 ≤ F 1 ≤ 1 − τ 1 + τ τ m + 1 Λ l ( τ ) , 0 ≤ F 2 ≤ ( 1 − τ ) 2 τ m + 1 Λ l ( τ ) , 0 < R ≤ 1 − τ 1 . Put j : = m − l j\mathrel{\mathop:}=m-l j : = m − l Lemma 6.8 , we have ∣ a ∣ ≤ M τ |a|\leq M_\tau ∣ a ∣ ≤ M τ ∣ δ ∣ ≤ 4 P τ M τ ( j + 2 ) τ j + 1 |\delta|\leq4P_\tau M_\tau(j+2)\tau^{j+1} ∣ δ ∣ ≤ 4 P τ M τ ( j + 2 ) τ j + 1 τ m + 1 Λ l ( τ ) = τ j + 1 ( 1 − τ l ) ( 1 − τ l + 1 ) ≥ ( 1 − τ ) ( 1 − τ 2 ) τ j + 1 \tau^{m+1}\Lambda_l(\tau)=\tau^{j+1}(1-\tau^l)(1-\tau^{l+1})\geq(1-\tau)(1-\tau^2)\tau^{j+1} τ m + 1 Λ l ( τ ) = τ j + 1 ( 1 − τ l ) ( 1 − τ l + 1 ) ≥ ( 1 − τ ) ( 1 − τ 2 ) τ j + 1 ∣ Q ∣ ≤ K τ τ m + 1 Λ l ( τ ) |Q|\leq K_\tau\tau^{m+1}\Lambda_l(\tau) ∣ Q ∣ ≤ K τ τ m + 1 Λ l ( τ )
K τ : = ( 1 + τ ) M τ 2 1 − τ + sup j ≥ 0 [ 8 P τ M τ 2 ( j + 2 ) τ j + 1 ( 1 − τ ) 2 + 16 P τ 2 M τ 2 ( j + 2 ) 2 τ j + 1 ( 1 − τ ) 2 ( 1 − τ 2 ) ] < ∞ . K_{\tau}\mathrel{\mathop:}= \frac{(1+\tau)M_{\tau}^{2}}{1-\tau}+\sup_{j\geq0}\left[\frac{8P_{\tau}M_{\tau}^{2}(j+2)\tau^{j+1}}{(1-\tau)^{2}}+\frac{16P_{\tau}^{2}M_{\tau}^{2}(j+2)^{2}\tau^{j+1}}{(1-\tau)^{2}(1-\tau^{2})}\right]<\infty. K τ : = 1 − τ ( 1 + τ ) M τ 2 + j ≥ 0 sup [ ( 1 − τ ) 2 8 P τ M τ 2 ( j + 2 ) τ j + 1 + ( 1 − τ ) 2 ( 1 − τ 2 ) 16 P τ 2 M τ 2 ( j + 2 ) 2 τ j + 1 ] < ∞. By (81) , part 1 and part 2 of Proposition 6.12 , we deduce that
ε m ( u l ) ≤ τ l 2 h l , ∞ ( τ ; τ ) ∞ ( 1 − τ 2 ) τ m K τ τ m + 1 Λ l ( τ ) = τ K τ ( τ ; τ ) ∞ ( 1 − τ ) ( 1 − τ 2 ) Λ l ( τ ) ∥ φ l ∥ 2 . \varepsilon_m(u_l)\leq\frac{\tau_l^2h_{l,\infty}}{(\tau;\tau)_\infty(1-\tau^2)\tau^m}K_\tau\tau^{m+1}\Lambda_l(\tau)=\frac{\tau K_\tau}{(\tau;\tau)_\infty(1-\tau)(1-\tau^2)}\Lambda_l(\tau)\lVert\varphi_l\rVert^2. ε m ( u l ) ≤ ( τ ; τ ) ∞ ( 1 − τ 2 ) τ m τ l 2 h l , ∞ K τ τ m + 1 Λ l ( τ ) = ( τ ; τ ) ∞ ( 1 − τ ) ( 1 − τ 2 ) τ K τ Λ l ( τ ) ∥ φ l ∥ 2 . For m = l − 1 m=l-1 m = l − 1 ( 1 − τ l ) τ 1 − l = τ Λ l ( τ ) 1 − τ ≤ τ Λ l ( τ ) 1 − τ (1-\tau^l)\tau^{1-l}=\frac{\tau\Lambda_l(\tau)}{1-\tau}\leq\frac{\tau\Lambda_l(\tau)}{1-\tau} ( 1 − τ l ) τ 1 − l = 1 − τ τ Λ l ( τ ) ≤ 1 − τ τ Λ l ( τ ) h l , l ≤ h l , ∞ ( τ ; τ ) ∞ h_{l,l}\leq\frac{h_{l,\infty}}{(\tau;\tau)_\infty} h l , l ≤ ( τ ; τ ) ∞ h l , ∞
ε l − 1 ( u l ) ≤ τ ( τ ; τ ) ∞ ( 1 − τ ) 2 ( 1 − τ 2 ) Λ l ( τ ) ∥ φ l ∥ 2 . \varepsilon_{l-1}(u_l)\leq\frac{\tau}{(\tau;\tau)_\infty(1-\tau)^2(1-\tau^2)}\Lambda_l(\tau)\lVert\varphi_l\rVert^2. ε l − 1 ( u l ) ≤ ( τ ; τ ) ∞ ( 1 − τ ) 2 ( 1 − τ 2 ) τ Λ l ( τ ) ∥ φ l ∥ 2 . Since E ( n ) ( u l ) = ε n − 1 ( u l ) + ε n ( u l ) E^{(n)}(u_l)=\varepsilon_{n-1}(u_l)+\varepsilon_n(u_l) E ( n ) ( u l ) = ε n − 1 ( u l ) + ε n ( u l )
C τ = 2 τ max { K τ ( 1 − τ ) , 1 } ( τ ; τ ) ∞ ( 1 − τ ) 2 ( 1 − τ 2 ) . C_\tau=\frac{2\tau\max\{K_\tau(1-\tau),1\}}{(\tau;\tau)_\infty(1-\tau)^2(1-\tau^2)}. C τ = ( τ ; τ ) ∞ ( 1 − τ ) 2 ( 1 − τ 2 ) 2 τ max { K τ ( 1 − τ ) , 1 } . We now pass to the Poisson integrals of the Podleś sphere. Recall that the left translations by the torus act on L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) β λ , 1 \beta_{\lambda,1} β λ , 1 (42) , and that their fixed point algebra is W ∗ ( A ) \mathrm{W}^*(A) W ∗ ( A ) Proposition 5.5 .
Proposition 6.31 For any bounded function f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C Θ ( f ( A ) ) \Theta(f(A)) Θ ( f ( A )) H f \mathcal{H}f H f
Proof. Since Θ \Theta Θ W ∗ ( A ) \mathrm{W}^*(A) W ∗ ( A ) H ∞ ( G ^ , P ) H^\infty(\widehat{G},\mathsf{P}) H ∞ ( G , P ) u u u u ∞ : = lim t u ( n t , k t ) u_\infty\mathrel{\mathop:}=\lim_t u(n_t,k_t) u ∞ : = lim t u ( n t , k t ) u ( v ) = E v [ u ∞ ] u(v)=\mathbb{E}_v[u_\infty] u ( v ) = E v [ u ∞ ] σ \sigma σ [42 Section 1] . For bounded harmonic functions u u u u ′ u' u ′ (50) converge pointwise to E v [ u ∞ u ∞ ′ ] \mathbb{E}_v[u_\infty u'_\infty] E v [ u ∞ u ∞ ′ ] ( u ⋅ u ′ ) ( v ) = E v [ u ∞ u ∞ ′ ] (u\mathbin{\cdot}u')(v)=\mathbb{E}_v[u_\infty u'_\infty] ( u ⋅ u ′ ) ( v ) = E v [ u ∞ u ∞ ′ ] I \mathcal{I} I σ \sigma σ u ↦ u ∞ u\mapsto u_\infty u ↦ u ∞ ∗ * ∗ H ∞ ( G ^ , P ) H^\infty(\widehat{G},\mathsf{P}) H ∞ ( G , P ) (50) , onto L ∞ ( I , P ( 0 , 0 ) ) L^\infty(\mathcal{I},\mathsf{P}_{(0,0)}) L ∞ ( I , P ( 0 , 0 ) ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 0 , 0 ) (0,0) ( 0 , 0 ) L ∞ ( I , P ( 0 , 0 ) ) L^\infty(\mathcal{I},\mathsf{P}_{(0,0)}) L ∞ ( I , P ( 0 , 0 ) ) W ∗ ( A ) \mathrm{W}^*(A) W ∗ ( A ) ∗ ^* ∗ p k p_k p k k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 Θ ( p k ) \Theta(p_k) Θ ( p k ) v ↦ P v ( Ω k ) v\mapsto\mathsf{P}_v(\Omega_k) v ↦ P v ( Ω k ) Ω k \Omega_k Ω k I \mathcal{I} I Ω k \Omega_k Ω k
P ( 0 , 0 ) ( Ω k ) = ε ^ ( Θ ( p k ) ) = h ( p k ) : = ( 1 − τ ) τ k . (82) \mathsf{P}_{(0,0)}(\Omega_k)=\widehat{\varepsilon}(\Theta(p_k))=h(p_k)\mathrel{\mathop:}=(1-\tau)\tau^k. \tag*{(82)} P ( 0 , 0 ) ( Ω k ) = ε ( Θ ( p k )) = h ( p k ) : = ( 1 − τ ) τ k . (82) The variable k ∞ k_\infty k ∞ I \mathcal{I} I j ∈ N 0 j\in\mathbb{N}_0 j ∈ N 0 { k ∞ = j } \{k_\infty=j\} { k ∞ = j } Ω k \Omega_k Ω k k ∈ S j ′ k\in S'_j k ∈ S j ′ ( S j ′ ) j ≥ 0 (S'_j)_{j\geq0} ( S j ′ ) j ≥ 0 N 0 \mathbb{N}_0 N 0 ∑ k ∈ S j ′ τ k = τ j \sum_{k\in S'_j}\tau^k=\tau^j ∑ k ∈ S j ′ τ k = τ j j j j j j j S j ′ = { j } S'_j=\{j\} S j ′ = { j } S i ′ = { i } S'_i=\{i\} S i ′ = { i } i < j i<j i < j j ∉ S j ′ j\notin S'_j j ∈ / S j ′ j ∈ S i ′ j\in S'_i j ∈ S i ′ i > j i>j i > j ∑ k ∈ S i ′ τ k ≥ τ j > τ i \sum_{k\in S'_i}\tau^k\geq\tau^j>\tau^i ∑ k ∈ S i ′ τ k ≥ τ j > τ i j ∈ S j ′ j\in S'_j j ∈ S j ′ ∑ k ∈ S j ′ τ k = τ j \sum_{k\in S'_j}\tau^k=\tau^j ∑ k ∈ S j ′ τ k = τ j S j ′ = { j } S'_j=\{j\} S j ′ = { j } Θ ( p k ) ( v ) = P v ( k ∞ = k ) = H 1 { k } ( v ) \Theta(p_k)(v)=\mathsf{P}_v(k_\infty=k)=\mathcal{H}\mathbf{1}_{\{k\}}(v) Θ ( p k ) ( v ) = P v ( k ∞ = k ) = H 1 { k } ( v ) k k k f ( A ) = ∑ k f ( k ) p k f(A)=\sum_k f(k)p_k f ( A ) = ∑ k f ( k ) p k ∗ ^* ∗ H \mathcal{H} H Θ ( f ( A ) ) = ∑ k f ( k ) H 1 { k } = H f \Theta(f(A))=\sum_k f(k)\mathcal{H}\mathbf{1}_{\{k\}}=\mathcal{H}f Θ ( f ( A )) = ∑ k f ( k ) H 1 { k } = H f
Recall that Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) u j 0 ( l ) u^{(l)}_{j0} u j 0 ( l ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 ∣ j ∣ ≤ l |j|\leq l ∣ j ∣ ≤ l L 2 ( S r 2 ) L^2(S_r^2) L 2 ( S r 2 ) [54 Theorem 1.4.3] and Lemma 5.2. Thus Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) Pol l ( S r 2 ) : = span { u j 0 ( l ) : ∣ j ∣ ≤ l } \operatorname{Pol}_l(S_r^2)\mathrel{\mathop:}=\operatorname{span}\{u^{(l)}_{j0}:|j|\leq l\} Pol l ( S r 2 ) : = span { u j 0 ( l ) : ∣ j ∣ ≤ l } l l l Pol ( S r 2 ) \operatorname{Pol}(S^2_r) Pol ( S r 2 ) u 00 ( l ) = φ 0 l ( A ) u_{00}^{(l)}=\varphi^l_0(A) u 00 ( l ) = φ 0 l ( A ) x ∈ L ∞ ( S r 2 ) x\in L^\infty(S^2_r) x ∈ L ∞ ( S r 2 ) x l x_l x l x x x Pol l ( S r 2 ) \operatorname{Pol}_l(S^2_r) Pol l ( S r 2 )
Corollary 6.32 For any integer n ≥ 0 n\ge0 n ≥ 0 x ∈ L ∞ ( S r 2 ) x\in L^\infty(S^2_r) x ∈ L ∞ ( S r 2 )
E ( n ) ( Θ ( x ) ) = ∑ l = 0 n + 1 e n ( l ) ∥ x l ∥ L 2 ( S r 2 ) 2 , where e n ( l ) ≔ E ( n ) ( H φ l τ ) ∥ φ l τ ∥ L N 0 2 ( ω ) 2 , \mathcal{E}^{(n)}(\Theta(x))=\sum_{l=0}^{n+1}e_n(l)\|x_l\|_{L^2(S^2_r)}^2,\qquad\text{where }e_n(l)\coloneqq\frac{\mathcal{E}^{(n)}(\mathcal{H}\varphi_l^\tau)}{\|\varphi_l^\tau\|_{L^2_{N_0}(\omega)}^2}, E ( n ) ( Θ ( x )) = l = 0 ∑ n + 1 e n ( l ) ∥ x l ∥ L 2 ( S r 2 ) 2 , where e n ( l ) : = ∥ φ l τ ∥ L N 0 2 ( ω ) 2 E ( n ) ( H φ l τ ) , and, for any l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 lim n → ∞ e n ( l ) = c τ Λ l ( τ ) \lim_{n\to\infty}e_n(l)=c_\tau\Lambda_l(\tau) lim n → ∞ e n ( l ) = c τ Λ l ( τ ) sup n e n ( l ) ≤ C τ Λ l ( τ ) \sup_n e_n(l)\le C_\tau\Lambda_l(\tau) sup n e n ( l ) ≤ C τ Λ l ( τ ) C τ C_\tau C τ x ∈ L ∞ ( S r 2 ) x\in L^\infty(S^2_r) x ∈ L ∞ ( S r 2 )
lim n → ∞ E ( n ) ( Θ ( x ) ) = c τ ∑ l ≥ 0 Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 ∈ [ 0 , ∞ ] , (83) \lim_{n\to\infty}\mathcal{E}^{(n)}(\Theta(x))=c_\tau\sum_{l\ge0}\Lambda_l(\tau)\|x_l\|_{L^2(S^2_r)}^2\in[0,\infty], \tag*{(83)} n → ∞ lim E ( n ) ( Θ ( x )) = c τ l ≥ 0 ∑ Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 ∈ [ 0 , ∞ ] , (83) and sup n ≥ 0 E ( n ) ( Θ ( x ) ) ≤ C τ ∑ l ≥ 0 Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 \sup_{n\ge0}\mathcal{E}^{(n)}(\Theta(x))\le C_\tau\sum_{l\ge0}\Lambda_l(\tau)\|x_l\|_{L^2(S^2_r)}^2 sup n ≥ 0 E ( n ) ( Θ ( x )) ≤ C τ ∑ l ≥ 0 Λ l ( τ ) ∥ x l ∥ L 2 ( S r 2 ) 2 lim n → ∞ E ( n ) ( H f ) = c τ E τ ( f ) \lim_{n\to\infty}\mathcal{E}^{(n)}(\mathcal{H}f)=c_\tau\mathcal{E}_\tau(f) lim n → ∞ E ( n ) ( H f ) = c τ E τ ( f ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) 1 1 1
Proof: Since P P P G G G ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) h ^ \widehat{h} h ( a , b ) ↦ [ 2 s + 1 ] r Tr ( K − 1 ( P ( a ∗ b ) − a ∗ b ) s ) (a,b)\mapsto[2s+1]_r\operatorname{Tr}(K^{-1}(P(a^*b)-a^*b)_s) ( a , b ) ↦ [ 2 s + 1 ] r Tr ( K − 1 ( P ( a ∗ b ) − a ∗ b ) s ) Θ \Theta Θ B n B_n B n L ∞ ( S r 2 ) L^\infty(S^2_r) L ∞ ( S r 2 ) Θ \Theta Θ x ↦ E ( n ) ( Θ ( x ) ) x\mapsto\mathcal{E}^{(n)}(\Theta(x)) x ↦ E ( n ) ( Θ ( x )) Θ \Theta Θ ℓ ∞ ( G ^ ) \ell^\infty(\widehat{G}) ℓ ∞ ( G ) s ′ ≤ s + 1 2 s'\le s+\frac12 s ′ ≤ s + 2 1 2 s + 1 = n + 1 2s+1=n+1 2 s + 1 = n + 1 E ( n ) ( a ) \mathcal{E}^{(n)}(a) E ( n ) ( a ) a a a s s s s ± 1 2 s\pm\frac12 s ± 2 1 E ( n ) ( Θ ( x ) ) = B n ( y , y ) \mathcal{E}^{(n)}(\Theta(x))=B_n(y,y) E ( n ) ( Θ ( x )) = B n ( y , y ) y = ∑ l ≤ n + 1 x l y=\sum_{l\le n+1}x_l y = ∑ l ≤ n + 1 x l B n B_n B n Pol l ( S r 2 ) × Pol l ′ ( S r 2 ) \operatorname{Pol}_l(S^2_r)\times\operatorname{Pol}_{l'}(S^2_r) Pol l ( S r 2 ) × Pol l ′ ( S r 2 ) l ≠ l ′ l\ne l' l = l ′ L 2 ( S r 2 ) L^2(S^2_r) L 2 ( S r 2 ) Pol l ( S r 2 ) \operatorname{Pol}_l(S^2_r) Pol l ( S r 2 ) x = φ l l ( A ) x=\varphi_l^l(A) x = φ l l ( A ) ∥ φ l l ( A ) ∥ L 2 ( S r 2 ) = ∥ φ l τ ∥ L N 0 2 ( ω ) \|\varphi_l^l(A)\|_{L^2(S^2_r)}=\|\varphi_l^\tau\|_{L^2_{N_0}(\omega)} ∥ φ l l ( A ) ∥ L 2 ( S r 2 ) = ∥ φ l τ ∥ L N 0 2 ( ω )
lim n → ∞ e n ( l ) = c τ E τ ( φ l τ ) ∥ φ l τ ∥ L N 0 2 ( ω ) 2 = c τ Λ l ( τ ) . \lim_{n\to\infty}e_n(l)=\frac{c_\tau\mathcal{E}_\tau(\varphi_l^\tau)}{\|\varphi_l^\tau\|_{L^2_{N_0}(\omega)}^2}=c_\tau\Lambda_l(\tau). n → ∞ lim e n ( l ) = ∥ φ l τ ∥ L N 0 2 ( ω ) 2 c τ E τ ( φ l τ ) = c τ Λ l ( τ ) . The bound on e n ( l ) e_n(l) e n ( l ) l ≥ 1 l\ge1 l ≥ 1 e n ( 0 ) = 0 e_n(0)=0 e n ( 0 ) = 0 H φ 0 τ = 1 \mathcal{H}\varphi_0^\tau=1 H φ 0 τ = 1 sup n E ( n ) ( Θ ( x ) ) \sup_n\mathcal{E}^{(n)}(\Theta(x)) sup n E ( n ) ( Θ ( x )) ∑ l Λ l ( τ ) ∥ x l ∥ 2 < ∞ \sum_l\Lambda_l(\tau)\|x_l\|^2<\infty ∑ l Λ l ( τ ) ∥ x l ∥ 2 < ∞ (83) follows from the dominated convergence theorem for series. Otherwise, Fatou’s lemma for series gives lim inf n E ( n ) ( Θ ( x ) ) = ∞ \liminf_n\mathcal{E}^{(n)}(\Theta(x))=\infty lim inf n E ( n ) ( Θ ( x )) = ∞
Let f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C f ( A ) f(A) f ( A ) f ( A ) l f(A)_l f ( A ) l φ l τ ( A ) \varphi_l^\tau(A) φ l τ ( A ) f ( A ) l = ⟨ f , e l ⟩ l 2 e l ( A ) f(A)_l=\langle f,e_l\rangle_{l^2}\,e_l(A) f ( A ) l = ⟨ f , e l ⟩ l 2 e l ( A ) e l = ∥ φ l τ ∥ − 1 φ l τ e_l=\|\varphi_l^\tau\|^{-1}\varphi_l^\tau e l = ∥ φ l τ ∥ − 1 φ l τ h ( ( φ l τ ) ∗ f ( A ) ) = ⟨ f , φ l τ ⟩ l N 0 2 ( ω ) h((\varphi_l^\tau)^*f(A))=\langle f,\varphi_l^\tau\rangle_{l^2_{N_0}(\omega)} h (( φ l τ ) ∗ f ( A )) = ⟨ f , φ l τ ⟩ l N 0 2 ( ω ) ∑ l Λ l ( τ ) ∥ f ( A ) l ∥ 2 = ∑ l Λ l ( τ ) ∣ ⟨ f , e l ⟩ ∣ 2 = E τ ( f ) \sum_l\Lambda_l(\tau)\|f(A)_l\|^2=\sum_l\Lambda_l(\tau)|\langle f,e_l\rangle|^2=\mathcal{E}_\tau(f) ∑ l Λ l ( τ ) ∥ f ( A ) l ∥ 2 = ∑ l Λ l ( τ ) ∣ ⟨ f , e l ⟩ ∣ 2 = E τ ( f ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] Θ ( f ( A ) ) = H f \Theta(f(A))=\mathcal{H}f Θ ( f ( A )) = H f x x x 1 1 1 x l ≠ 0 x_l\ne0 x l = 0 l ≥ 1 l\ge1 l ≥ 1 (83) is positive and E G ^ ( Θ ( x ) ) = ∑ n E ( n ) ( Θ ( x ) ) = ∞ \mathcal{E}_{\widehat{G}}(\Theta(x))=\sum_n\mathcal{E}^{(n)}(\Theta(x))=\infty E G ( Θ ( x )) = ∑ n E ( n ) ( Θ ( x )) = ∞ Θ \Theta Θ H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P )
Comparison of the two boundary energies
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d f : N 0 → C f:\mathbb{N}_0\to\mathbb{C} f : N 0 → C F ( x ) = f ( k ) F(x)=f(k) F ( x ) = f ( k ) ∥ x ∥ K = Q − k \lVert x\rVert_K=Q^{-k} ∥ x ∥ K = Q − k O d O^d O d f ( A ) f(A) f ( A ) Theorem 5.6 .
Corollary 6.33 1. The harmonic measure of the tree T \mathsf{T} T
For any f ∈ ℓ N 0 2 ( w ) f\in\ell^2_{\mathbb{N}_0}(w) f ∈ ℓ N 0 2 ( w ) E τ ( f ) \mathcal{E}_\tau(f) E τ ( f ) lim n E ( n ) ( H f ) \lim_n\mathcal{E}^{(n)}(\mathcal{H}f) lim n E ( n ) ( H f ) f f f Corollary 6.32 , satisfy
2 ( 1 − τ ) 2 1 + τ E τ ( f ) ≤ E T ( F ) ≤ 2 ( 1 − τ ) [ 1 ( 1 − τ ) 3 + τ ( 1 + τ ) 2 ( 1 − τ ) 2 ] E τ ( f ) . \frac{2(1-\tau)^2}{1+\tau}\mathcal{E}_\tau(f)\leq\mathcal{E}_{\mathsf{T}}(F)\leq2(1-\tau)\left[\frac{1}{(1-\tau)^3}+\frac{\tau}{(1+\tau)^2(1-\tau)^2}\right]\mathcal{E}_\tau(f). 1 + τ 2 ( 1 − τ ) 2 E τ ( f ) ≤ E T ( F ) ≤ 2 ( 1 − τ ) [ ( 1 − τ ) 3 1 + ( 1 + τ ) 2 ( 1 − τ ) 2 τ ] E τ ( f ) . In particular, these forms have the same domain.
Proof. 1. Under the Haar measure of O d O^d O d ( 1 − τ ) τ k (1-\tau)\tau^k ( 1 − τ ) τ k (7) . The quantum case is Theorem 6.10.
By Theorem 6.1 and Proposition 2.1 , we have E T ( F ) = 2 ( 1 − τ ) [ E R ( f ) + κ ∥ f − E 0 f ∥ 2 ] \mathcal{E}_{\mathsf{T}}(F)=2(1-\tau)\left[\mathcal{E}_{R}(f)+\kappa\lVert f-E_0f\rVert^2\right] E T ( F ) = 2 ( 1 − τ ) [ E R ( f ) + κ ∥ f − E 0 f ∥ 2 ] α = d \alpha=d α = d κ = 1 1 + τ \kappa=\frac{1}{1+\tau} κ = 1 + τ 1 Theorem 4.1 with α = d \alpha=d α = d 1 − τ 1 + τ E τ ( f ) ≤ E R ( f ) ≤ ( 1 − τ ) − 3 E τ ( f ) \frac{1-\tau}{1+\tau}\mathcal{E}_\tau(f)\leq\mathcal{E}_{R}(f)\leq(1-\tau)^{-3}\mathcal{E}_\tau(f) 1 + τ 1 − τ E τ ( f ) ≤ E R ( f ) ≤ ( 1 − τ ) − 3 E τ ( f ) ∥ f − E 0 f ∥ 2 ≤ Λ 1 ( τ ) − 1 E τ ( f ) \lVert f-E_0f\rVert^2\leq\Lambda_1(\tau)^{-1}\mathcal{E}_\tau(f) ∥ f − E 0 f ∥ 2 ≤ Λ 1 ( τ ) − 1 E τ ( f ) Λ 1 ( τ ) = ( 1 − τ ) 2 ( 1 + τ ) τ \Lambda_1(\tau)=\frac{(1-\tau)^2(1+\tau)}{\tau} Λ 1 ( τ ) = τ ( 1 − τ ) 2 ( 1 + τ )
Thus the two operators compared in Theorem 4.1 for α = d \alpha=d α = d d d d S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Proposition 6.4 . Their conductances do not decay with the level, so that each level contributes a nearest-neighbour form in k k k
Remark 6.34 One could expect the total energy of the quantum Poisson integrals to be finite and comparable with the Taibleson energy. Corollary 6.32 shows that this fails: no element of H ∞ ( G ^ , P ) H^\infty(\widehat{G},P) H ∞ ( G , P ) 1 1 1 E ( n ) \mathcal{E}^{(n)} E ( n )
Quantum Markov semigroups on the Podleś sphere
Tail expectations and an exact lifting
In this subsection 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 ρ > 1 \rho>1 ρ > 1 c c c κ \kappa κ λ n \lambda_n λ n a n ( t ) a_n(t) a n ( t ) (6) and (2.14. If τ = Q − d \tau=Q^{-d} τ = Q − d ρ = Q α \rho=Q^\alpha ρ = Q α Proposition 5.5 , it is convenient to work in B ( ℓ N 0 2 ) B(\ell^2_{\mathbb{N}_0}) B ( ℓ N 0 2 ) ω τ \omega_\tau ω τ θ \theta θ B ( ℓ N 0 2 ) B(\ell^2_{\mathbb{N}_0}) B ( ℓ N 0 2 ) L ∞ ( S r 2 ) L^\infty(S^2_r) L ∞ ( S r 2 ) p n = e n n p_n=e_{nn} p n = e nn P n = ∑ k ≥ n e k k P_n=\sum_{k\geq n}e_{kk} P n = ∑ k ≥ n e k k h = ω τ h=\omega_\tau h = ω τ C ( S r 2 ) = K ( ℓ N 0 2 ) + C 1 C(S^2_r)=\mathcal{K}(\ell^2_{\mathbb{N}_0})+\mathbb{C}1 C ( S r 2 ) = K ( ℓ N 0 2 ) + C 1 n ≥ 0 n\geq0 n ≥ 0
Φ n ( x ) : = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n , M n : = ( 1 − P n ) B ( ℓ N 0 2 ) ( 1 − P n ) ⊕ C P n . (84) \Phi_n(x)\mathrel{\mathop:}=(1-P_n)x(1-P_n)+\tau^{-n}h(P_nxP_n)P_n,\qquad\mathcal{M}_n\mathrel{\mathop:}=(1-P_n)B(\ell^2_{\mathbb{N}_0})(1-P_n)\oplus\mathbb{C}P_n. \tag*{(84)} Φ n ( x ) : = ( 1 − P n ) x ( 1 − P n ) + τ − n h ( P n x P n ) P n , M n : = ( 1 − P n ) B ( ℓ N 0 2 ) ( 1 − P n ) ⊕ C P n . (84) Lemma 7.1 The map Φ n \Phi_n Φ n h h h M n \mathcal{M}_n M n 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 ) Φ 0 ( x ) = h ( x ) 1 \Phi_0(x)=h(x)1 Φ 0 ( x ) = h ( x ) 1 h h h L 2 L^2 L 2 Φ n \Phi_n Φ n Φ n \Phi_n Φ n C ( S q 2 ) C(S_{q}^{2}) C ( S q 2 ) Φ n ( x ) → x \Phi_n(x)\to x Φ n ( x ) → x x ∈ C ( S q 2 ) x\in C(S_{q}^{2}) x ∈ C ( S q 2 ) τ = Q − d \tau=Q^{-d} τ = Q − d
Φ n J = J E n . (85) \Phi_n J=J E_n. \tag*{(85)} Φ n J = J E n . (85) Proof: Pinching by P n P_n P n 1 − P n 1-P_n 1 − P n x ↦ τ − n h ( P n x P n ) x\mapsto\tau^{-n}h(P_nxP_n) x ↦ τ − n h ( P n x P n ) h ( P n ) = τ n h(P_n)=\tau^n h ( P n ) = τ n (84) is consequently normal and unital completely positive, and it is idempotent and bimodular over M n \mathcal{M}_n M n
Each P n P_n P n D D D ( 1 − P n ) x P n (1-P_n)xP_n ( 1 − P n ) x P n P n x ( 1 − P n ) P_nx(1-P_n) P n x ( 1 − P n ) h Φ n = h h\Phi_n=h h Φ n = h M n ⊂ M n + 1 \mathcal{M}_n\subset\mathcal{M}_{n+1} M n ⊂ M n + 1 p n p_n p n i ≤ j i\leq j i ≤ j Φ i Φ j \Phi_i\Phi_j Φ i Φ j h h h M i \mathcal{M}_i M i Φ j \Phi_j Φ j M i ⊂ M j \mathcal{M}_i\subset\mathcal{M}_j M i ⊂ M j M i \mathcal{M}_i M i M i \mathcal{M}_i M i σ h = Ad ( D i s ) \sigma^h=\operatorname{Ad}(D^{is}) σ h = Ad ( D i s ) P i P_i P i D D D [69 Theorem IX.4.2] , we obtain Φ i Φ j = Φ i \Phi_i\Phi_j=\Phi_i Φ i Φ j = Φ i Φ j Φ i = Φ i \Phi_j\Phi_i=\Phi_i Φ j Φ i = Φ i σ s h = Ad ( D i s ) \sigma_s^h=\operatorname{Ad}(D^{is}) σ s h = Ad ( D i s ) P n P_n P n D D D σ h \sigma^h σ h
Realize the GNS space by x ↦ x D 1 2 x\mapsto xD^{\frac{1}{2}} x ↦ x D 2 1 L 2 L^2 L 2 x ↦ D 1 4 x D 1 4 x\mapsto D^{\frac{1}{4}}xD^{\frac{1}{4}} x ↦ D 4 1 x D 4 1 S 2 S^2 S 2
Φ n ( 2 ) ξ = ( 1 − P n ) ξ ( 1 − P n ) + τ − n ⟨ ξ , D 1 2 P n ⟩ S 2 D 1 2 P n . (86) \Phi_n^{(2)}\xi=(1-P_n)\xi(1-P_n)+\tau^{-n}\left\langle\xi,D^{\frac{1}{2}}P_n\right\rangle_{S^2}D^{\frac{1}{2}}P_n. \tag*{(86)} Φ n ( 2 ) ξ = ( 1 − P n ) ξ ( 1 − P n ) + τ − n ⟨ ξ , D 2 1 P n ⟩ S 2 D 2 1 P n . (86) Since ∥ D 1 2 P n ∥ S 2 2 = h ( P n ) = τ n \left\|D^{\frac{1}{2}}P_n\right\|_{S^2}^{2}=h(P_n)=\tau^n D 2 1 P n S 2 2 = h ( P n ) = τ n S 2 S^2 S 2 Φ n ( 2 ) \Phi_n^{(2)} Φ n ( 2 ) n n n Φ n ( 2 ) → I \Phi_n^{(2)}\to I Φ n ( 2 ) → I τ = Q − d \tau=Q^{-d} τ = Q − d J f Jf J f (84) is the normalized average in (18) , which proves (85) .
Finally, let x x x P n → 0 P_n\to0 P n → 0 ∥ P n x ∥ → 0 \|P_nx\|\to0 ∥ P n x ∥ → 0 ∥ x P n ∥ → 0 \|xP_n\|\to0 ∥ x P n ∥ → 0 ( 1 − P n ) x ( 1 − P n ) → x (1-P_n)x(1-P_n)\to x ( 1 − P n ) x ( 1 − P n ) → x ∣ τ − n h ( P n x P n ) ∣ = ∣ ( 1 − τ ) ∑ k ≥ n τ k − n x k k ∣ ≤ sup k ≥ n ∣ x k k ∣ |\tau^{-n}h(P_nxP_n)|=|(1-\tau)\sum_{k\geq n}\tau^{k-n}x_{kk}|\leq\sup_{k\geq n}|x_{kk}| ∣ τ − n h ( P n x P n ) ∣ = ∣ ( 1 − τ ) ∑ k ≥ n τ k − n x k k ∣ ≤ sup k ≥ n ∣ x k k ∣ Φ n ( x ) → x \Phi_n(x)\to x Φ n ( x ) → x Φ n ( x ) ∈ K ( ℓ 0 2 ) + C P n ⊂ C ( S q 2 ) \Phi_n(x)\in\mathcal{K}(\ell^2_0)+\mathbb{C}P_n\subset C(S_q^2) Φ n ( x ) ∈ K ( ℓ 0 2 ) + C P n ⊂ C ( S q 2 ) Φ n ( 1 ) = 1 \Phi_n(1)=1 Φ n ( 1 ) = 1
Theorem 7.2 For t > 0 t>0 t > 0
T t ≔ ∑ n ≥ 0 a n ( t ) Φ n (87) T_t\coloneqq\sum_{n\geq0}a_n(t)\Phi_n \tag*{(87)} T t : = n ≥ 0 ∑ a n ( t ) Φ n (87) defines a semigroup of normal unital completely positive, h h h L ∞ ( S q 2 ) L^\infty(S_q^2) L ∞ ( S q 2 ) σ \sigma σ L 2 L^2 L 2 h h h ( β λ ) (\beta_\lambda) ( β λ ) L 2 L^2 L 2
L = c ( I − Φ 0 ) + ( ρ − 1 ) ∑ n ≥ 1 ρ n ( I − Φ n ) , (88) \mathcal{L}=c(I-\Phi_0)+(\rho-1)\sum_{n\geq1}\rho^n(I-\Phi_n), \tag*{(88)} L = c ( I − Φ 0 ) + ( ρ − 1 ) n ≥ 1 ∑ ρ n ( I − Φ n ) , (88) initially on ⋃ n Ran Φ n ( 2 ) \bigcup_n\operatorname{Ran}\Phi_n^{(2)} ⋃ n Ran Φ n ( 2 ) d n = Φ n + 1 ( 2 ) − Φ n ( 2 ) d_n=\Phi_{n+1}^{(2)}-\Phi_n^{(2)} d n = Φ n + 1 ( 2 ) − Φ n ( 2 )
L = ∑ n ≥ 0 λ n d n , dom L = { ξ : ∑ n ≥ 0 λ n 2 ∥ d n ξ ∥ 2 2 < ∞ } . (89) \mathcal{L}=\sum_{n\geq0}\lambda_n d_n,\qquad
\operatorname{dom}\mathcal{L}=\left\{\xi:\sum_{n\geq0}\lambda_n^2\|d_n\xi\|_2^2<\infty\right\}. \tag*{(89)} L = n ≥ 0 ∑ λ n d n , dom L = { ξ : n ≥ 0 ∑ λ n 2 ∥ d n ξ ∥ 2 2 < ∞ } . (89) If τ = Q − d \tau= Q^{-d} τ = Q − d ρ = Q α \rho= Q^\alpha ρ = Q α
T t J = J T t , L J 2 = J 2 R r a d α , \mathcal{T}_t J = J\mathcal{T}_t,\qquad\mathcal{L}J_2 = J_2R_{\mathrm{rad}}^\alpha, T t J = J T t , L J 2 = J 2 R rad α , where the generator equality includes equality of domains on the radial subspace. Finally, each T t \mathcal{T}_t T t C ( S r 2 ) C(S_r^2) C ( S r 2 ) ( T t ) (\mathcal{T}_t) ( T t ) C ( S r 2 ) C(S_r^2) C ( S r 2 )
Proof: The coefficients in (19) form a probability distribution. Hence (87) converges in operator norm in the space of bounded maps, is normal, unital completely positive, and preserves h h h
The projections Φ 0 ( 2 ) , d 0 , d 1 , … \Phi_0^{(2)},d_0,d_1,\ldots Φ 0 ( 2 ) , d 0 , d 1 , … Ran d k \operatorname{Ran}d_k Ran d k Φ n ( 2 ) \Phi_n^{(2)} Φ n ( 2 ) n ≤ k n\leq k n ≤ k n > k n>k n > k (87) acts there by ∑ n > k a n ( t ) = e − t λ k \sum_{n>k}a_n(t)=e^{-t\lambda_k} ∑ n > k a n ( t ) = e − t λ k L 2 L^2 L 2 (89) . Formula (88) follows from c + ( ρ − 1 ) ∑ n = 1 k ρ n = λ k c+(\rho-1)\sum_{n=1}^k\rho^n=\lambda_k c + ( ρ − 1 ) ∑ n = 1 k ρ n = λ k
For any x ∈ L ∞ ( S r 2 ) x\in L^\infty(S_r^2) x ∈ L ∞ ( S r 2 ) (84) converges strongly* to x x x ∥ x ∥ \lVert x\rVert ∥ x ∥ P n → 0 P_n\to0 P n → 0 Φ n ( x ) → x \Phi_n(x)\to x Φ n ( x ) → x σ \sigma σ N N N n ≤ N n\leq N n ≤ N 1 − e − t λ N 1-e^{-t\lambda_N} 1 − e − t λ N t → 0 + t\to0^+ t → 0 + N N N N N N x x x σ \sigma σ σ h \sigma^h σ h β λ , 1 \beta_{\lambda,1} β λ , 1 P n P_n P n h h h (86) shows that each symmetric L 2 L^2 L 2
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d ρ = Q α \rho=Q^\alpha ρ = Q α (85) and the mixture formulas (20) and (87) imply T t J = J T t \mathcal{T}_tJ=J\mathcal{T}_t T t J = J T t J 2 J_2 J 2 d n d_n d n (89) gives the generator and domain assertion.
Finally, T t \mathcal{T}_t T t C ( S r 2 ) C(S_r^2) C ( S r 2 ) (87) converges in norm. Let x ∈ C ( S r 2 ) x\in C(S_r^2) x ∈ C ( S r 2 ) ε > 0 \varepsilon>0 ε > 0 N N N ∥ Φ n ( x ) − x ∥ ≤ ε \lVert\Phi_n(x)-x\rVert\leq\varepsilon ∥ Φ n ( x ) − x ∥ ≤ ε n > N n>N n > N
∥ T t ( x ) − x ∥ ≤ 2 ∥ x ∥ ∑ n ≤ N a n ( t ) + ε = 2 ∥ x ∥ ( 1 − e − t λ N ) + ε , \lVert\mathcal{T}_t(x)-x\rVert\leq2\lVert x\rVert\sum_{n\leq N}a_n(t)+\varepsilon=2\lVert x\rVert(1-e^{-t\lambda_N})+\varepsilon, ∥ T t ( x ) − x ∥ ≤ 2 ∥ x ∥ n ≤ N ∑ a n ( t ) + ε = 2 ∥ x ∥ ( 1 − e − t λ N ) + ε , which is at most 2 ε 2\varepsilon 2 ε t t t C ( S r 2 ) C(S_r^2) C ( S r 2 )
The associated closed quadratic form is
Q ( ξ ) = c ∥ ( I − Φ 0 ( 2 ) ) ξ ∥ 2 2 + ( ρ − 1 ) ∑ n ≥ 1 ρ n ∥ ( I − Φ n ( 2 ) ) ξ ∥ 2 2 = ∑ n ≥ 0 λ n ∥ d n ξ ∥ 2 2 . \mathcal{Q}(\xi)=c\left\lVert(I-\Phi_0^{(2)})\xi\right\rVert_2^2+(\rho-1)\sum_{n\geq1}\rho^n\left\lVert(I-\Phi_n^{(2)})\xi\right\rVert_2^2=\sum_{n\geq0}\lambda_n\left\lVert d_n\xi\right\rVert_2^2. Q ( ξ ) = c ( I − Φ 0 ( 2 ) ) ξ 2 2 + ( ρ − 1 ) n ≥ 1 ∑ ρ n ( I − Φ n ( 2 ) ) ξ 2 2 = n ≥ 0 ∑ λ n ∥ d n ξ ∥ 2 2 . The identity follows first on finite filtration ranges and then by monotone convergence.
Remark 7.3 The expectations (84) make sense on the whole algebra M = L ∞ ( S U r ( 2 ) ) \mathcal{M}=L^\infty(\mathrm{SU}_r(2)) M = L ∞ ( SU r ( 2 )) P n = ∑ k ≥ n e k k ⊗ 1 P_n=\sum_{k\geq n}e_{kk}\otimes1 P n = ∑ k ≥ n e k k ⊗ 1 C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) T t ( b ) = ∑ n ≥ 1 e − t λ n 1 − r 2 n e n − 1 , n ⊗ 1 \mathcal{T}_t(b)=\sum_{n\geq1}e^{-t\lambda_n}\sqrt{1-r^{2n}}e_{n-1,n}\otimes1 T t ( b ) = ∑ n ≥ 1 e − t λ n 1 − r 2 n e n − 1 , n ⊗ 1 ∥ T t ( b ) − b ∥ = sup n ( 1 − e − t λ n ) 1 − r 2 n = 1 \lVert\mathcal{T}_t(b)-b\rVert=\sup_n(1-e^{-t\lambda_n})\sqrt{1-r^{2n}}=1 ∥ T t ( b ) − b ∥ = sup n ( 1 − e − t λ n ) 1 − r 2 n = 1 t > 0 t>0 t > 0 L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) b b b C ( S r 2 ) C(S_r^2) C ( S r 2 )
Noncommutative L p L^p L p
We denote by L p ( S r 2 ) L^p(S_r^2) L p ( S r 2 ) [34 , 70 ] of ( L ∞ ( S r 2 ) , h ) (L^\infty(S_r^2),h) ( L ∞ ( S r 2 ) , h ) 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ d h d_h d h h h h Φ n \Phi_n Φ n
Φ n ( p ) ( d h 1 2 p x d h 1 2 p ) = d h 1 2 p Φ n ( x ) d h 1 2 p , x ∈ L ∞ ( S r 2 ) . \Phi_n^{(p)}\left(d_h^{\frac{1}{2p}}x d_h^{\frac{1}{2p}}\right)=d_h^{\frac{1}{2p}}\Phi_n(x)d_h^{\frac{1}{2p}},\qquad x\in L^\infty(S_r^2). Φ n ( p ) ( d h 2 p 1 x d h 2 p 1 ) = d h 2 p 1 Φ n ( x ) d h 2 p 1 , x ∈ L ∞ ( S r 2 ) . According to [35 Theorem 5.1 and Example 5.8] , this extension is a contractive projection. Since Φ n \Phi_n Φ n [35 Proposition 5.5] . We put δ n ( p ) : = Φ n + 1 ( p ) − Φ n ( p ) \delta_n^{(p)}\mathrel{\mathop:}=\Phi_{n+1}^{(p)}-\Phi_n^{(p)} δ n ( p ) : = Φ n + 1 ( p ) − Φ n ( p ) n ≥ 0 n\ge0 n ≥ 0
Theorem 7.4 Suppose that 1 < p < ∞ 1<p<\infty 1 < p < ∞ C p C_p C p p p p
For any bounded complex sequence ( m ∗ , m 0 , m 1 , … ) (m_*,m_0,m_1,\ldots) ( m ∗ , m 0 , m 1 , … )
M m ξ = m ∗ Φ 0 ( p ) ξ + ∑ n ≥ 0 m n δ n ( p ) ξ (90) M_m\xi=m_*\Phi_0^{(p)}\xi+\sum_{n\ge0}m_n\delta_n^{(p)}\xi \tag*{(90)} M m ξ = m ∗ Φ 0 ( p ) ξ + n ≥ 0 ∑ m n δ n ( p ) ξ (90) converges in L p ( S r 2 ) L^p(S_r^2) L p ( S r 2 ) ξ ∈ L p ( S r 2 ) \xi\in L^p(S_r^2) ξ ∈ L p ( S r 2 )
∥ M m ∥ L p ( S r 2 ) → L p ( S r 2 ) ≤ C p max { ∣ m ∗ ∣ , sup n ≥ 0 ∣ m n ∣ } . (91) \|M_m\|_{L^p(S_r^2)\to L^p(S_r^2)}\le C_p\max\left\{|m_*|,\sup_{n\ge0}|m_n|\right\}. \tag*{(91)} ∥ M m ∥ L p ( S r 2 ) → L p ( S r 2 ) ≤ C p max { ∣ m ∗ ∣ , n ≥ 0 sup ∣ m n ∣ } . (91) The same constant works after replacing L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 ) h h h Φ n \Phi_n Φ n M N ( C ) ⊗ ‾ L ∞ ( S r 2 ) M_N(\mathbb{C})\overline{\otimes}L^\infty(S_r^2) M N ( C ) ⊗ L ∞ ( S r 2 ) tr N ⊗ h \operatorname{tr}_N\otimes h tr N ⊗ h Id M N ⊗ Φ n \operatorname{Id}_{M_N}\otimes\Phi_n Id M N ⊗ Φ n N ≥ 1 N\ge1 N ≥ 1
The maps T t T_t T t L p ( S r 2 ) L^p(S_r^2) L p ( S r 2 ) L p \mathcal{L}_p L p ξ ↦ ∑ n ≥ 0 λ n δ n ( p ) ξ \xi\mapsto\sum_{n\ge0}\lambda_n\delta_n^{(p)}\xi ξ ↦ ∑ n ≥ 0 λ n δ n ( p ) ξ ⋃ N ≥ 0 Ran Φ N ( p ) \bigcup_{N\ge0}\operatorname{Ran}\Phi_N^{(p)} ⋃ N ≥ 0 Ran Φ N ( p )
dom L p = { ξ ∈ L p ( S r 2 ) : ∑ n ≥ 0 λ n δ n ( p ) ξ converges in L p ( S r 2 ) } . (92) \operatorname{dom}\mathcal{L}_p=\left\{\xi\in L^p(S_r^2):\sum_{n\ge0}\lambda_n\delta_n^{(p)}\xi\text{ converges in }L^p(S_r^2)\right\}. \tag*{(92)} dom L p = { ξ ∈ L p ( S r 2 ) : n ≥ 0 ∑ λ n δ n ( p ) ξ converges in L p ( S r 2 ) } . (92) On X p = Ker Φ 0 ( p ) X_p=\operatorname{Ker}\Phi_0^{(p)} X p = Ker Φ 0 ( p ) B p B_p B p L p \mathcal{L}_p L p H ∞ ( Σ θ ) H^\infty(\Sigma_\theta) H ∞ ( Σ θ ) 0 < θ < π 0<\theta<\pi 0 < θ < π Σ θ = { z ∈ C ∖ { 0 } : ∣ arg z ∣ < θ } \Sigma_\theta=\{z\in\mathbb{C}\setminus\{0\}:|\arg z|<\theta\} Σ θ = { z ∈ C ∖ { 0 } : ∣ arg z ∣ < θ } f ∈ H ∞ ( Σ θ ) f\in H^\infty(\Sigma_\theta) f ∈ H ∞ ( Σ θ )
f ( B p ) ξ = ∑ n ≥ 0 f ( λ n ) δ n ( p ) ξ , ∥ f ( B p ) ∥ X p → X p ≤ C p ∥ f ∥ ∞ . (93) f(B_p)\xi=\sum_{n\ge0}f(\lambda_n)\delta_n^{(p)}\xi,\qquad\|f(B_p)\|_{X_p\to X_p}\le C_p\|f\|_\infty. \tag*{(93)} f ( B p ) ξ = n ≥ 0 ∑ f ( λ n ) δ n ( p ) ξ , ∥ f ( B p ) ∥ X p → X p ≤ C p ∥ f ∥ ∞ . (93) These estimates hold with the same constants on the matrix tensor extensions of part 1. The fixed subspace Ran Φ 0 ( p ) \operatorname{Ran}\Phi_0^{(p)} Ran Φ 0 ( p )
Proof: 1. In the tracial case, the noncommutative Burkholder–Gundy inequalities [60 Theorem 2.1] imply that martingale difference sequences are unconditional in L p L^p L p 1 < p < ∞ 1<p<\infty 1 < p < ∞ κ p \kappa_p κ p p p p [60 Remark 2.4] . By the reduction method of [35 ] , this unconditionality extends to the Haagerup spaces of a σ \sigma σ φ \varphi φ φ \varphi φ σ \sigma σ [35 Theorem 6.3] . Here the filtration is ( M n ) n ≥ 0 (\mathcal{M}_n)_{n\ge0} ( M n ) n ≥ 0 ( Φ j ( p ) ξ − Φ 0 ( p ) ξ ) 0 ≤ j ≤ N + 1 (\Phi^{(p)}_j\xi-\Phi^{(p)}_0\xi)_{0\le j\le N+1} ( Φ j ( p ) ξ − Φ 0 ( p ) ξ ) 0 ≤ j ≤ N + 1 2 κ p 2\kappa_p 2 κ p ∥ Φ N + 1 ( p ) ξ − Φ 0 ( p ) ξ ∥ p ≤ 2 ∥ ξ ∥ p \left\|\Phi^{(p)}_{N+1}\xi-\Phi^{(p)}_0\xi\right\|_p\le2\|\xi\|_p Φ N + 1 ( p ) ξ − Φ 0 ( p ) ξ p ≤ 2∥ ξ ∥ p (90) satisfy (7.8), for instance with C p = 1 + 4 κ p C_p=1+4\kappa_p C p = 1 + 4 κ p
We next justify convergence. The spaces Ran Φ N ( p ) \operatorname{Ran}\Phi^{(p)}_N Ran Φ N ( p ) L p ( S q 2 ) L^p(S^2_q) L p ( S q 2 ) ∂ h 1 2 p x ∂ h 1 2 p \partial_h^{\frac{1}{2p}}x\partial_h^{\frac{1}{2p}} ∂ h 2 p 1 x ∂ h 2 p 1 x x x Φ N ( p ) \Phi^{(p)}_N Φ N ( p ) N N N Φ N ( p ) ξ → ξ \Phi^{(p)}_N\xi\to\xi Φ N ( p ) ξ → ξ L p ( S q 2 ) L^p(S^2_q) L p ( S q 2 ) N ′ > N N'>N N ′ > N N N N N ′ − 1 N'-1 N ′ − 1 Ran Φ N ( p ) \operatorname{Ran}\Phi^{(p)}_N Ran Φ N ( p ) ξ \xi ξ C p sup n ∣ m n ∣ ∥ ξ − Φ N ( p ) ξ ∥ p C_p\sup_n|m_n|\left\|\xi-\Phi^{(p)}_N\xi\right\|_p C p sup n ∣ m n ∣ ξ − Φ N ( p ) ξ p tr N ⊗ h \operatorname{tr}_N\otimes h tr N ⊗ h
The probability mixture (87) , with each expectation replaced by its extension, defines a contraction on L p ( S q 2 ) L^p(S^2_q) L p ( S q 2 ) Ran Φ 0 ( p ) \operatorname{Ran}\Phi^{(p)}_0 Ran Φ 0 ( p ) e − t λ n e^{-t\lambda_n} e − t λ n Ran δ n ( p ) \operatorname{Ran}\delta^{(p)}_n Ran δ n ( p ) L p ( S q 2 ) L^p(S^2_q) L p ( S q 2 ) − L p -\mathcal{L}_p − L p Φ N ( p ) \Phi^{(p)}_N Φ N ( p ) Φ N ( p ) L p ξ = L p Φ N ( p ) ξ \Phi^{(p)}_N\mathcal{L}_p\xi=\mathcal{L}_p\Phi^{(p)}_N\xi Φ N ( p ) L p ξ = L p Φ N ( p ) ξ ξ ∈ dom L p \xi\in\operatorname{dom}\mathcal{L}_p ξ ∈ dom L p L p ξ \mathcal{L}_p\xi L p ξ ∑ n < N λ n δ n ( p ) ξ \sum_{n<N}\lambda_n\delta^{(p)}_n\xi ∑ n < N λ n δ n ( p ) ξ (92) . Conversely, if that series converges, the vectors Φ N ( p ) ξ \Phi^{(p)}_N\xi Φ N ( p ) ξ ξ \xi ξ L p \mathcal{L}_p L p (92) .
On X p X_p X p λ n − 1 \lambda_n^{-1} λ n − 1 B p B_p B p z ∉ [ 0 , ∞ ) z\notin[0,\infty) z ∈ / [ 0 , ∞ ) R z ξ ≔ ∑ n ≥ 0 ( z − λ n ) − 1 δ n ( p ) ξ R_z\xi\coloneqq\sum_{n\ge0}(z-\lambda_n)^{-1}\delta^{(p)}_n\xi R z ξ : = ∑ n ≥ 0 ( z − λ n ) − 1 δ n ( p ) ξ λ n ( z − λ n ) − 1 \lambda_n(z-\lambda_n)^{-1} λ n ( z − λ n ) − 1 (92) , R z ξ ∈ dom B p R_z\xi\in\operatorname{dom}B_p R z ξ ∈ dom B p ( z − B p ) R z ξ = ξ (z-B_p)R_z\xi=\xi ( z − B p ) R z ξ = ξ θ > 0 \theta>0 θ > 0
sup z ∉ Σ θ ‾ ∥ z R z ∥ ≤ C p sup z ∉ Σ θ ‾ ∣ z z − u ∣ < ∞ . (94) \sup_{z\notin\overline{\Sigma_\theta}}\|zR_z\|\le C_p\sup_{z\notin\overline{\Sigma_\theta}}\left|\frac{z}{z-u}\right|<\infty. \tag*{(94)} z ∈ / Σ θ sup ∥ z R z ∥ ≤ C p z ∈ / Σ θ sup z − u z < ∞. (94) This proves that B p B_p B p [36 Chapter 2] and its extension to bounded holomorphic functions [36 Section 5.1] . For a decaying holomorphic function f f f ( f ( λ n ) ) (f(\lambda_n)) ( f ( λ n )) X p X_p X p e ε ( z ) = z z + ε 1 1 + ε z e_\varepsilon(z)=\frac{z}{z+\varepsilon}\frac{1}{1+\varepsilon z} e ε ( z ) = z + ε z 1 + ε z 1 ε > 0 \varepsilon>0 ε > 0 Σ θ \Sigma_\theta Σ θ 0 < e ε ( λ n ) ≤ 1 0<e_\varepsilon(\lambda_n)\le1 0 < e ε ( λ n ) ≤ 1 λ n \lambda_n λ n ( ( f e ε ) ( λ n ) ) ((fe_\varepsilon)(\lambda_n)) (( f e ε ) ( λ n )) C p ∥ f ∥ ∞ C_p\|f\|_\infty C p ∥ f ∥ ∞ X p X_p X p (93) . This is the bounded extension of the sectorial calculus and proves the assertion. The argument is unchanged on each matrix tensor extension.
Remark 7.5 By [35 Remark 1.2] , the Haagerup space of ( B ( ℓ N 0 2 ) , ω τ ) (\mathrm{B}(\ell^2_{N_0}),\omega_\tau) ( B ( ℓ N 0 2 ) , ω τ ) S p S^p S p B ( ℓ N 0 2 ) \mathrm{B}(\ell^2_{N_0}) B ( ℓ N 0 2 ) x ↦ D 1 2 p x D 1 2 p x\mapsto D^{\frac{1}{2p}}xD^{\frac{1}{2p}} x ↦ D 2 p 1 x D 2 p 1
Φ n ( p ) ( X ) = ( 1 − P n ) X ( 1 − P n ) + τ − n Tr ( D 1 − 1 p P n X P n ) D 1 p P n , X ∈ S p , \Phi^{(p)}_n(X)=(1-P_n)X(1-P_n)+\tau^{-n}\operatorname{Tr}\left(D^{1-\frac{1}{p}}P_nXP_n\right)D^{\frac{1}{p}}P_n,\qquad X\in S^p, Φ n ( p ) ( X ) = ( 1 − P n ) X ( 1 − P n ) + τ − n Tr ( D 1 − p 1 P n X P n ) D p 1 P n , X ∈ S p , since Tr ( D P n x P n ) = Tr ( D 1 − 1 p P n D 1 2 p x D 1 2 p P n ) \operatorname{Tr}(DP_nxP_n)=\operatorname{Tr}(D^{1-\frac{1}{p}}P_nD^{\frac{1}{2p}}xD^{\frac{1}{2p}}P_n) Tr ( D P n x P n ) = Tr ( D 1 − p 1 P n D 2 p 1 x D 2 p 1 P n ) Theorem 7.4 is thus a statement on explicit contractive projections of S p S^p S p τ = Q − d \tau=Q^{-d} τ = Q − d J J J J p J_p J p L p L^p L p J p f = ∂ h 1 2 p J ( f ) ∂ h 1 2 p J_pf=\partial_h^{\frac{1}{2p}}J(f)\partial_h^{\frac{1}{2p}} J p f = ∂ h 2 p 1 J ( f ) ∂ h 2 p 1 (85) implies Φ n ( p ) J p = J p E n \Phi_n^{(p)}J_p=J_pE_n Φ n ( p ) J p = J p E n
Zonal convolution semigroups
In this subsection 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 E \mathbb{E} E W ∗ ( A ) W^*(A) W ∗ ( A ) Proposition 5.5 and the identification C ∗ ( A ) = C ( X τ ) C^*(A)=C(X_\tau) C ∗ ( A ) = C ( X τ )
Proposition 7.6 Let ( T t ) t ≥ 0 (T_t)_{t\geq0} ( T t ) t ≥ 0 C ( X τ ) C(X_\tau) C ( X τ ) ψ : N 0 → [ 0 , ∞ ) \psi:\mathbb{N}_0\to[0,\infty) ψ : N 0 → [ 0 , ∞ ) T t φ l τ = e − t ψ ( l ) φ l τ T_t\varphi_l^\tau=e^{-t\psi(l)}\varphi_l^\tau T t φ l τ = e − t ψ ( l ) φ l τ l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 t ≥ 0 t\geq0 t ≥ 0 t ≥ 0 t\geq0 t ≥ 0 μ t \mu_t μ t X τ X_\tau X τ ∫ F d μ t = ( T t F ) ( 0 ) \int F\,\mathrm{d}\mu_t=(T_tF)(0) ∫ F d μ t = ( T t F ) ( 0 ) ω t ≔ μ t ∘ E \omega_t\coloneqq\mu_t\circ\mathbb{E} ω t : = μ t ∘ E S t ≔ ( Id ⊗ ω t ) ∘ Δ \mathcal{S}_t\coloneqq(\operatorname{Id}\otimes\omega_t)\circ\Delta S t : = ( Id ⊗ ω t ) ∘ Δ
Each ω t \omega_t ω t C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) β λ , μ \beta_{\lambda,\mu} β λ , μ S t \mathcal{S}_t S t C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) h h h S t β λ , μ = β λ , μ S t \mathcal{S}_t\beta_{\lambda,\mu}=\beta_{\lambda,\mu}\mathcal{S}_t S t β λ , μ = β λ , μ S t
For any l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_0 l ∈ 2 1 N 0 j , k j,k j , k S t ( u j k ( l ) ) = δ k 0 e − t ψ ( l ) u j 0 ( l ) \mathcal{S}_t(u_{jk}^{(l)})=\delta_{k0}e^{-t\psi(l)}u_{j0}^{(l)} S t ( u j k ( l ) ) = δ k 0 e − t ψ ( l ) u j 0 ( l ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 S t ( u j k ( l ) ) = 0 \mathcal{S}_t(u_{jk}^{(l)})=0 S t ( u j k ( l ) ) = 0 S t \mathcal{S}_t S t C ( S r 2 ) C(S_r^2) C ( S r 2 ) S t ( F ( A ) ) = ( T t F ) ( A ) \mathcal{S}_t(F(A))=(T_tF)(A) S t ( F ( A )) = ( T t F ) ( A ) F ∈ C ( X τ ) F\in C(X_\tau) F ∈ C ( X τ ) S t \mathcal{S}_t S t C ( S r 2 ) C(S_r^2) C ( S r 2 )
For any x , y ∈ Pol ( S r 2 ) x,y\in\operatorname{Pol}(S_r^2) x , y ∈ Pol ( S r 2 ) h ( S t ( y ) ∗ x ) = h ( y ∗ S t ( x ) ) h(\mathcal{S}_t(y)^*x)=h(y^*\mathcal{S}_t(x)) h ( S t ( y ) ∗ x ) = h ( y ∗ S t ( x )) S t \mathcal{S}_t S t h h h Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 )
If μ t ( { 0 } ) = 0 \mu_t(\{0\})=0 μ t ({ 0 }) = 0 t > 0 t>0 t > 0 ω t \omega_t ω t S t \mathcal{S}_t S t M M M L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 )
Proof. 1. The map E \mathbb{E} E C ( S U r ( 2 ) ) C(\mathrm{SU}_r(2)) C ( SU r ( 2 )) C ∗ ( A ) C^*(A) C ∗ ( A ) Proposition 5.5 , and μ t \mu_t μ t C ∗ ( A ) = C ( X τ ) C^*(A)=C(X_\tau) C ∗ ( A ) = C ( X τ ) ω t \omega_t ω t E ∘ β λ , μ = E \mathbb{E}\circ\beta_{\lambda,\mu}=\mathbb{E} E ∘ β λ , μ = E Id ⊗ ω t \operatorname{Id}\otimes\omega_t Id ⊗ ω t S t \mathcal{S}_t S t [54 Theorem 1.2.1] , h ∘ S t = ( h ⊗ ω t ) Δ = ω t ( 1 ) h = h h\circ\mathcal{S}_t=(h\otimes\omega_t)\Delta=\omega_t(1)h=h h ∘ S t = ( h ⊗ ω t ) Δ = ω t ( 1 ) h = h (42) and coassociativity,
S t β λ , μ = ( ε λ ⊗ Id ⊗ ω t ∘ β 1 , μ ) ∘ Δ ( 2 ) , β λ , μ S t = ( ε λ ⊗ Id ⊗ ω t ∘ β μ , 1 ) ∘ Δ ( 2 ) , \mathcal{S}_t\beta_{\lambda,\mu}=(\varepsilon_\lambda\otimes\operatorname{Id}\otimes\omega_t\circ\beta_{1,\mu})\circ\Delta^{(2)},\qquad\beta_{\lambda,\mu}\mathcal{S}_t=(\varepsilon_\lambda\otimes\operatorname{Id}\otimes\omega_t\circ\beta_{\mu,1})\circ\Delta^{(2)}, S t β λ , μ = ( ε λ ⊗ Id ⊗ ω t ∘ β 1 , μ ) ∘ Δ ( 2 ) , β λ , μ S t = ( ε λ ⊗ Id ⊗ ω t ∘ β μ , 1 ) ∘ Δ ( 2 ) , where Δ ( 2 ) = ( Δ ⊗ Id ) Δ \Delta^{(2)}=(\Delta\otimes\operatorname{Id})\Delta Δ ( 2 ) = ( Δ ⊗ Id ) Δ ( ω t ⊗ ε μ ) Δ = ω t ∘ β 1 , μ (\omega_t\otimes\varepsilon_\mu)\Delta=\omega_t\circ\beta_{1,\mu} ( ω t ⊗ ε μ ) Δ = ω t ∘ β 1 , μ ( ε μ ⊗ ω t ) Δ = ω t ∘ β μ , 1 (\varepsilon_\mu\otimes\omega_t)\Delta=\omega_t\circ\beta_{\mu,1} ( ε μ ⊗ ω t ) Δ = ω t ∘ β μ , 1 (42) . Since ω t \omega_t ω t
By Lemma 5.2, E ( u m k ( l ) ) = ∫ T 2 λ − 2 m μ − 2 k d λ d μ u m k ( l ) \mathbb{E}(u_{mk}^{(l)})=\int_{\mathbb{T}^2}\lambda^{-2m}\mu^{-2k}\,\mathrm{d}\lambda\,\mathrm{d}\mu\,u_{mk}^{(l)} E ( u mk ( l ) ) = ∫ T 2 λ − 2 m μ − 2 k d λ d μ u mk ( l ) u 00 ( l ) u_{00}^{(l)} u 00 ( l ) m = k = 0 m=k=0 m = k = 0 Proposition 5.3 , μ t ( φ l τ ) = ( T t φ l τ ) ( 0 ) = e − t ψ ( l ) \mu_t(\varphi_l^\tau)=(T_t\varphi_l^\tau)(0)=e^{-t\psi(l)} μ t ( φ l τ ) = ( T t φ l τ ) ( 0 ) = e − t ψ ( l ) ω t ( u m k ( l ) ) = δ m 0 δ k 0 e − t ψ ( l ) \omega_t(u_{mk}^{(l)})=\delta_{m0}\delta_{k0}e^{-t\psi(l)} ω t ( u mk ( l ) ) = δ m 0 δ k 0 e − t ψ ( l ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 ω t ( u m k ( l ) ) = 0 \omega_t(u_{mk}^{(l)})=0 ω t ( u mk ( l ) ) = 0 l ∉ N 0 l\notin\mathbb{N}_0 l ∈ / N 0 S t ( u j k ( l ) ) = ∑ m u j m ( l ) ω t ( u m k ( l ) ) \mathcal{S}_t(u_{jk}^{(l)})=\sum_m u_{jm}^{(l)}\omega_t(u_{mk}^{(l)}) S t ( u j k ( l ) ) = ∑ m u j m ( l ) ω t ( u mk ( l ) ) S t \mathcal{S}_t S t β 1 , μ \beta_{1,\mu} β 1 , μ C ( S r 2 ) C(S_r^2) C ( S r 2 ) F ↦ S t ( F ( A ) ) F\mapsto\mathcal{S}_t(F(A)) F ↦ S t ( F ( A )) F ↦ ( T t F ) ( A ) F\mapsto(T_tF)(A) F ↦ ( T t F ) ( A ) φ l τ \varphi_l^\tau φ l τ C ( X τ ) C(X_\tau) C ( X τ ) Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) S t \mathcal{S}_t S t ( u j 0 ( l ) ) (u_{j0}^{(l)}) ( u j 0 ( l ) ) S t ( x ) → x \mathcal{S}_t(x)\to x S t ( x ) → x x ∈ Pol ( S r 2 ) x\in\operatorname{Pol}(S_r^2) x ∈ Pol ( S r 2 ) Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) S t \mathcal{S}_t S t C ( S r 2 ) C(S_r^2) C ( S r 2 )
By the orthogonality relations [54 Theorem 1.4.3 (ii)] , h ( ( u j ′ 0 ( l ′ ) ) ∗ u j 0 ( l ) ) = 0 h((u_{j'0}^{(l')})^*u_{j0}^{(l)})=0 h (( u j ′ 0 ( l ′ ) ) ∗ u j 0 ( l ) ) = 0 l ≠ l ′ l\ne l' l = l ′ l = l ′ l=l' l = l ′ j ≠ j ′ j\ne j' j = j ′ h h h β λ , 1 \beta_{\lambda,1} β λ , 1 ( u j ′ 0 ( l ) ) ∗ u j 0 ( l ) (u_{j'0}^{(l)})^*u_{j0}^{(l)} ( u j ′ 0 ( l ) ) ∗ u j 0 ( l )
λ 2 j ′ − 2 j \lambda^{2j'-2j} λ 2 j ′ − 2 j S t \mathcal{S}_t S t u j 0 ( l ) u_{j_0}^{(l)} u j 0 ( l ) [21 (11.3)] , the modular group acts on u j 0 ( l ) u_{j_0}^{(l)} u j 0 ( l ) r 2 i s j r^{2isj} r 2 i s j
Suppose that μ t ( { 0 } ) = 0 \mu_t(\{0\})=0 μ t ({ 0 }) = 0 k ≥ 0 k \ge0 k ≥ 0 ϕ k \phi_k ϕ k x ↦ ⟨ x ( e k ⊗ 1 ) , e k ⊗ 1 ⟩ x \mapsto\langle x(e_k \otimes1),e_k \otimes1\rangle x ↦ ⟨ x ( e k ⊗ 1 ) , e k ⊗ 1 ⟩ M \mathcal{M} M β λ , μ \beta_{\lambda,\mu} β λ , μ (40) , so that ϕ k ( x ) = ϕ k ( E ( x ) ) = E ( x ) ( τ k ) \phi_k(x)=\phi_k(E(x))=E(x)(\tau^k) ϕ k ( x ) = ϕ k ( E ( x )) = E ( x ) ( τ k ) ω t = ∑ k μ t ( { τ k } ) ϕ k \omega_t=\sum_k\mu_t(\{\tau^k\})\phi_k ω t = ∑ k μ t ({ τ k }) ϕ k Δ \Delta Δ ∗ * ∗ M \mathcal{M} M M ⊗ M \mathcal{M}\otimes\mathcal{M} M ⊗ M ( Id ⊗ ω t ) ∘ Δ (\operatorname{Id}\otimes\omega_t)\circ\Delta ( Id ⊗ ω t ) ∘ Δ β 1 , μ \beta_{1,\mu} β 1 , μ σ \sigma σ L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 )
Theorem 7.7 Let 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 ( T t ) (T_t) ( T t ) e − t L τ e^{-tL_\tau} e − t L τ η = τ \eta=\tau η = τ μ t \mu_t μ t S t = ( Id ⊗ μ t ∘ E ) ∘ Δ \mathcal{S}_t=(\operatorname{Id}\otimes\mu_t\circ E)\circ\Delta S t = ( Id ⊗ μ t ∘ E ) ∘ Δ ψ ( l ) = Λ l ( τ ) \psi(l)=\Lambda_l(\tau) ψ ( l ) = Λ l ( τ )
The restriction of ( S t ) (\mathcal{S}_t) ( S t ) C ( S r 2 ) C(S_r^2) C ( S r 2 ) S t ( u j 0 ( l ) ) = e − t Λ l ( τ ) u j 0 ( l ) \mathcal{S}_t(u_{j_0}^{(l)})=e^{-t\Lambda_l(\tau)}u_{j_0}^{(l)} S t ( u j 0 ( l ) ) = e − t Λ l ( τ ) u j 0 ( l ) C ∗ ( A ) C^*(A) C ∗ ( A ) T t r T_t^r T t r
The maps S t \mathcal{S}_t S t L ∞ ( S r 2 ) L^\infty(S_r^2) L ∞ ( S r 2 )
If τ = Q − d \tau=Q^{-d} τ = Q − d S t J = J T t r \mathcal{S}_tJ=JT_t^r S t J = J T t r L τ L_\tau L τ R r a d d R_{\mathrm{rad}}^d R rad d Theorem 1.1 for α = d \alpha=d α = d
Proof : By Proposition 3.3, ( T t r ) (T_t^r) ( T t r ) C ( Y τ ) = C ( X τ ) C(Y_\tau)=C(X_\tau) C ( Y τ ) = C ( X τ ) T t r ϕ l = e − t Λ l ( τ ) ϕ l T_t^r\phi_l=e^{-t\Lambda_l(\tau)}\phi_l T t r ϕ l = e − t Λ l ( τ ) ϕ l sup X τ ∣ ϕ l ∣ ≤ 1 \sup_{X_\tau}|\phi_l|\le1 sup X τ ∣ ϕ l ∣ ≤ 1 μ t r ( { 0 } ) = 0 \mu_t^r(\{0\})=0 μ t r ({ 0 }) = 0 t > 0 t>0 t > 0 e − t Λ l ( τ ) e^{-t\Lambda_l(\tau)} e − t Λ l ( τ ) ( u j 0 ( l ) ) (u_{j_0}^{(l)}) ( u j 0 ( l ) ) ( L ∞ ( S r 2 ) , h ) (L^\infty(S_r^2),h) ( L ∞ ( S r 2 ) , h ) S t ( x ) ∗ S t ( x ) ≤ S t ( x ∗ x ) \mathcal{S}_t(x)^*\mathcal{S}_t(x)\le\mathcal{S}_t(x^*x) S t ( x ) ∗ S t ( x ) ≤ S t ( x ∗ x ) h ∘ S t = h h\circ\mathcal{S}_t=h h ∘ S t = h α = d \alpha=d α = d η = τ \eta=\tau η = τ
Corollary 7.8 Let 0 < s ≤ 1 0<s\le1 0 < s ≤ 1 ( ν t ( s ) ) t ≥ 0 (\nu_t^{(s)})_{t\ge0} ( ν t ( s ) ) t ≥ 0 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) e − t λ s e^{-t\lambda^s} e − t λ s [64 Chapter 5] , with ν t ( s ) = δ t \nu_t^{(s)}=\delta_t ν t ( s ) = δ t T t ( s ) : = ∫ [ 0 , ∞ ) T u d ν t ( s ) ( u ) T_t^{(s)}\mathrel{\mathop:}=\int_{[0,\infty)}T_u\,d\nu_t^{(s)}(u) T t ( s ) : = ∫ [ 0 , ∞ ) T u d ν t ( s ) ( u ) C ( X τ ) C(X_\tau) C ( X τ ) T t ( s ) ϕ l = e − t Λ l ( τ ) s ϕ l T_t^{(s)}\phi_l=e^{-t\Lambda_l(\tau)^s}\phi_l T t ( s ) ϕ l = e − t Λ l ( τ ) s ϕ l L 2 L^2 L 2 L τ s L_\tau^s L τ s ψ ( l ) = Λ l ( τ ) s \psi(l)=\Lambda_l(\tau)^s ψ ( l ) = Λ l ( τ ) s S t ( s ) = ∫ S u d ν t ( s ) ( u ) \mathcal{S}_t^{(s)}=\int\mathcal{S}_u\,d\nu_t^{(s)}(u) S t ( s ) = ∫ S u d ν t ( s ) ( u ) τ = Q − d \tau=Q^{-d} τ = Q − d s = α d s=\frac{\alpha}{d} s = d α 0 < α ≤ d 0<\alpha\le d 0 < α ≤ d L τ s L_\tau^{s} L τ s R r a d α R_{\mathrm{rad}}^\alpha R rad α
Proof : For F ∈ C ( X τ ) F\in C(X_\tau) F ∈ C ( X τ ) u ↦ T u r F u\mapsto T_u^rF u ↦ T u r F T t ( s ) F T_t^{(s)}F T t ( s ) F C ( X τ ) C(X_\tau) C ( X τ ) T t ( s ) T_t^{(s)} T t ( s ) ν t ( s ) ∗ ν t ′ ( s ) = ν t + t ′ ( s ) \nu_t^{(s)}*\nu_{t'}^{(s)}=\nu_{t+t'}^{(s)} ν t ( s ) ∗ ν t ′ ( s ) = ν t + t ′ ( s ) ( T t r ) (T_t^r) ( T t r ) ν t ( s ) → δ 0 \nu_t^{(s)}\to\delta_0 ν t ( s ) → δ 0 t → 0 + t\to0+ t → 0 + ∫ e − u λ d ν t ( s ) ( u ) = e − t λ s \int e^{-u\lambda}\,d\nu_t^{(s)}(u)=e^{-t\lambda^s} ∫ e − u λ d ν t ( s ) ( u ) = e − t λ s T t ( s ) ϕ l = e − t Λ l ( τ ) s ϕ l T_t^{(s)}\phi_l=e^{-t\Lambda_l(\tau)^s}\phi_l T t ( s ) ϕ l = e − t Λ l ( τ ) s ϕ l ℓ 2 ( N 0 , ω ) \ell^2(\mathbb{N}_0,\omega) ℓ 2 ( N 0 , ω ) T t ( s ) T_t^{(s)} T t ( s ) e − t L τ s e^{-tL_\tau^s} e − t L τ s [64 Chapter 12] . The law at the origin is μ t ( s ) = ∫ μ u d ν t ( s ) ( u ) \mu_t^{(s)}=\int\mu_u\,d\nu_t^{(s)}(u) μ t ( s ) = ∫ μ u d ν t ( s ) ( u ) 0 0 0 t > 0 t>0 t > 0 μ t τ ( { 0 } ) = 0 \mu_t^\tau(\{0\})=0 μ t τ ({ 0 }) = 0 u > 0 u>0 u > 0 ν t ( s ) ( { 0 } ) = lim Λ → ∞ e − t Λ s = 0 \nu_t^{(s)}(\{0\})=\lim_{\Lambda\to\infty}e^{-t\Lambda^s}=0 ν t ( s ) ({ 0 }) = lim Λ → ∞ e − t Λ s = 0 μ t τ ∘ E = ∫ ω u d ν t ( s ) ( u ) \mu_t^\tau\circ E=\int\omega_u\,\mathrm{d}\nu_t^{(s)}(u) μ t τ ∘ E = ∫ ω u d ν t ( s ) ( u ) S t ( s ) \mathcal{S}_t^{(s)} S t ( s ) τ = Q − d \tau=Q^{-d} τ = Q − d s = α d s=\frac{\alpha}{d} s = d α 0 < α ≤ d 0<\alpha\le d 0 < α ≤ d ϑ = d \vartheta=d ϑ = d α < 3 d \alpha<3d α < 3 d
Proposition 7.9 Let 0 < η < 1 0<\eta<1 0 < η < 1 ∣ φ l η ( x ) ∣ ≤ 1 |\varphi_l^\eta(x)|\le1 ∣ φ l η ( x ) ∣ ≤ 1 l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 x ∈ Y η x\in Y_\eta x ∈ Y η ( T t ) t > 0 (T_t)_{t>0} ( T t ) t > 0 C ( Y η ) C(Y_\eta) C ( Y η ) ψ : N 0 → [ 0 , ∞ ) \psi:\mathbb{N}_0\to[0,\infty) ψ : N 0 → [ 0 , ∞ ) T t φ l η = e − t ψ ( l ) φ l η T_t\varphi_l^\eta=e^{-t\psi(l)}\varphi_l^\eta T t φ l η = e − t ψ ( l ) φ l η l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 t > 0 t>0 t > 0
ψ ( l ) ≤ K τ , η ψ ( 1 ) Λ l ( η ) , l ≥ 1 , where K τ , η ≔ max { 2 , A ′ } ( 1 − τ ) ( 1 − η ) ( 1 − τ η ) 2 (95) \psi(l)\le K_{\tau,\eta}\psi(1)\Lambda_l(\eta),\qquad l\ge1,\qquad\text{where }K_{\tau,\eta}\coloneqq\frac{\max\{2,A'\}(1-\tau)}{(1-\eta)(1-\tau\eta)^2} \tag*{(95)} ψ ( l ) ≤ K τ , η ψ ( 1 ) Λ l ( η ) , l ≥ 1 , where K τ , η : = ( 1 − η ) ( 1 − τ η ) 2 max { 2 , A ′ } ( 1 − τ ) (95) and A ′ A' A ′ K τ ≔ K τ , τ K_\tau\coloneqq K_{\tau,\tau} K τ : = K τ , τ
Proof. Let l ≥ 1 l\ge1 l ≥ 1 1 − φ l η ( x ) ≤ max { 2 , A ′ } η − l x 1-\varphi_l^\eta(x)\le\max\{2,A'\}\eta^{-l}x 1 − φ l η ( x ) ≤ max { 2 , A ′ } η − l x x ∈ Y η x\in Y_\eta x ∈ Y η x ≤ η l x\le\eta^l x ≤ η l x = η k x=\eta^k x = η k k < l k<l k < l x η − l ≥ η − 1 > 1 x\eta^{-l}\ge\eta^{-1}>1 x η − l ≥ η − 1 > 1 ∣ φ l η ( x ) ∣ ≤ 1 |\varphi_l^\eta(x)|\le1 ∣ φ l η ( x ) ∣ ≤ 1 1 − φ l η ( x ) ≤ 2 ≤ 2 η − l x 1-\varphi_l^\eta(x)\le2\le2\eta^{-l}x 1 − φ l η ( x ) ≤ 2 ≤ 2 η − l x
For t > 0 t>0 t > 0 F ↦ ( T t F ) ( 0 ) F\mapsto(T_tF)(0) F ↦ ( T t F ) ( 0 ) C ( Y η ) C(Y_\eta) C ( Y η ) μ t \mu_t μ t Y η Y_\eta Y η ∫ φ l η d μ t = e − t ψ ( l ) \int\varphi_l^\eta\,\mathrm{d}\mu_t=e^{-t\psi(l)} ∫ φ l η d μ t = e − t ψ ( l ) φ l η ( 0 ) = 1 \varphi_l^\eta(0)=1 φ l η ( 0 ) = 1 x = 1 − φ 1 η ( x ) 1 − τ η x=\frac{1-\varphi_1^\eta(x)}{1-\tau\eta} x = 1 − τ η 1 − φ 1 η ( x ) ∫ x d μ t ( x ) = 1 − τ 1 − τ η ( 1 − e − t ψ ( 1 ) ) . \int x\,\mathrm{d}\mu_t(x)=\frac{1-\tau}{1-\tau\eta}(1-e^{-t\psi(1)}). ∫ x d μ t ( x ) = 1 − τ η 1 − τ ( 1 − e − t ψ ( 1 ) ) .
1 − e − t ψ ( l ) = ∫ Y η ( 1 − φ l η ) d μ t ≤ max { 2 , A ′ } η − l 1 − τ 1 − τ η ( 1 − e − t ψ ( 1 ) ) . 1-e^{-t\psi(l)}=\int_{Y_\eta}(1-\varphi_l^\eta)\,\mathrm{d}\mu_t\le\max\{2,A'\}\eta^{-l}\frac{1-\tau}{1-\tau\eta}(1-e^{-t\psi(1)}). 1 − e − t ψ ( l ) = ∫ Y η ( 1 − φ l η ) d μ t ≤ max { 2 , A ′ } η − l 1 − τ η 1 − τ ( 1 − e − t ψ ( 1 ) ) . Dividing by t t t t → 0 + t\to0^+ t → 0 + ψ ( l ) ≤ max { 2 , A ′ } ( 1 − τ ) 1 − τ η η − l ψ ( 1 ) \psi(l)\le\frac{\max\{2,A'\}(1-\tau)}{1-\tau\eta}\eta^{-l}\psi(1) ψ ( l ) ≤ 1 − τ η m a x { 2 , A ′ } ( 1 − τ ) η − l ψ ( 1 ) (95) .
For η = τ \eta=\tau η = τ Proposition 7.9 is satisfied by Proposition 5.3, since Y τ = X τ Y_\tau=X_\tau Y τ = X τ K τ = max { 2 , A ′ } ( 1 − τ 2 ) 2 K_\tau=\frac{\max\{2,A'\}}{(1-\tau^2)^2} K τ = ( 1 − τ 2 ) 2 m a x { 2 , A ′ }
Proposition 7.9 is a Lévy–Khintchine type bound: the exponents of a positive semigroup which is diagonal in the zonal spherical functions grow at most like the Casimir eigenvalues. We now identify the semigroups to which it applies on the Podleś sphere.
Theorem 7.10 Let ( S t ) t ≥ 0 (S_t)_{t\ge0} ( S t ) t ≥ 0 C ( S r 2 ) C(S_r^2) C ( S r 2 )
Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ on C ( S r 2 ) , t ≥ 0. \Delta\circ S_t=(\operatorname{Id}\otimes S_t)\circ\Delta\quad\text{on }C(S_r^2),\quad t\ge0. Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ on C ( S r 2 ) , t ≥ 0. There exists a function ψ : N 0 → [ 0 , ∞ ) \psi:\mathbb{N}_0\to[0,\infty) ψ : N 0 → [ 0 , ∞ ) S t ( u j 0 ( l ) ) = e − t ψ ( l ) u j 0 ( l ) S_t(u_{j0}^{(l)})=e^{-t\psi(l)}u_{j0}^{(l)} S t ( u j 0 ( l ) ) = e − t ψ ( l ) u j 0 ( l ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 ∣ j ∣ ≤ l |j|\le l ∣ j ∣ ≤ l t ≥ 0 t\ge0 t ≥ 0 S t S_t S t C ∗ ( A ) C^*(A) C ∗ ( A ) T t T_t T t C ∗ ( A ) = C ( X τ ) C^*(A)=C(X_\tau) C ∗ ( A ) = C ( X τ ) T t φ l τ = e − t ψ ( l ) φ l τ T_t\varphi_l^\tau=e^{-t\psi(l)}\varphi_l^\tau T t φ l τ = e − t ψ ( l ) φ l τ ψ ( l ) ≤ K τ ψ ( 1 ) Λ l ( τ ) \psi(l)\le K_\tau\psi(1)\Lambda_l(\tau) ψ ( l ) ≤ K τ ψ ( 1 ) Λ l ( τ ) l ≥ 1 l\ge1 l ≥ 1
Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d α > 0 \alpha>0 α > 0 f ↦ ∑ l ≥ 1 ψ ( l ) ∣ ⟨ f , e l ⟩ ∣ 2 f\mapsto\sum_{l\ge1}\psi(l)|\langle f,e_l\rangle|^2 f ↦ ∑ l ≥ 1 ψ ( l ) ∣ ⟨ f , e l ⟩ ∣ 2 ( T t ) (T_t) ( T t ) ℓ 0 2 ( w ) \ell_0^2(w) ℓ 0 2 ( w ) R r a d α R_{\mathrm{rad}}^\alpha R rad α α ≤ d \alpha\le d α ≤ d α ≤ d \alpha\le d α ≤ d ( S t α d ) (\mathcal{S}_t^{\frac{\alpha}{d}}) ( S t d α )
For any s > 1 s>1 s > 1 ( e − t L τ s ) t ≥ 0 (e^{-tL_\tau^s})_{t\ge0} ( e − t L τ s ) t ≥ 0 ℓ 0 2 ( w ) \ell_0^2(w) ℓ 0 2 ( w )
Proof. 1. By (42) , we have β 1 , μ = ( Id ⊗ ε μ ) ∘ Δ \beta_{1,\mu}=(\operatorname{Id}\otimes\varepsilon_\mu)\circ\Delta β 1 , μ = ( Id ⊗ ε μ ) ∘ Δ β μ , 1 = ( ε μ ⊗ Id ) ∘ Δ \beta_{\mu,1}=(\varepsilon_\mu\otimes\operatorname{Id})\circ\Delta β μ , 1 = ( ε μ ⊗ Id ) ∘ Δ
Δ ∘ β 1 , μ = ( Id ⊗ β 1 , μ ) ∘ Δ , ( β 1 , μ ⊗ Id ) ∘ Δ = ( Id ⊗ β μ , 1 ) ∘ Δ . (96) \Delta\circ\beta_{1,\mu}=(\operatorname{Id}\otimes\beta_{1,\mu})\circ\Delta,\qquad(\beta_{1,\mu}\otimes\operatorname{Id})\circ\Delta=(\operatorname{Id}\otimes\beta_{\mu,1})\circ\Delta. \tag*{(96)} Δ ∘ β 1 , μ = ( Id ⊗ β 1 , μ ) ∘ Δ , ( β 1 , μ ⊗ Id ) ∘ Δ = ( Id ⊗ β μ , 1 ) ∘ Δ. (96) By the first identity, Δ \Delta Δ C ( S r 2 ) C(S_r^2) C ( S r 2 ) Id ⊗ β 1 , μ \operatorname{Id}\otimes\beta_{1,\mu} Id ⊗ β 1 , μ C ( S U r ( 2 ) ) ⊗ min C ( S r 2 ) C(SU_r(2))\otimes_{\min} C(S_r^2) C ( S U r ( 2 )) ⊗ m i n C ( S r 2 ) Id ⊗ E R \operatorname{Id}\otimes E_R Id ⊗ E R ω t : = ε ∘ S t \omega_t\mathrel{\mathop:}=\varepsilon\circ S_t ω t : = ε ∘ S t C ( S r 2 ) C(S_r^2) C ( S r 2 ) Id ⊗ ε \operatorname{Id}\otimes\varepsilon Id ⊗ ε ( Id ⊗ ε ) ∘ Δ = Id (\operatorname{Id}\otimes\varepsilon)\circ\Delta=\operatorname{Id} ( Id ⊗ ε ) ∘ Δ = Id S t = ( Id ⊗ ω t ) ∘ Δ S_t=(\operatorname{Id}\otimes\omega_t)\circ\Delta S t = ( Id ⊗ ω t ) ∘ Δ C ( S r 2 ) C(S_r^2) C ( S r 2 ) S t ( x ) S_t(x) S t ( x ) β 1 , μ \beta_{1,\mu} β 1 , μ x ∈ C ( S r 2 ) x\in C(S_r^2) x ∈ C ( S r 2 ) (96) gives
( Id ⊗ ω t ) Δ ( x ) = β 1 , μ ( S t ( x ) ) = ( Id ⊗ ω t ∘ β μ , 1 ) Δ ( x ) , x ∈ C ( S r 2 ) , μ ∈ T . (\operatorname{Id}\otimes\omega_t)\Delta(x)=\beta_{1,\mu}(S_t(x))=(\operatorname{Id}\otimes\omega_t\circ\beta_{\mu,1})\Delta(x),\qquad x\in C(S_r^2),\ \mu\in\mathbb{T}. ( Id ⊗ ω t ) Δ ( x ) = β 1 , μ ( S t ( x )) = ( Id ⊗ ω t ∘ β μ , 1 ) Δ ( x ) , x ∈ C ( S r 2 ) , μ ∈ T . where we used that β μ , 1 \beta_{\mu,1} β μ , 1 C ( S r 2 ) C(S_r^2) C ( S r 2 ) x = u j 0 ( l ) x=u^{(l)}_{j0} x = u j 0 ( l ) ω t ( u k 0 ( l ) ) = μ − 2 k ω t ( u k 0 ( l ) ) \omega_t(u^{(l)}_{k0})=\mu^{-2k}\omega_t(u^{(l)}_{k0}) ω t ( u k 0 ( l ) ) = μ − 2 k ω t ( u k 0 ( l ) ) μ ∈ T \mu\in\mathbb{T} μ ∈ T ω t ( u k 0 ( l ) ) = 0 \omega_t(u^{(l)}_{k0})=0 ω t ( u k 0 ( l ) ) = 0 k ≠ 0 k\ne0 k = 0
S t ( u j 0 ( l ) ) = ∑ k u j k ( l ) ω t ( u k 0 ( l ) ) = ω t ( u 00 ( l ) ) u j 0 ( l ) . S_t(u^{(l)}_{j0})=\sum_k u^{(l)}_{jk}\omega_t(u^{(l)}_{k0})=\omega_t(u^{(l)}_{00})u^{(l)}_{j0}. S t ( u j 0 ( l ) ) = k ∑ u j k ( l ) ω t ( u k 0 ( l ) ) = ω t ( u 00 ( l ) ) u j 0 ( l ) . The number ω t ( u 00 ( l ) ) = ω t ( φ l τ ( A ) ) \omega_t(u^{(l)}_{00})=\omega_t(\varphi_l^\tau(A)) ω t ( u 00 ( l ) ) = ω t ( φ l τ ( A )) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] t t t t t t 1 1 1 t = 0 t=0 t = 0 e − t ψ ( l ) e^{-t\psi(l)} e − t ψ ( l ) ψ ( l ) ∈ [ 0 , ∞ ) \psi(l)\in[0,\infty) ψ ( l ) ∈ [ 0 , ∞ ) C ( X τ ) C(X_\tau) C ( X τ ) S t S_t S t C ∗ ( A ) C^*(A) C ∗ ( A ) T t φ l τ = e − t ψ ( l ) φ l τ T_t\varphi_l^\tau=e^{-t\psi(l)}\varphi_l^\tau T t φ l τ = e − t ψ ( l ) φ l τ T t T_t T t η = τ \eta=\tau η = τ
Suppose that the forms are equivalent. Then e l ∈ dom E R e_l\in\operatorname{dom}E_R e l ∈ dom E R l ≥ 1 l\ge1 l ≥ 1 ϑ = d \vartheta=d ϑ = d c , c ′ > 0 c,c'>0 c , c ′ > 0 ψ ( l ) ≥ c E R ( e l ) ≥ c ′ Λ l ( τ ) α d \psi(l)\ge cE_R(e_l)\ge c'\Lambda_l(\tau)^{\frac{\alpha}{d}} ψ ( l ) ≥ c E R ( e l ) ≥ c ′ Λ l ( τ ) d α l ≥ 1 l\ge1 l ≥ 1 Λ l ( τ ) α d − 1 \Lambda_l(\tau)^{\frac{\alpha}{d}-1} Λ l ( τ ) d α − 1 l l l Λ l ( τ ) → ∞ \Lambda_l(\tau)\to\infty Λ l ( τ ) → ∞ α ≤ d \alpha\le d α ≤ d S u = ( Id ⊗ ω u ) ∘ Δ S_u=(\operatorname{Id}\otimes\omega_u)\circ\Delta S u = ( Id ⊗ ω u ) ∘ Δ
Δ ∘ ( Id ⊗ ω u ) ∘ Δ = ( Id ⊗ Id ⊗ ω u ) ∘ ( Δ ⊗ Id ) ∘ Δ = ( Id ⊗ Id ⊗ ω u ) ∘ ( Id ⊗ Δ ) ∘ Δ = ( Id ⊗ S u ) ∘ Δ . \Delta\circ(\operatorname{Id}\otimes\omega_u)\circ\Delta=(\operatorname{Id}\otimes\operatorname{Id}\otimes\omega_u)\circ(\Delta\otimes\operatorname{Id})\circ\Delta=(\operatorname{Id}\otimes\operatorname{Id}\otimes\omega_u)\circ(\operatorname{Id}\otimes\Delta)\circ\Delta=(\operatorname{Id}\otimes S_u)\circ\Delta. Δ ∘ ( Id ⊗ ω u ) ∘ Δ = ( Id ⊗ Id ⊗ ω u ) ∘ ( Δ ⊗ Id ) ∘ Δ = ( Id ⊗ Id ⊗ ω u ) ∘ ( Id ⊗ Δ ) ∘ Δ = ( Id ⊗ S u ) ∘ Δ. Integrating with respect to ν t ( s ) \nu_t^{(s)} ν t ( s ) S t ( s ) S_t^{(s)} S t ( s ) s = α d ≤ 1 s=\frac{\alpha}{d}\le1 s = d α ≤ 1
Suppose that the operators T t = e − t L τ s T_t=e^{-tL_\tau^s} T t = e − t L τ s L τ 1 = 0 L_\tau1=0 L τ 1 = 0 φ l τ \varphi_l^\tau φ l τ e − t Λ l ( τ ) s φ l τ e^{-t\Lambda_l(\tau)^s}\varphi_l^\tau e − t Λ l ( τ ) s φ l τ C ( X τ ) C(X_\tau) C ( X τ ) η = τ \eta=\tau η = τ ψ ( l ) = Λ l ( τ ) s \psi(l)=\Lambda_l(\tau)^s ψ ( l ) = Λ l ( τ ) s Λ l ( τ ) s − 1 \Lambda_l(\tau)^{s-1} Λ l ( τ ) s − 1 l l l s > 1 s>1 s > 1
Remark 7.11 Theorem 7.10 and Theorem 4.6 together describe how the order α \alpha α α ≤ d \alpha\le d α ≤ d C ( S r 2 ) C(S_r^2) C ( S r 2 ) d < α < 3 d d<\alpha<3d d < α < 3 d L τ α d L_\tau^{\frac{\alpha}{d}} L τ d α − L τ α d -L_\tau^{\frac{\alpha}{d}} − L τ d α α ≥ 3 d \alpha\ge3d α ≥ 3 d L τ L_\tau L τ
Remark 7.12 The radial Taibleson semigroup is not of convolution type. More precisely, let ω \omega ω C ( S U r ( 2 ) ) \mathrm{C}(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) S = ( Id ⊗ ω ) ∘ Δ S=(\operatorname{Id}\otimes\omega)\circ\Delta S = ( Id ⊗ ω ) ∘ Δ S = ( ω ⊗ Id ) ∘ Δ S=(\omega\otimes\operatorname{Id})\circ\Delta S = ( ω ⊗ Id ) ∘ Δ S ( u j k ( l ) ) S(u_{jk}^{(l)}) S ( u j k ( l ) ) C ( l ) C(l) C ( l ) C ( 0 ) ⊕ C ( 1 ) C(0)\oplus C(1) C ( 0 ) ⊕ C ( 1 ) S S S Proposition 5.3 and (27) , A = 1 − u 00 ( 1 ) 1 + τ A=\frac{1-u_{00}^{(1)}}{1+\tau} A = 1 + τ 1 − u 00 ( 1 ) S ( A ) ∈ C ∗ ( A ) S(A)\in C^*(A) S ( A ) ∈ C ∗ ( A ) S ( A ) S(A) S ( A ) β λ , μ \beta_{\lambda,\mu} β λ , μ C ( 0 ) ⊕ C ( 1 ) C(0)\oplus C(1) C ( 0 ) ⊕ C ( 1 ) span { 1 , u 00 ( 1 ) } = span { 1 , A } \operatorname{span}\{1,u_{00}^{(1)}\}=\operatorname{span}\{1,A\} span { 1 , u 00 ( 1 ) } = span { 1 , A } τ = Q − d \tau=Q^{-d} τ = Q − d ⟨ A , J 2 g n ⟩ \langle A,J_{2g_n}\rangle ⟨ A , J 2 g n ⟩ Proposition 4.7 , so that J T t J − 1 ( A ) ∈ span { 1 , A } JT_tJ^{-1}(A)\in\operatorname{span}\{1,A\} J T t J − 1 ( A ) ∈ span { 1 , A } e − t λ n e^{-t\lambda_n} e − t λ n n n n J T t J − 1 JT_tJ^{-1} J T t J − 1 C ∗ ( A ) C^*(A) C ∗ ( A ) t > 0 t>0 t > 0 Theorem 7.7 is the zonal convolution semigroup whose energy is equivalent to the Taibleson energy of order d d d
The frontier between the Podles sphere and SUr(2)
In this section 0 < r < 1 0<r<1 0 < r < 1 τ = r 2 \tau=r^2 τ = r 2 p k = e k k ⊗ 1 ∈ C ∗ ( A ) p_k=e_{kk}\otimes1\in C^*(A) p k = e k k ⊗ 1 ∈ C ∗ ( A ) b ( e n ⊗ 1 ) = 1 − τ n e n − 1 ⊗ 1 b(e_n\otimes1)=\sqrt{1-\tau^n}e_{n-1}\otimes1 b ( e n ⊗ 1 ) = 1 − τ n e n − 1 ⊗ 1 1 1 1 L 2 ( T ) L^2(\mathbb{T}) L 2 ( T )
A coefficient estimate for completely positive maps
Lemma 8.1 Let S : C ( S U r ( 2 ) ) → C ( S U r ( 2 ) ) S:C(\mathrm{SU}_{r}(2))\to C(\mathrm{SU}_{r}(2)) S : C ( SU r ( 2 )) → C ( SU r ( 2 )) K K K N 0 \mathbb{N}_0 N 0 S ( p k ) = ∑ i K ( i , k ) p i S(p_k)=\sum_i K(i,k)p_i S ( p k ) = ∑ i K ( i , k ) p i k k k i , j ≥ 0 i,j\geq0 i , j ≥ 0
∣ ⟨ S ( b ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ ∣ ≤ ∑ k ≥ 1 1 − τ k K ( i , k − 1 ) K ( j , k ) . (97) \left|\langle S(b)(e_j\otimes1),e_i\otimes1\rangle\right|
\leq\sum_{k\geq1}\sqrt{1-\tau^k}\sqrt{K(i,k-1)K(j,k)}. \tag*{(97)} ∣ ⟨ S ( b ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ ∣ ≤ k ≥ 1 ∑ 1 − τ k K ( i , k − 1 ) K ( j , k ) . (97) In particular, if both K ( n − 1 , ⋅ ) K(n-1,\cdot) K ( n − 1 , ⋅ ) K ( n , ⋅ ) K(n,\cdot) K ( n , ⋅ ) ℓ 0 1 \ell^1_0 ℓ 0 1 μ \mu μ n → ∞ n\to\infty n → ∞
lim sup n → ∞ ∣ ⟨ S ( b ) ( e n ⊗ 1 ) , e n − 1 ⊗ 1 ⟩ ∣ ≤ F ( μ ) : d e f = ∑ k ≥ 1 μ ( k − 1 ) μ ( k ) . (98) \limsup_{n\to\infty}\left|\langle S(b)(e_n\otimes1),e_{n-1}\otimes1\rangle\right|
\leq F(\mu)\mathrel{\mathop:^{\mathrm{def}}}
=\sum_{k\geq1}\sqrt{\mu(k-1)\mu(k)}. \tag*{(98)} n → ∞ lim sup ∣ ⟨ S ( b ) ( e n ⊗ 1 ) , e n − 1 ⊗ 1 ⟩ ∣ ≤ F ( μ ) : def = k ≥ 1 ∑ μ ( k − 1 ) μ ( k ) . (98) No normal extension of S S S
Proof: By Stinespring’s theorem, there exist a Hilbert space K \mathcal{K} K Π \Pi Π C ( S U r ( 2 ) ) C(\mathrm{SU}_{r}(2)) C ( SU r ( 2 )) K \mathcal{K} K V : ℓ N 0 2 ⊗ L 2 ( T ) → K V:\ell^2_{\mathbb{N}_0}\otimes L^2(\mathbb{T})\to\mathcal{K} V : ℓ N 0 2 ⊗ L 2 ( T ) → K S ( x ) = V ∗ Π ( x ) V S(x)=V^*\Pi(x)V S ( x ) = V ∗ Π ( x ) V η i : d e f = V ( e i ⊗ 1 ) \eta_i\mathrel{\mathop:^{\mathrm{def}}}=V(e_i\otimes1) η i : def = V ( e i ⊗ 1 )
∥ Π ( p k ) η i ∥ 2 = ⟨ S ( p k ) ( e i ⊗ 1 ) , e i ⊗ 1 ⟩ = K ( i , k ) . (99) \|\Pi(p_k)\eta_i\|^2=\langle S(p_k)(e_i\otimes1),e_i\otimes1\rangle=K(i,k). \tag*{(99)} ∥Π ( p k ) η i ∥ 2 = ⟨ S ( p k ) ( e i ⊗ 1 ) , e i ⊗ 1 ⟩ = K ( i , k ) . (99) Since ∑ k K ( i , k ) = 1 = ∥ η i ∥ 2 \sum_k K(i,k)=1=\|\eta_i\|^2 ∑ k K ( i , k ) = 1 = ∥ η i ∥ 2 Π ( p k ) \Pi(p_k) Π ( p k ) η i = ∑ k ≥ 0 Π ( p k ) η i \eta_i=\sum_{k\geq0}\Pi(p_k)\eta_i η i = ∑ k ≥ 0 Π ( p k ) η i Π \Pi Π Π ( p k ) \Pi(p_k) Π ( p k ) η i \eta_i η i
The relations b p 0 = 0 bp_0=0 b p 0 = 0 b p k = 1 − τ k ( e k − 1 , k ⊗ 1 ) bp_k=\sqrt{1-\tau^k}(e_{k-1,k}\otimes1) b p k = 1 − τ k ( e k − 1 , k ⊗ 1 ) k ≥ 1 k\geq1 k ≥ 1 b p k = p k − 1 b p k bp_k=p_{k-1}bp_k b p k = p k − 1 b p k
⟨ S ( b ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ = ⟨ Π ( b ) η j , η i ⟩ = ∑ k ≥ 1 ⟨ Π ( b p k ) Π ( p k ) η j , Π ( p k − 1 ) η i ⟩ . \langle S(b)(e_j\otimes1),e_i\otimes1\rangle
=\langle\Pi(b)\eta_j,\eta_i\rangle
=\sum_{k\geq1}\langle\Pi(bp_k)\Pi(p_k)\eta_j,\Pi(p_{k-1})\eta_i\rangle. ⟨ S ( b ) ( e j ⊗ 1 ) , e i ⊗ 1 ⟩ = ⟨ Π ( b ) η j , η i ⟩ = k ≥ 1 ∑ ⟨ Π ( b p k ) Π ( p k ) η j , Π ( p k − 1 ) η i ⟩ . Each summand is bounded by ∥ b p k ∥ ∥ Π ( p k ) η j ∥ ∥ Π ( p k − 1 ) η i ∥ \|bp_k\|\,\|\Pi(p_k)\eta_j\|\,\|\Pi(p_{k-1})\eta_i\| ∥ b p k ∥ ∥Π ( p k ) η j ∥ ∥Π ( p k − 1 ) η i ∥ (97) by (99) . If two probability rows converge in ℓ 1 \ell^1 ℓ 1 ℓ 2 \ell^2 ℓ 2 ∣ a − b ∣ 2 ≤ ∣ a − b ∣ |\sqrt{a}-\sqrt{b}|^2\leq|a-b| ∣ a − b ∣ 2 ≤ ∣ a − b ∣ 1 − τ k ≤ 1 \sqrt{1-\tau^k}\leq1 1 − τ k ≤ 1 (98) .
Lemma 8.2. For any probability measure μ \mu μ N 0 \mathbb{N}_0 N 0
F ( μ ) ≤ 1 − ( sup k μ ( k ) ) 2 . (100) F(\mu) \le\sqrt{1-\left(\sup_k\mu(k)\right)^2}. \tag*{(100)} F ( μ ) ≤ 1 − ( k sup μ ( k ) ) 2 . (100) Proof: Let μ + ( 0 ) = 0 \mu^+(0)=0 μ + ( 0 ) = 0 μ + ( k ) = μ ( k − 1 ) \mu^+(k)=\mu(k-1) μ + ( k ) = μ ( k − 1 ) k ≥ 1 k\ge1 k ≥ 1 { 0 , … , j } \{0,\ldots,j\} { 0 , … , j } μ \mu μ μ + \mu^+ μ + μ ( j ) \mu(j) μ ( j ) 1 2 ∥ μ − μ + ∥ 1 ≥ sup j μ ( j ) \frac{1}{2}\lVert\mu-\mu^+\rVert_1\ge\sup_j\mu(j) 2 1 ∥ μ − μ + ∥ 1 ≥ sup j μ ( j ) p , q p,q p , q N 0 \mathbb{N}_0 N 0
1 2 ∑ k ∣ p k − q k ∣ ≤ 1 2 ( ∑ k ( p k − q k ) 2 ) 1 2 ( ∑ k ( p k + q k ) 2 ) 1 2 = 1 − ( ∑ k p k q k ) 2 . \frac{1}{2}\sum_k\lvert p_k-q_k\rvert\le\frac{1}{2}\left(\sum_k(\sqrt{p_k}-\sqrt{q_k})^2\right)^{\frac{1}{2}}\left(\sum_k(\sqrt{p_k}+\sqrt{q_k})^2\right)^{\frac{1}{2}}=\sqrt{1-\left(\sum_k\sqrt{p_kq_k}\right)^2}. 2 1 k ∑ ∣ p k − q k ∣ ≤ 2 1 ( k ∑ ( p k − q k ) 2 ) 2 1 ( k ∑ ( p k + q k ) 2 ) 2 1 = 1 − ( k ∑ p k q k ) 2 . Applying this to p = μ p=\mu p = μ q = μ + q=\mu^+ q = μ + (100) .
Theorem 8.3. Let S : C ( S U r ( 2 ) ) → C ( S U r ( 2 ) ) S:C(\mathrm{SU}_r(2))\to C(\mathrm{SU}_r(2)) S : C ( SU r ( 2 )) → C ( SU r ( 2 )) S ( C ∗ ( A ) ) ⊂ C ∗ ( A ) S(C^*(A))\subset C^*(A) S ( C ∗ ( A )) ⊂ C ∗ ( A ) S S S C ∗ ( A ) = C ( X τ ) C^*(A)=C(X_\tau) C ∗ ( A ) = C ( X τ ) N 0 \mathbb{N}_0 N 0 K K K K ( n , ⋅ ) K(n,\cdot) K ( n , ⋅ ) ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 μ \mu μ n → ∞ n\to\infty n → ∞
∥ S ( b ) − b ∥ ≥ 1 − F ( μ ) ≥ 1 − 1 − ( sup k μ ( k ) ) 2 . (101) \lVert S(b)-b\rVert\ge1-F(\mu)\ge1-\sqrt{1-\left(\sup_k\mu(k)\right)^2}. \tag*{(101)} ∥ S ( b ) − b ∥ ≥ 1 − F ( μ ) ≥ 1 − 1 − ( k sup μ ( k ) ) 2 . (101) Proof: The hypothesis gives S ( p k ) = ∑ i K ( i , k ) p i S(p_k)=\sum_iK(i,k)p_i S ( p k ) = ∑ i K ( i , k ) p i p k p_k p k τ k \tau^k τ k Lemma 8.1 ,
∥ S ( b ) − b ∥ ≥ lim sup n → ∞ ∣ 1 − τ n − ⟨ S ( b ) ( e n ⊗ 1 ) , e n − 1 ⊗ 1 ⟩ ∣ ≥ 1 − F ( μ ) . \lVert S(b)-b\rVert\ge\limsup_{n\to\infty}\left\lvert\sqrt{1-\tau^n}-\langle S(b)(e_n\otimes1),e_{n-1}\otimes1\rangle\right\rvert\ge1-F(\mu). ∥ S ( b ) − b ∥ ≥ n → ∞ lim sup 1 − τ n − ⟨ S ( b ) ( e n ⊗ 1 ) , e n − 1 ⊗ 1 ⟩ ≥ 1 − F ( μ ) . Lemma 8.2 gives the second inequality.
The Taibleson dynamics
Corollary 8.4. Suppose that τ = Q − d \tau=Q^{-d} τ = Q − d S t : C ( S U r ( 2 ) ) → C ( S U r ( 2 ) ) S_t:C(\mathrm{SU}_r(2))\to C(\mathrm{SU}_r(2)) S t : C ( SU r ( 2 )) → C ( SU r ( 2 )) t > 0 t>0 t > 0 J T t J − 1 JT_tJ^{-1} J T t J − 1 C ∗ ( A ) C^*(A) C ∗ ( A ) t → 0 + t\to0^+ t → 0 +
∥ S ρ − j ( b ) − b ∥ ≥ 1 − 1 − δ T 2 > 0 , j ≥ 1 , (102) \lVert S_{\rho^{-j}}(b)-b\rVert\ge1-\sqrt{1-\delta_T^2}>0,\qquad j\ge1, \tag*{(102)} ∥ S ρ − j ( b ) − b ∥ ≥ 1 − 1 − δ T 2 > 0 , j ≥ 1 , (102) where δ T = ( 1 − τ ) ( e − τ − 1 − e − ρ − 1 ) \delta_T=(1-\tau)(e^{-\tau^{-1}}-e^{-\rho^{-1}}) δ T = ( 1 − τ ) ( e − τ − 1 − e − ρ − 1 )
Proof: By Lemma 2.6 , the rows of the transition matrix of T t T_t T t ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 μ t \mu_t μ t sup k μ ρ − j ( k ) ≥ δ T \sup_k\mu_{\rho^{-j}}(k)\ge\delta_T sup k μ ρ − j ( k ) ≥ δ T Theorem 8.3 gives (102) . Since ρ − j → 0 \rho^{-j}\to0 ρ − j → 0
There is no contradiction with Theorem 7.2 . The conditional-expectation lifting is strongly continuous on C ( S r 2 ) C(S_r^2) C ( S r 2 ) b b b 0 ∈ X τ 0\in X_\tau 0 ∈ X τ S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) Lemma 5.1 ), the radial semigroup has the entrance law μ t \mu_t μ t ρ − j \rho^{-j} ρ − j b b b Lemma 8.1 detects exactly this requirement.
The Casimir dynamics
Let 0 < η < 1 0<\eta<1 0 < η < 1 μ t η \mu_t^\eta μ t η ( T t η ) (T_t^\eta) ( T t η ) − L η -L_\eta − L η Proposition 3.3 . If ∣ φ l η ∣ ≤ 1 |\varphi_l^\eta|\leq1 ∣ φ l η ∣ ≤ 1 Y η Y_\eta Y η l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 Proposition 3.3 , μ t η \mu_t^\eta μ t η N 0 \mathbb{N}_0 N 0 t > 0 t>0 t > 0 P t η ( n , ⋅ ) P_t^\eta(n,\cdot) P t η ( n , ⋅ ) ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 Proposition 5.3 , this is the case for η = τ \eta=\tau η = τ
Proposition 8.5 Let 0 < η < 1 0<\eta<1 0 < η < 1 ∣ φ l η ( x ) ∣ ≤ 1 |\varphi_l^\eta(x)|\leq1 ∣ φ l η ( x ) ∣ ≤ 1 l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 x ∈ Y η x\in Y_\eta x ∈ Y η
c 0 ( τ , η ) ≔ ( 1 − τ ) η ( 1 − τ η 3 ) 2 ( 1 + η ) ( 1 − τ η ) ( 1 − τ η 2 ) . c_0(\tau,\eta)\mathrel{\mathop{\coloneqq}} \frac{(1-\tau)\eta(1-\tau\eta^3)}{2(1+\eta)(1-\tau\eta)(1-\tau\eta^2)}. c 0 ( τ , η ) : = 2 ( 1 + η ) ( 1 − τ η ) ( 1 − τ η 2 ) ( 1 − τ ) η ( 1 − τ η 3 ) . N 0 ( τ , η ) ≔ 1 + log 16 c 0 ( τ , η ) log 1 η , δ C ( τ , η ) ≔ c 0 ( τ , η ) 8 N 0 ( τ , η ) . N_0(\tau,\eta)\mathrel{\mathop{\coloneqq}}1+\frac{\log\frac{16}{c_0(\tau,\eta)}}{\log\frac{1}{\eta}},\qquad
\delta_C(\tau,\eta)\mathrel{\mathop{\coloneqq}}\frac{c_0(\tau,\eta)}{8N_0(\tau,\eta)}. N 0 ( τ , η ) : = 1 + log η 1 log c 0 ( τ , η ) 16 , δ C ( τ , η ) : = 8 N 0 ( τ , η ) c 0 ( τ , η ) . For any 0 < t ≤ Λ 1 ( η ) − 1 0<t\leq\Lambda_1(\eta)^{-1} 0 < t ≤ Λ 1 ( η ) − 1 sup k μ t η ( k ) ≥ δ C ( τ , η ) \sup_k\mu_t^\eta(k)\geq\delta_C(\tau,\eta) sup k μ t η ( k ) ≥ δ C ( τ , η ) c 0 ( τ , τ ) = τ ( 1 + τ 2 ) 2 ( 1 + τ ) 2 ( 1 + τ + τ 2 ) c_0(\tau,\tau)=\frac{\tau(1+\tau^2)}{2(1+\tau)^2(1+\tau+\tau^2)} c 0 ( τ , τ ) = 2 ( 1 + τ ) 2 ( 1 + τ + τ 2 ) τ ( 1 + τ 2 )
Proof : We consider the random variable x x x Y η Y_\eta Y η μ t ≔ μ t η \mu_t\mathrel{\mathop{\coloneqq}}\mu_t^\eta μ t : = μ t η m 1 = ∫ x d μ t m_1=\int x\,\mathrm{d}\mu_t m 1 = ∫ x d μ t m 2 = ∫ x 2 d μ t m_2=\int x^2\,\mathrm{d}\mu_t m 2 = ∫ x 2 d μ t Λ 1 = Λ 1 ( η ) \Lambda_1=\Lambda_1(\eta) Λ 1 = Λ 1 ( η ) Λ 2 = Λ 2 ( η ) \Lambda_2=\Lambda_2(\eta) Λ 2 = Λ 2 ( η ) c 0 = c 0 ( τ , η ) c_0=c_0(\tau,\eta) c 0 = c 0 ( τ , η ) [44 (14.12.1)] , as in the proof of Lemma 4.4 , we have
φ 1 η ( x ) = 1 − 1 − τ η 1 − τ x , φ 2 η ( x ) = 1 − ( 1 + η ) ( 1 − τ η 2 ) η ( 1 − τ ) x + c 2 , 2 x 2 , \varphi_1^\eta(x)=1-\frac{1-\tau\eta}{1-\tau}x,\qquad
\varphi_2^\eta(x)=1-\frac{(1+\eta)(1-\tau\eta^2)}{\eta(1-\tau)}x+c_{2,2}x^2, φ 1 η ( x ) = 1 − 1 − τ 1 − τ η x , φ 2 η ( x ) = 1 − η ( 1 − τ ) ( 1 + η ) ( 1 − τ η 2 ) x + c 2 , 2 x 2 , c 2 , 2 ≔ ( 1 − τ η 2 ) ( 1 − τ η 3 ) η ( 1 − τ ) ( 1 − τ η ) . c_{2,2}\mathrel{\mathop{\coloneqq}}\frac{(1-\tau\eta^2)(1-\tau\eta^3)}{\eta(1-\tau)(1-\tau\eta)}. c 2 , 2 : = η ( 1 − τ ) ( 1 − τ η ) ( 1 − τ η 2 ) ( 1 − τ η 3 ) . and (4) gives Λ 2 Λ 1 = ( 1 + η ) ( 1 − τ η 2 ) η ( 1 − τ η ) \frac{\Lambda_2}{\Lambda_1}=\frac{(1+\eta)(1-\tau\eta^2)}{\eta(1-\tau\eta)} Λ 1 Λ 2 = η ( 1 − τ η ) ( 1 + η ) ( 1 − τ η 2 ) (29) for n = 1 n=1 n = 1
m 1 = 1 − τ 1 − τ η ( 1 − e − t Λ 1 ) . m_1=\frac{1-\tau}{1-\tau\eta}(1-e^{-t\Lambda_1}). m 1 = 1 − τ η 1 − τ ( 1 − e − t Λ 1 ) . Since 1 − e − u ≥ u − u 2 2 ≥ u 2 1-e^{-u}\geq u-\frac{u^2}{2}\geq\frac{u}{2} 1 − e − u ≥ u − 2 u 2 ≥ 2 u 0 ≤ u ≤ 1 0\leq u\leq1 0 ≤ u ≤ 1 m 1 ≥ ( 1 − τ ) t Λ 1 2 ( 1 − τ η ) m_1\geq\frac{(1-\tau)t\Lambda_1}{2(1-\tau\eta)} m 1 ≥ 2 ( 1 − τ η ) ( 1 − τ ) t Λ 1 t Λ 1 ≤ 1 t\Lambda_1\leq1 t Λ 1 ≤ 1 (29) for n = 2 n=2 n = 2 m 1 m_1 m 1
c 2 , 2 m 2 = e − t Λ 2 − 1 + Λ 2 Λ 1 ( 1 − e − t Λ 1 ) . c_{2,2}m_2=e^{-t\Lambda_2}-1+\frac{\Lambda_2}{\Lambda_1}(1-e^{-t\Lambda_1}). c 2 , 2 m 2 = e − t Λ 2 − 1 + Λ 1 Λ 2 ( 1 − e − t Λ 1 ) . For u ≥ 0 u\geq0 u ≥ 0 e − u ≤ 1 − u + u 2 2 e^{-u}\leq1-u+\frac{u^2}{2} e − u ≤ 1 − u + 2 u 2 1 − e − u ≤ u 1-e^{-u}\leq u 1 − e − u ≤ u m 2 ≤ t 2 Λ 2 2 2 c 2 , 2 m_2\leq\frac{t^2\Lambda_2^2}{2c_{2,2}} m 2 ≤ 2 c 2 , 2 t 2 Λ 2 2 μ t ( x > m 1 2 ) ≥ m 1 2 4 m 2 \mu_t(x>\frac{m_1}{2})\geq\frac{m_1^2}{4m_2} μ t ( x > 2 m 1 ) ≥ 4 m 2 m 1 2
μ t ( x > m 1 2 ) ≥ ( 1 − τ ) 2 c 2 , 2 8 ( 1 − τ η ) 2 Λ 1 2 Λ 2 2 = c 0 4 . \mu_t\left(x>\frac{m_1}{2}\right)\geq\frac{(1-\tau)^2c_{2,2}}{8(1-\tau\eta)^2}\frac{\Lambda_1^2}{\Lambda_2^2}=\frac{c_0}{4}. μ t ( x > 2 m 1 ) ≥ 8 ( 1 − τ η ) 2 ( 1 − τ ) 2 c 2 , 2 Λ 2 2 Λ 1 2 = 4 c 0 . On the other hand, Markov’s inequality gives μ t ( x > 8 m 1 c 0 ) ≤ c 0 8 \mu_t(x>\frac{8m_1}{c_0})\leq\frac{c_0}{8} μ t ( x > c 0 8 m 1 ) ≤ 8 c 0 μ t ( m 1 2 < x ≤ 8 m 1 c 0 ) ≥ c 0 8 \mu_t(\frac{m_1}{2}<x\leq\frac{8m_1}{c_0})\geq\frac{c_0}{8} μ t ( 2 m 1 < x ≤ c 0 8 m 1 ) ≥ 8 c 0 η k \eta^k η k ( m 1 2 , 8 m 1 c 0 ] (\frac{m_1}{2},\frac{8m_1}{c_0}] ( 2 m 1 , c 0 8 m 1 ] k k k log 16 c 0 log 1 η \frac{\log\frac{16}{c_0}}{\log\frac{1}{\eta}} l o g η 1 l o g c 0 16 N 0 ( τ , η ) N_0(\tau,\eta) N 0 ( τ , η ) μ t ( k ) ≥ δ C ( τ , η ) \mu_t(k)\geq\delta_C(\tau,\eta) μ t ( k ) ≥ δ C ( τ , η ) c 0 ( τ , τ ) c_0(\tau,\tau) c 0 ( τ , τ ) 1 − τ 4 = ( 1 − τ ) ( 1 + τ ) ( 1 + τ 2 ) 1-\tau^4=(1-\tau)(1+\tau)(1+\tau^2) 1 − τ 4 = ( 1 − τ ) ( 1 + τ ) ( 1 + τ 2 ) 1 − τ 3 = ( 1 − τ ) ( 1 + τ + τ 2 ) 1-\tau^3=(1-\tau)(1+\tau+\tau^2) 1 − τ 3 = ( 1 − τ ) ( 1 + τ + τ 2 )
Theorem 8.6 Put Λ 1 2 ≔ r − 1 ( 1 − r ) ( 1 − r 3 ) \Lambda_{\frac{1}{2}} \coloneqq r^{-1}(1-r)(1-r^3) Λ 2 1 : = r − 1 ( 1 − r ) ( 1 − r 3 ) (47) .
For any 0 < t ≤ Λ 1 ( τ ) − 1 0<t\leq\Lambda_1(\tau)^{-1} 0 < t ≤ Λ 1 ( τ ) − 1 S : C ( S U r ( 2 ) ) → C ( S U r ( 2 ) ) S:\mathrm{C}(\mathrm{SU}_r(2))\to\mathrm{C}(\mathrm{SU}_r(2)) S : C ( SU r ( 2 )) → C ( SU r ( 2 )) S ( F ( A ) ) = ( T t T F ) ( A ) S(F(A))=(T_t^T F)(A) S ( F ( A )) = ( T t T F ) ( A ) F ∈ C ( X τ ) F\in\mathrm{C}(X_\tau) F ∈ C ( X τ )
∥ S ( b ) − b ∥ ≥ 1 − 1 − δ C ( τ , τ ) 2 . \lVert S(b)-b\rVert\geq1-\sqrt{1-\delta_{\mathrm{C}}(\tau,\tau)^2}. ∥ S ( b ) − b ∥ ≥ 1 − 1 − δ C ( τ , τ ) 2 . Let t 0 ≔ min { Λ 1 ( τ ) − 1 , 1 2 Λ 1 2 − 1 ( 1 − 1 − δ C ( τ , τ ) 2 ) } t_0\coloneqq\min\left\{\Lambda_1(\tau)^{-1},\frac{1}{2}\Lambda_{\frac{1}{2}}^{-1}\left(1-\sqrt{1-\delta_{\mathrm{C}}(\tau,\tau)^2}\right)\right\} t 0 : = min { Λ 1 ( τ ) − 1 , 2 1 Λ 2 1 − 1 ( 1 − 1 − δ C ( τ , τ ) 2 ) } 0 < t ≤ t 0 0<t\leq t_0 0 < t ≤ t 0 U t \mathcal{U}_t U t Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_r(2)) Pol ( SU r ( 2 )) U t ( u j k ( l ) ) = e − t Λ l u j k ( l ) \mathcal{U}_t(u_{jk}^{(l)})=e^{-t\Lambda_l}u_{jk}^{(l)} U t ( u j k ( l ) ) = e − t Λ l u j k ( l ) l ∈ 1 2 N 0 l\in\frac{1}{2}\mathbb{N}_0 l ∈ 2 1 N 0 C ( S U r ( 2 ) ) \mathrm{C}(\mathrm{SU}_r(2)) C ( SU r ( 2 )) Pol ( S r 2 ) \operatorname{Pol}(S_r^2) Pol ( S r 2 ) S t \mathcal{S}_t S t Theorem 7.7 .
Proof : 1. By the discussion preceding Proposition 8.5, the restriction of S S S C ∗ ( A ) \mathrm{C}^*(A) C ∗ ( A ) Theorem 8.3 with μ = μ t τ \mu=\mu_t^\tau μ = μ t τ η = τ \eta=\tau η = τ (101) give the estimate.
Suppose that U t \mathcal{U}_t U t U ~ t \widetilde{\mathcal{U}}_t U t C ( S U r ( 2 ) ) \mathrm{C}(\mathrm{SU}_r(2)) C ( SU r ( 2 )) Lemma 5.1 , the counit extends to a character ε \varepsilon ε C ( S U r ( 2 ) ) \mathrm{C}(\mathrm{SU}_r(2)) C ( SU r ( 2 )) ψ t ≔ ε ∘ U ~ t \psi_t\coloneqq\varepsilon\circ\widetilde{\mathcal{U}}_t ψ t : = ε ∘ U t S ≔ ( Id ⊗ ψ t ) ∘ Δ S\coloneqq(\operatorname{Id}\otimes\psi_t)\circ\Delta S : = ( Id ⊗ ψ t ) ∘ Δ C ( S U r ( 2 ) ) \mathrm{C}(\mathrm{SU}_r(2)) C ( SU r ( 2 ))
S ( u j k ( l ) ) = ∑ m u j m ( l ) ψ t ( u m k ( l ) ) = ∑ m u j m ( l ) e − t Λ l ε ( u m k ( l ) ) = e − t Λ l u j k ( l ) . S(u_{jk}^{(l)})=\sum_m u_{jm}^{(l)}\psi_t(u_{mk}^{(l)})=\sum_m u_{jm}^{(l)}e^{-t\Lambda_l}\varepsilon(u_{mk}^{(l)})=e^{-t\Lambda_l}u_{jk}^{(l)}. S ( u j k ( l ) ) = m ∑ u j m ( l ) ψ t ( u mk ( l ) ) = m ∑ u j m ( l ) e − t Λ l ε ( u mk ( l ) ) = e − t Λ l u j k ( l ) . Thus S = U t S=\mathcal{U}_t S = U t Pol ( S U r ( 2 ) ) \operatorname{Pol}(\mathrm{SU}_r(2)) Pol ( SU r ( 2 )) S ( φ l τ ( A ) ) = e − t Λ l ( τ ) φ l τ ( A ) = ( T t T φ l τ ) ( A ) S(\varphi_l^\tau(A))=e^{-t\Lambda_l(\tau)}\varphi_l^\tau(A)=(T_t^T\varphi_l^\tau)(A) S ( φ l τ ( A )) = e − t Λ l ( τ ) φ l τ ( A ) = ( T t T φ l τ ) ( A ) Proposition 5.3 , so that S ( F ( A ) ) = ( T t T F ) ( A ) S(F(A))=(T_t^T F)(A) S ( F ( A )) = ( T t T F ) ( A ) F ∈ C ( X τ ) F\in\mathrm{C}(X_\tau) F ∈ C ( X τ ) b = u − 1 2 , − 1 2 ( 1 2 ) b=u_{-\frac{1}{2},-\frac{1}{2}}^{(\frac{1}{2})} b = u − 2 1 , − 2 1 ( 2 1 ) S ( b ) = e − t Λ 1 2 b S(b)=e^{-t\Lambda_{\frac{1}{2}}}b S ( b ) = e − t Λ 2 1 b
∥ S ( b ) − b ∥ = 1 − e − t Λ 1 2 ≤ t Λ 1 2 < 1 − 1 − δ C ( τ , τ ) 2 , \lVert S(b)-b\rVert=1-e^{-t\Lambda_{\frac{1}{2}}}\leq t\Lambda_{\frac{1}{2}}<1-\sqrt{1-\delta_{\mathrm{C}}(\tau,\tau)^2}, ∥ S ( b ) − b ∥ = 1 − e − t Λ 2 1 ≤ t Λ 2 1 < 1 − 1 − δ C ( τ , τ ) 2 , since ∥ b ∥ = 1 \lVert b\rVert=1 ∥ b ∥ = 1 Proposition 7.6 , since both maps act on u j 0 ( l ) u_{j0}^{(l)} u j 0 ( l ) e − t Λ l ( τ ) e^{-t\Lambda_l(\tau)} e − t Λ l ( τ )
:::
The crystal limit
In this section we return to the local field, so that τ = Q − d \tau=\mathbb{Q}^{-d} τ = Q − d α > 0 \alpha>0 α > 0 0 < η < 1 0<\eta<1 0 < η < 1 L η L_\eta L η ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) n ≥ 0 n\geq0 n ≥ 0 k ≥ 0 k\geq0 k ≥ 0
C n η ( k ) ≔ ∏ i = 0 n − 1 1 − η k − i 1 − η n − i , C_n^\eta(k)\coloneqq\prod_{i=0}^{n-1}\frac{1-\eta^{k-i}}{1-\eta^{n-i}}, C n η ( k ) : = i = 0 ∏ n − 1 1 − η n − i 1 − η k − i , with C 0 η = 1 C_0^\eta=1 C 0 η = 1 U n , x q = ( q x ; q − 1 ) n U_{n,x}^q=(q^x;q^{-1})_n U n , x q = ( q x ; q − 1 ) n [48 (2.21)] , we have C n η ( k ) = U n , k η U n , n η C_n^\eta(k)=\frac{U_{n,k}^\eta}{U_{n,n}^\eta} C n η ( k ) = U n , n η U n , k η q q q [48 Section 2] .
Lemma 9.1 1. For any n ≥ 0 n \ge0 n ≥ 0 C n η C_n^\eta C n η n n n η k \eta^k η k C n η ( k ) = 0 C_n^\eta(k)=0 C n η ( k ) = 0 k < n k<n k < n C n η ( n ) = 1 C_n^\eta(n)=1 C n η ( n ) = 1 η → 0 + \eta\to0^+ η → 0 + C n η C_n^\eta C n η ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) 1 { k ≥ n } = 1 B n \mathbf{1}_{\{k\ge n\}}=\mathbf{1}_{B_n} 1 { k ≥ n } = 1 B n
Let ( e ~ n η ) n ≥ 0 (\widetilde e_n^\eta)_{n\ge0} ( e n η ) n ≥ 0 ( C n η ) n ≥ 0 (C_n^\eta)_{n\ge0} ( C n η ) n ≥ 0 e ~ n η = ± e n η \widetilde e_n^\eta=\pm e_n^\eta e n η = ± e n η L η L_\eta L η Λ n ( η ) \Lambda_n(\eta) Λ n ( η ) η → 0 + \eta\to0^+ η → 0 + e ~ 0 η = 1 \widetilde e_0^\eta=1 e 0 η = 1 e ~ n η → g n − 1 \widetilde e_n^\eta\to g_{n-1} e n η → g n − 1 ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) n ≥ 1 n\ge1 n ≥ 1
Proof. 1. Writing 1 − η k − i = 1 − η − i x 1-\eta^{k-i}=1-\eta^{-i}x 1 − η k − i = 1 − η − i x x = η k x=\eta^k x = η k C n η C_n^\eta C n η n n n x x x k < n k<n k < n i = k i=k i = k k = n k=n k = n k ≥ n k\ge n k ≥ n ( 0 , ( 1 − η ) − 1 ] (0,(1-\eta)^{-1}] ( 0 , ( 1 − η ) − 1 ] η → 0 + \eta\to0^+ η → 0 + 0 ≤ C n η ≤ 2 n 0\le C_n^\eta\le2^n 0 ≤ C n η ≤ 2 n η ≤ 1 2 \eta\le\frac12 η ≤ 2 1 C n η → 1 { k ≥ n } C_n^\eta\to\mathbf{1}_{\{k\ge n\}} C n η → 1 { k ≥ n } w w w ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w )
By part 1, C 0 η , … , C n η C_0^\eta,\ldots,C_n^\eta C 0 η , … , C n η P n P_n P n e ~ n η \widetilde e_n^\eta e n η P n P_n P n P n − 1 P_{n-1} P n − 1 e ~ n η = ± e n η \widetilde e_n^\eta=\pm e_n^\eta e n η = ± e n η 1 B 0 , 1 B 1 , … \mathbf{1}_{B_0},\mathbf{1}_{B_1},\ldots 1 B 0 , 1 B 1 , … e ~ n η \widetilde e_n^\eta e n η n n n ( 1 B m ) m ≥ 0 (\mathbf{1}_{B_m})_{m\ge0} ( 1 B m ) m ≥ 0 n ≥ 1 n\ge1 n ≥ 1 1 B n − τ 1 B n − 1 \mathbf{1}_{B_n}-\tau\mathbf{1}_{B_{n-1}} 1 B n − τ 1 B n − 1 span { 1 B 0 , … , 1 B n } \operatorname{span}\{\mathbf{1}_{B_0},\ldots,\mathbf{1}_{B_n}\} span { 1 B 0 , … , 1 B n } 1 1 1 1 B n \mathbf{1}_{B_n} 1 B n 1 B m \mathbf{1}_{B_m} 1 B m m < n m<n m < n ⟨ 1 B n − τ 1 B n − 1 , 1 B m ⟩ = τ n − τ τ n − 1 = 0 \langle\mathbf{1}_{B_n}-\tau\mathbf{1}_{B_{n-1}},\mathbf{1}_{B_m}\rangle=\tau^n-\tau\tau^{n-1}=0 ⟨ 1 B n − τ 1 B n − 1 , 1 B m ⟩ = τ n − τ τ n − 1 = 0 g n − 1 g_{n-1} g n − 1 (13) . This proves part 2.
Theorem 9.2 For 0 < η ≤ Q − α 0<\eta\le Q^{-\alpha} 0 < η ≤ Q − α s ( η ) = α log Q log 1 η ∈ ( 0 , 1 ] s(\eta)=\frac{\alpha\log Q}{\log\frac{1}{\eta}}\in(0,1] s ( η ) = l o g η 1 α l o g Q ∈ ( 0 , 1 ]
The semigroup ( e − t L η s ( η ) ) t ≥ 0 (e^{-tL_\eta^{s(\eta)}})_{t\ge0} ( e − t L η s ( η ) ) t ≥ 0 ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) C ( Y η ) C(Y_\eta) C ( Y η )
We have
lim η → 0 + ∥ ( L η s ( η ) + I ) − 1 − ( M r a d α + I ) − 1 ∥ = 0 , lim η → 0 + ∥ e − t L η s ( η ) − e − t M r a d α ∥ = 0 ( t > 0 ) , \lim_{\eta\to0^+}\left\|(L_\eta^{s(\eta)}+I)^{-1}-(M_{\mathrm{rad}}^\alpha+I)^{-1}\right\|=0,\qquad
\lim_{\eta\to0^+}\left\|e^{-tL_\eta^{s(\eta)}}-e^{-tM_{\mathrm{rad}}^\alpha}\right\|=0\quad(t>0), η → 0 + lim ( L η s ( η ) + I ) − 1 − ( M rad α + I ) − 1 = 0 , η → 0 + lim e − t L η s ( η ) − e − t M rad α = 0 ( t > 0 ) , where M r a d α = R r a d α + κ ( I − E 0 ) M_{\mathrm{rad}}^\alpha=R_{\mathrm{rad}}^\alpha+\kappa(I-E_0) M rad α = R rad α + κ ( I − E 0 ) M α M^\alpha M α Proposition 2.1 .
Proof. 1. This follows from Proposition 3.3 and subordination with respect to the Bernstein function λ ↦ λ s ( η ) \lambda\mapsto\lambda^{s(\eta)} λ ↦ λ s ( η )
By Theorem 2.3 and Proposition 2.1 , M r a d α M_{\mathrm{rad}}^\alpha M rad α e 0 0 = d e f 1 e_0^0\mathrel{\overset{\mathrm{def}}{=}}1 e 0 0 = def 1 e n 0 = d e f g n − 1 e_n^0\mathrel{\overset{\mathrm{def}}{=}}g_{n-1} e n 0 = def g n − 1 ( n ≥ 1 ) (n\ge1) ( n ≥ 1 ) ν 0 = 0 \nu_0=0 ν 0 = 0 ν n = λ n − 1 + κ = ρ n \nu_n=\lambda_{n-1}+\kappa=\rho^n ν n = λ n − 1 + κ = ρ n Lemma 9.1 , L η s ( η ) L_\eta^{s(\eta)} L η s ( η ) ( e ~ n η ) (\widetilde e_n^\eta) ( e n η ) ν n ( η ) = d e f Λ n ( η ) s ( η ) \nu_n(\eta)\mathrel{\overset{\mathrm{def}}{=}}\Lambda_n(\eta)^{s(\eta)} ν n ( η ) = def Λ n ( η ) s ( η ) s ( η ) log 1 η = α log Q s(\eta)\log\frac{1}{\eta}=\alpha\log Q s ( η ) log η 1 = α log Q η − n s ( η ) = ρ n \eta^{-ns(\eta)}=\rho^n η − n s ( η ) = ρ n
ν n ( η ) = ρ n ( 1 − η n ) s ( η ) ( 1 − τ η n ) s ( η ) . \nu_n(\eta)=\rho^n(1-\eta^n)^{s(\eta)}(1-\tau\eta^n)^{s(\eta)}. ν n ( η ) = ρ n ( 1 − η n ) s ( η ) ( 1 − τ η n ) s ( η ) . Hence ν n ( η ) → ρ n \nu_n(\eta)\to\rho^n ν n ( η ) → ρ n η → 0 + \eta\to0^+ η → 0 + n n n 0 < s ( η ) ≤ 1 0<s(\eta)\le1 0 < s ( η ) ≤ 1 ( 0 , 1 ] (0,1] ( 0 , 1 ] ν n ( η ) ≥ c ∗ ρ n \nu_n(\eta)\ge c_*\rho^n ν n ( η ) ≥ c ∗ ρ n n ≥ 1 n\ge1 n ≥ 1 0 < η ≤ Q − α 0<\eta\le Q^{-\alpha} 0 < η ≤ Q − α
c ∗ = d e f ( 1 − Q − α ) ( 1 − τ ) . c_*\mathrel{\overset{\mathrm{def}}{=}}(1-Q^{-\alpha})(1-\tau). c ∗ = def ( 1 − Q − α ) ( 1 − τ ) . Let Π n η \Pi_n^\eta Π n η Π n 0 \Pi_n^0 Π n 0 C e ~ n η \mathbb{C}\widetilde e_n^\eta C e n η C e n 0 \mathbb{C}e_n^0 C e n 0 Lemma 9.1 , Π n η → Π n 0 \Pi_n^\eta\to\Pi_n^0 Π n η → Π n 0 n n n f f f λ ↦ ( 1 + λ ) − 1 \lambda\mapsto(1+\lambda)^{-1} λ ↦ ( 1 + λ ) − 1 λ ↦ e − t λ \lambda\mapsto e^{-t\lambda} λ ↦ e − t λ t > 0 t>0 t > 0 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) N ≥ 1 N\ge1 N ≥ 1
∥ f ( L η s ( η ) ) − f ( M r a d α ) ∥ ≤ ∑ n ≤ N ∥ f ( ν n ( η ) ) Π n η − f ( ν n ) Π n 0 ∥ + f ( c ∗ ρ N + 1 ) + f ( ρ N + 1 ) , (103) \left\|f(L_\eta^{s(\eta)})-f(M_{\mathrm{rad}}^\alpha)\right\|
\le\sum_{n\le N}\left\|f(\nu_n(\eta))\Pi_n^\eta-f(\nu_n)\Pi_n^0\right\|+f(c_*\rho^{N+1})+f(\rho^{N+1}), \tag*{(103)} f ( L η s ( η ) ) − f ( M rad α ) ≤ n ≤ N ∑ f ( ν n ( η )) Π n η − f ( ν n ) Π n 0 + f ( c ∗ ρ N + 1 ) + f ( ρ N + 1 ) , (103) since the parts of the two operators on the orthogonal complements of the first N + 1 N+1 N + 1 sup n > N f ( ν n ( η ) ) ≤ f ( c ∗ ρ N + 1 N + 1 ) \sup_{n>N} f(\nu_n(\eta)) \leq f(c_* \rho_{N+1}^{N+1}) sup n > N f ( ν n ( η )) ≤ f ( c ∗ ρ N + 1 N + 1 ) f ( ρ N + 1 ) f(\rho_{N+1}) f ( ρ N + 1 ) ε > 0 \varepsilon>0 ε > 0 N N N ε \varepsilon ε η \eta η ε \varepsilon ε
Remark 9.3 The family ( L η ) 0 < η < 1 (L_\eta)_{0<\eta<1} ( L η ) 0 < η < 1 L τ L_\tau L τ Theorem 7.7 , and, for η = Q − α \eta=Q^{-\alpha} η = Q − α R r a d α R_{\mathrm{rad}}^\alpha R rad α Theorem 4.1 ; for η = Q − α \eta=Q^{-\alpha} η = Q − α s ( η ) = 1 s(\eta)=1 s ( η ) = 1 s ( η ) s(\eta) s ( η ) Λ n ( η ) → ∞ \Lambda_n(\eta)\to\infty Λ n ( η ) → ∞ n ≥ 1 n\geq1 n ≥ 1 η → 0 + \eta\to0^+ η → 0 + L η L_\eta L η p a b ; q ( x ) = p n ( q − 1 a , q − 1 b ; q ) p_a^{b;q}(x)=p_n(q^{-1}a,q^{-1}b;q) p a b ; q ( x ) = p n ( q − 1 a , q − 1 b ; q ) [48 (2.2)] , the eigenfunctions of L η L_\eta L η φ n η ( η k ) = p n τ , η ; η ( k ) \varphi_n^\eta(\eta^k)=p_n^{\tau,\eta;\eta}(k) φ n η ( η k ) = p n τ , η ; η ( k ) Lemma 9.1 is a variant of the limit theorem of Haran [38 ] and Koornwinder–Onn [48 Theorem 2.1] , in which the parameter b = η b=\eta b = η q = η q=\eta q = η p n τ , 0 ; 0 p_n^{\tau,0;0} p n τ , 0 ; 0 G L d ( O ) \mathrm{GL}_d(\mathcal{O}) GL d ( O ) O d \mathcal{O}^d O d [48 Section 5.2] , in accordance with the remark following Theorem 2.3 . See also the ladder structures of [39 Chapter 4] .
Quantum projective spaces and the orders d / ( n − 1 ) d/(n-1) d / ( n − 1 )
In this section, n ≥ 2 n\geq2 n ≥ 2 0 < q < 1 0<q<1 0 < q < 1 L η L_\eta L η
η = q 2 and τ = τ n : = q 2 ( n − 1 ) , (104) \eta=q^2 \quad\text{and}\quad\tau=\tau_n \mathrel{\mathop:}=q^{2(n-1)}, \tag*{(104)} η = q 2 and τ = τ n : = q 2 ( n − 1 ) , (104) is the zonal part of the Casimir operator of the standard quantum complex projective space C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 n = 2 n=2 n = 2 § 5 , with q = r q=r q = r η = τ \eta=\tau η = τ Theorem 4.1 , this shows that the Taibleson operator of order α = d n − 1 \alpha=\frac{d}{n-1} α = n − 1 d O d \mathcal{O}^d O d C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 q 2 = Q − d / ( n − 1 ) q^2=Q^{-d/(n-1)} q 2 = Q − d / ( n − 1 ) C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 [57 ] . Throughout this section, the parameter τ \tau τ τ n \tau_n τ n w k = ( 1 − τ n ) τ n k w_k=(1-\tau_n)\tau_n^k w k = ( 1 − τ n ) τ n k φ l η ( x ) = p l ( x ; τ n η − 1 , 1 ; η ) = p l ( x ; q 2 ( n − 2 ) , 1 ; q 2 ) \varphi_l^\eta(x)=p_l(x;\tau_n\eta^{-1},1;\eta)=p_l(x;q^{2(n-2)},1;q^2) φ l η ( x ) = p l ( x ; τ n η − 1 , 1 ; η ) = p l ( x ; q 2 ( n − 2 ) , 1 ; q 2 ) Y η = { q 2 k : k ≥ 0 } ∪ { 0 } Y_\eta=\{q^{2k}: k\geq0\}\cup\{0\} Y η = { q 2 k : k ≥ 0 } ∪ { 0 }
Zonal spherical functions
Let U q ( n ) U_q(n) U q ( n ) ∗ * ∗ Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) [57 ] . We denote by h n h_n h n ϵ \epsilon ϵ Δ \Delta Δ C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) C ∗ C^* C ∗ Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) U q ( n ) U_q(n) U q ( n ) U ( n ) U(n) U ( n ) [57 ] , the argument of [54 Theorem 2.7.14] shows that U q ( n ) U_q(n) U q ( n ) [7 ] . Hence the universal and the reduced C ∗ C^* C ∗ U q ( n ) U_q(n) U q ( n ) [54 Theorem 2.7.10] , and h n h_n h n C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) C ∗ C^* C ∗ [8 ] . Let K n K_n K n U q ( n − 1 ) × T U_q(n-1)\times\mathbb{T} U q ( n − 1 ) × T σ = ∞ \sigma=\infty σ = ∞ [25 ] , and let π K : C ( U q ( n ) ) → C ( K n ) \pi_K:C(U_q(n))\to C(K_n) π K : C ( U q ( n )) → C ( K n ) h K h_K h K K n K_n K n
C ( C P q n − 1 ) : = { x ∈ C ( U q ( n ) ) : ( Id ⊗ π K ) Δ ( x ) = x ⊗ 1 } , C(\mathbb{C}P_q^{n-1}) \mathrel{\mathop:}=\{x\in C(U_q(n)):(\operatorname{Id}\otimes\pi_K)\Delta(x)=x\otimes1\}, C ( C P q n − 1 ) : = { x ∈ C ( U q ( n )) : ( Id ⊗ π K ) Δ ( x ) = x ⊗ 1 } , and we put Pol ( C P q n − 1 ) ≔ Pol ( U q ( n ) ) ∩ C ( C P q n − 1 ) \operatorname{Pol}(\mathbb{C P}_{q}^{n-1}) \coloneqq\operatorname{Pol}(U_q(n)) \cap C(\mathbb{C P}_{q}^{n-1}) Pol ( CP q n − 1 ) : = Pol ( U q ( n )) ∩ C ( CP q n − 1 )
E K ≔ ( h K π K ⊗ Id ⊗ h K π K ) ∘ ( Δ ⊗ Id ) ∘ Δ (105) E_K \coloneqq(h_K\pi_K \otimes\operatorname{Id} \otimes h_K\pi_K) \circ(\Delta\otimes\operatorname{Id}) \circ\Delta \tag*{(105)} E K : = ( h K π K ⊗ Id ⊗ h K π K ) ∘ ( Δ ⊗ Id ) ∘ Δ (105) is a unital completely positive idempotent onto the K n K_n K n C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) h n h_n h n
The irreducible unitary corepresentations of U q ( n ) U_q(n) U q ( n ) U ( n ) U(n) U ( n ) K n K_n K n π l \pi_l π l l ( ε 1 − ε n ) l(\varepsilon_1-\varepsilon_n) l ( ε 1 − ε n ) l ∈ N 0 l \in\mathbb{N}_0 l ∈ N 0 K n K_n K n π l \pi_l π l [57 ] , see also [25 Section 5] . For any l ∈ N 0 l \in\mathbb{N}_0 l ∈ N 0 ( e i ) (e_i) ( e i ) π l \pi_l π l K n K_n K n e 0 e_0 e 0 u i j ( l ) u_{ij}^{(l)} u ij ( l ) π l \pi_l π l ∥ u i j ( l ) ∥ ≤ 1 \left\|u_{ij}^{(l)}\right\| \leq1 u ij ( l ) ≤ 1 π \pi π u i j π u_{ij}^{\pi} u ij π ( h K π K ( u i j π ) ) i , j (h_K\pi_K(u_{ij}^{\pi}))_{i,j} ( h K π K ( u ij π ) ) i , j K n K_n K n π \pi π
E K ( u i j ( l ) ) = δ i 0 δ j 0 u 00 ( l ) , E K ( u i j π ) = 0 if π is not equivalent to some π l . (106) E_K(u_{ij}^{(l)})=\delta_{i0}\delta_{j0}u_{00}^{(l)}, \qquad E_K(u_{ij}^{\pi})=0\text{ if }\pi\text{ is not equivalent to some }\pi_l. \tag*{(106)} E K ( u ij ( l ) ) = δ i 0 δ j 0 u 00 ( l ) , E K ( u ij π ) = 0 if π is not equivalent to some π l . (106) The zonal spherical function of π l \pi_l π l φ l ( n ) ≔ u 00 ( l ) \varphi_l^{(n)} \coloneqq u_{00}^{(l)} φ l ( n ) : = u 00 ( l ) K n K_n K n ε ( φ l ( n ) ) = 1 \varepsilon(\varphi_l^{(n)})=1 ε ( φ l ( n ) ) = 1 [54 Theorem 1.5.3] and (106) , the K n K_n K n Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) span { φ l ( n ) : l ∈ N 0 } \operatorname{span}\{\varphi_l^{(n)}:l\in\mathbb{N}_0\} span { φ l ( n ) : l ∈ N 0 } Pol ( C P q n − 1 ) = span { u i 0 ( l ) : l ∈ N 0 , i } \operatorname{Pol}(\mathbb{C P}_{q}^{n-1})=\operatorname{span}\{u_{i0}^{(l)}:l\in\mathbb{N}_0,\ i\} Pol ( CP q n − 1 ) = span { u i 0 ( l ) : l ∈ N 0 , i }
Let ( x i j ) 1 ≤ i , j ≤ n (x_{ij})_{1\leq i,j\leq n} ( x ij ) 1 ≤ i , j ≤ n U q ( n ) U_q(n) U q ( n ) [57 ] , and put
x n ≔ ∑ i = 1 n − 1 x n i x n i ∗ = 1 − x n n x n n ∗ . x_n \coloneqq\sum_{i=1}^{n-1}x_{ni}x_{ni}^{*}=1-x_{nn}x_{nn}^{*}. x n : = i = 1 ∑ n − 1 x ni x ni ∗ = 1 − x nn x nn ∗ . This is the element ζ n − 1 \zeta_{n-1} ζ n − 1 [57 (4.26)] , written with the elements z i = x n i z_i=x_{ni} z i = x ni w i = z i ∗ w_i=z_i^* w i = z i ∗ [57 (4.6)] , and the second equality is [57 (4.9.d)] . The element x n x_n x n ε ( x n ) = 0 \varepsilon(x_n)=0 ε ( x n ) = 0 ε ( x n i ) = δ n i \varepsilon(x_{ni})=\delta_{ni} ε ( x ni ) = δ ni [57 Proposition 4.2] , the irreducible corepresentation of highest weight ℓ ε 1 − m ε n \ell\varepsilon_1-m\varepsilon_n ℓ ε 1 − m ε n ℓ , m ∈ N 0 \ell,m\in\mathbb{N}_0 ℓ , m ∈ N 0 U q ( n − 1 ) U_q(n-1) U q ( n − 1 ) ( ℓ − m ) ε n (\ell-m)\varepsilon_n ( ℓ − m ) ε n T \mathbb{T} T K n K_n K n ℓ = m \ell=m ℓ = m [57 Theorem 4.7] , the corresponding matrix coefficient φ ℓ , m \varphi^{\ell,m} φ ℓ , m ε ( φ ℓ , m ) = 1 \varepsilon(\varphi^{\ell,m})=1 ε ( φ ℓ , m ) = 1 [57 (4.34.a)] . For ℓ = m = l \ell=m=l ℓ = m = l [57 (4.33)] of the little q q q
φ l ( n ) = P l ( n − 2 , 0 ) ( x n ; q 2 ) = p l ( x n ; q 2 ( n − 2 ) , 1 ; q 2 ) = φ l η ( x n ) , l ∈ N 0 , (107) \varphi_l^{(n)}=P_l^{(n-2,0)}(x_n;q^2)=p_l(x_n;q^{2(n-2)},1;q^2)=\varphi_l^\eta(x_n),\qquad l\in\mathbb{N}_0, \tag*{(107)} φ l ( n ) = P l ( n − 2 , 0 ) ( x n ; q 2 ) = p l ( x n ; q 2 ( n − 2 ) , 1 ; q 2 ) = φ l η ( x n ) , l ∈ N 0 , (107) where p l ( ⋅ ; a , b ; q 2 ) p_l(\cdot;a,b;q^2) p l ( ⋅ ; a , b ; q 2 ) q 2 q^2 q 2 [44 Section 14.12] . Indeed, both polynomials are equal to 2 φ 1 ( q − 2 l , q 2 ( l + n − 1 ) ; q 2 ( n − 1 ) ; q 2 , q 2 x ) {}_2\varphi_1(q^{-2l},q^{2(l+n-1)};q^{2(n-1)};q^2,q^{2}x) 2 φ 1 ( q − 2 l , q 2 ( l + n − 1 ) ; q 2 ( n − 1 ) ; q 2 , q 2 x ) [57 ] uses left invariant elements of the quotient by U q ( n − 1 ) U_q(n-1) U q ( n − 1 ) φ l ( n ) \varphi_l^{(n)} φ l ( n ) K n K_n K n (107) does not depend on this convention. See also [29 Theorem 3.3.1] , where x n x_n x n Q n − 1 Q_{n-1} Q n − 1 q q q [25 Section 5] . By Lemma 4.4, we have x n = 1 − τ n 1 − τ n ( 1 − φ 1 ( n ) ) x_n=\frac{1-\tau_n}{1-\tau_n}\left(1-\varphi_1^{(n)}\right) x n = 1 − τ n 1 − τ n ( 1 − φ 1 ( n ) ) x n x_n x n K n K_n K n n = 2 n=2 n = 2 (107) reduces to Proposition 5.3, with x 2 = A x_2=A x 2 = A q = r q=r q = r
Lemma 10.1 The spectrum of x n x_n x n Y n Y_n Y n x n x_n x n
h n ( F ( x n ) ) = ( 1 − τ n ) ∑ k ≥ 0 τ n k F ( q 2 k ) = ∑ k ≥ 0 w k F ( η k ) , F ∈ C ( Y n ) . h_n(F(x_n))=(1-\tau_n)\sum_{k\geq0}\tau_n^kF(q^{2k})=\sum_{k\geq0}w_kF(\eta^k),\qquad F\in C(Y_n). h n ( F ( x n )) = ( 1 − τ n ) k ≥ 0 ∑ τ n k F ( q 2 k ) = k ≥ 0 ∑ w k F ( η k ) , F ∈ C ( Y n ) . Moreover, we have sup l ≥ 0 sup Y η ∣ φ l η ∣ ≤ 1 \sup_{l\geq0}\sup_{Y_\eta}|\varphi_l^\eta|\leq1 sup l ≥ 0 sup Y η ∣ φ l η ∣ ≤ 1
Proof: Let μ \mu μ x n x_n x n h n h_n h n x n x_n x n [54 Theorem 1.4.3] , h n ( u 00 ( l ) ) = 0 h_n(u_{00}^{(l)})=0 h n ( u 00 ( l ) ) = 0 l ≥ 1 l\geq1 l ≥ 1 π l \pi_l π l (107) , we deduce that ∫ φ l η d μ = 0 \int\varphi_l^\eta\,\mathrm{d}\mu=0 ∫ φ l η d μ = 0 l ≥ 1 l\geq1 l ≥ 1 ν : = ∑ k ≥ 0 w k η δ η k \nu\mathrel{\mathop:}=\sum_{k\geq0}w_k^\eta\delta_{\eta_k} ν : = ∑ k ≥ 0 w k η δ η k φ l η \varphi_l^\eta φ l η ℓ N 0 2 ( w ) \ell^2_{N_0}(w) ℓ N 0 2 ( w ) φ 0 η = 1 \varphi_0^\eta=1 φ 0 η = 1 ( φ l η ) l ≥ 0 (\varphi_l^\eta)_{l\geq0} ( φ l η ) l ≥ 0 μ \mu μ ν \nu ν R \mathbb{R} R μ = ν \mu=\nu μ = ν h n h_n h n C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) x n x_n x n μ \mu μ Y η Y_\eta Y η ∥ φ l η ( x n ) ∥ = sup Y η ∣ φ l η ∣ \lVert\varphi_l^\eta(x_n)\rVert=\sup_{Y_\eta}|\varphi_l^\eta| ∥ φ l η ( x n )∥ = sup Y η ∣ φ l η ∣ ∥ φ l η ∥ = ∥ u 00 ( l ) ∥ ≤ 1 \lVert\varphi_l^\eta\rVert=\lVert u_{00}^{(l)}\rVert\leq1 ∥ φ l η ∥ = ∥ u 00 ( l ) ∥ ≤ 1 (107) .
Remark 10.2 Lemma 10.1 agrees with the computations of [57 ] . By [57 Theorem 4.6] , the Haar state on the polynomials in ζ 1 , … , ζ n − 1 \zeta_1,\ldots,\zeta_{n-1} ζ 1 , … , ζ n − 1 [57 (4.27)] , see [27 (17.2.46)] for the Jackson integral. For a polynomial F F F ζ n − 1 \zeta_{n-1} ζ n − 1 n − 2 n-2 n − 2 ( 1 − q 2 ) n − 2 ( q 2 ; q 2 ) n − 2 − 1 ζ n − 1 n − 2 (1-q^2)^{n-2}(q^2;q^2)_{n-2}^{-1}\zeta_{n-1}^{n-2} ( 1 − q 2 ) n − 2 ( q 2 ; q 2 ) n − 2 − 1 ζ n − 1 n − 2 [57 p. 78] , so that
h n ( F ( x n ) ) = 1 − q 2 ( n − 1 ) 1 − q 2 ∫ 0 1 F ( ζ ) ζ n − 2 d q 2 ζ = ( 1 − τ n ) ∑ k ≥ 0 τ n k F ( q 2 k ) . h_n(F(x_n))=\frac{1-q^{2(n-1)}}{1-q^2}\int_0^1F(\zeta)\zeta^{n-2}\,\mathrm{d}_{q^2}\zeta=(1-\tau_n)\sum_{k\geq0}\tau_n^kF(q^{2k}). h n ( F ( x n )) = 1 − q 2 1 − q 2 ( n − 1 ) ∫ 0 1 F ( ζ ) ζ n − 2 d q 2 ζ = ( 1 − τ n ) k ≥ 0 ∑ τ n k F ( q 2 k ) . by [57 (4.24)] . Moreover, [57 (4.48)] with ℓ = m = ℓ ′ = m ′ = l \ell=m=\ell'=m'=l ℓ = m = ℓ ′ = m ′ = l h n ( ( φ l ( n ) ) ∗ φ l ( n ) ) = d l − 1 h_n((\varphi_l^{(n)})^*\varphi_l^{(n)})=d_l^{-1} h n (( φ l ( n ) ) ∗ φ l ( n ) ) = d l − 1
d l = q − 2 l ( n − 1 ) 1 − q 2 ( 2 l + n − 1 ) 1 − q 2 ( n − 1 ) ( ( q 2 ( n − 1 ) ; q 2 ) l ( q 2 ; q 2 ) l ) 2 . d_l=q^{-2l(n-1)}\frac{1-q^{2(2l+n-1)}}{1-q^{2(n-1)}}\left(\frac{(q^{2(n-1)};q^2)_l}{(q^2;q^2)_l}\right)^2. d l = q − 2 l ( n − 1 ) 1 − q 2 ( n − 1 ) 1 − q 2 ( 2 l + n − 1 ) ( ( q 2 ; q 2 ) l ( q 2 ( n − 1 ) ; q 2 ) l ) 2 . is the quantum dimension of π l \pi_l π l η = q 2 \eta=q^2 η = q 2 τ = τ n \tau=\tau_n τ = τ n ∥ φ l η ∥ ℓ N 0 2 ( w ) 2 \lVert\varphi_l^\eta\rVert_{\ell^2_{N_0}(w)}^2 ∥ φ l η ∥ ℓ N 0 2 ( w ) 2 (107) is [57 (4.49)] .
The Casimir operator and the energy bridge
Let C C C ∑ i , j q 2 ( n − i ) L i j + S ( L j i + ) \sum_{i,j}q^{2(n-i)}L_{ij}^+S(L_{ji}^+) ∑ i , j q 2 ( n − i ) L ij + S ( L j i + ) U q g l n U_q\mathfrak{gl}_n U q gl n [25 (5.5)] . It acts on the irreducible module of highest weight λ \lambda λ χ λ ( C ) = ∑ k = 1 n q 2 ( λ k + n − k ) \chi_\lambda(C)=\sum_{k=1}^n q^{2(\lambda_k+n-k)} χ λ ( C ) = ∑ k = 1 n q 2 ( λ k + n − k ) [25 Section 5] , and in particular on the module of π l \pi_l π l
χ l : = q 2 ( l + n − 1 ) + q − 2 l + q 2 − q 2 n − 2 1 − q 2 , (108) \chi_l\mathrel{\mathop:}=q^{2(l+n-1)}+q^{-2l}+\frac{q^2-q^{2n-2}}{1-q^2}, \tag*{(108)} χ l : = q 2 ( l + n − 1 ) + q − 2 l + 1 − q 2 q 2 − q 2 n − 2 , (108) see [25 (5.7)] . A direct computation gives, for any l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0
χ l − χ 0 = q − 2 l + q 2 ( l + n − 1 ) − 1 − q 2 ( n − 1 ) = q − 2 l ( 1 − q 2 l ) ( 1 − q 2 ( l + n − 1 ) ) = (1.4) Λ l ( η ) . (109) \chi_l-\chi_0=q^{-2l}+q^{2(l+n-1)}-1-q^{2(n-1)}=q^{-2l}(1-q^{2l})(1-q^{2(l+n-1)})\mathrel{\mathop=^{\text{\scriptsize(1.4)}}}\Lambda_l(\eta). \tag*{(109)} χ l − χ 0 = q − 2 l + q 2 ( l + n − 1 ) − 1 − q 2 ( n − 1 ) = q − 2 l ( 1 − q 2 l ) ( 1 − q 2 ( l + n − 1 ) ) = (1.4) Λ l ( η ) . (109) Through the duality between U q g l n U_q\mathfrak{gl}_n U q gl n Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) Ω ( n ) : x ↦ ( Id ⊗ C ) Δ ( x ) \Omega^{(n)}:x\mapsto(\operatorname{Id}\otimes C)\Delta(x) Ω ( n ) : x ↦ ( Id ⊗ C ) Δ ( x ) π l \pi_l π l χ l \chi_l χ l n = 2 n=2 n = 2 (109) is (47) with r = q r=q r = q
Theorem 10.3 Let n ≥ 2 n\geq2 n ≥ 2 0 < q < 1 0<q<1 0 < q < 1 η = q 2 \eta=q^2 η = q 2 τ n = q 2 ( n − 1 ) \tau_n=q^{2(n-1)} τ n = q 2 ( n − 1 )
For any polynomial F F F ( Ω ( n ) − χ 0 ) ( F ( x n ) ) = ( L η F ) ( x n ) (\Omega^{(n)}-\chi_0)(F(x_n))=(L_\eta F)(x_n) ( Ω ( n ) − χ 0 ) ( F ( x n )) = ( L η F ) ( x n ) L η L_\eta L η (3) with τ = τ n \tau=\tau_n τ = τ n η \eta η L η L_\eta L η C P q n − 1 \mathbb{C}\mathbb{P}^{n-1}_q C P q n − 1 (109) .
Let ( T t η ) (T_t^\eta) ( T t η ) e − t L η e^{-tL_\eta} e − t L η Proposition 3.3 , and μ t η \mu_t^\eta μ t η
S t ( n ) ≔ ( Id ⊗ μ t η ∘ E K ) ∘ Δ {\mathcal S}_t^{(n)}\coloneqq(\operatorname{Id}\otimes\mu_t^\eta\circ E_K)\circ\Delta S t ( n ) : = ( Id ⊗ μ t η ∘ E K ) ∘ Δ restrict to a Haar-preserving strongly continuous semigroup of unital completely positive maps on C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 )
S t ( n ) ( u i 0 ( l ) ) = e − t Λ l ( η ) u i 0 ( l ) , {\mathcal S}_t^{(n)}(u_{i0}^{(l)})=e^{-t\Lambda_l(\eta)}u_{i0}^{(l)}, S t ( n ) ( u i 0 ( l ) ) = e − t Λ l ( η ) u i 0 ( l ) , and S t ( n ) ( F ( x n ) ) = ( T t η F ) ( x n ) {\mathcal S}_t^{(n)}(F(x_n))=(T_t^\eta F)(x_n) S t ( n ) ( F ( x n )) = ( T t η F ) ( x n ) F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η )
Suppose that τ n = Q − d \tau_n=Q^{-d} τ n = Q − d q 2 = Q − α q^2=Q^{-\alpha} q 2 = Q − α α = d n − 1 \alpha=\frac{d}{n-1} α = n − 1 d
J n : L ∞ ( O d ) r a d → W ∗ ( x n ) , J n ( f ( − log Q ∥ ⋅ ∥ K ) ) = ∑ k ≥ 0 f ( k ) 1 { η k } ( x n ) J_n:L^\infty(\mathcal{O}^d)_{\mathrm{rad}}\to W^*(x_n),\qquad
J_n(f(-\log_Q\lVert\cdot\rVert_K))=\sum_{k\geq0}f(k)1_{\{\eta^k\}}(x_n) J n : L ∞ ( O d ) rad → W ∗ ( x n ) , J n ( f ( − log Q ∥ ⋅ ∥ K )) = k ≥ 0 ∑ f ( k ) 1 { η k } ( x n ) is a normal unital state-preserving ∗ * ∗ h n h_n h n E η \mathcal{E}_\eta E η L η L_\eta L η ( S t ( n ) ) ({\mathcal S}_t^{(n)}) ( S t ( n ) ) E R \mathcal{E}_R E R R r a d α R^\alpha_{\mathrm{rad}} R rad α c 1 c_1 c 1 c 2 c_2 c 2 Theorem 4.1 . In particular, for n = d + 1 n=d+1 n = d + 1 O d \mathcal{O}^d O d C P q d \mathbb{C}\mathbb{P}^d_q C P q d q = Q − 1 2 q=Q^{-\frac12} q = Q − 2 1
Proof: 1. By (107) , we have Ω ( n ) ( φ l η ( x n ) ) = Ω ( n ) ( u 00 ( l ) ) = χ l φ l η ( x n ) \Omega^{(n)}(\varphi_l^\eta(x_n))=\Omega^{(n)}(u_{00}^{(l)})=\chi_l\varphi_l^\eta(x_n) Ω ( n ) ( φ l η ( x n )) = Ω ( n ) ( u 00 ( l ) ) = χ l φ l η ( x n ) L η φ l η = Λ l ( η ) φ l η L_\eta\varphi_l^\eta=\Lambda_l(\eta)\varphi_l^\eta L η φ l η = Λ l ( η ) φ l η Λ l ( η ) = χ l − χ 0 \Lambda_l(\eta)=\chi_l-\chi_0 Λ l ( η ) = χ l − χ 0 (109) . Since ( φ l η ) l ≥ 0 (\varphi_l^\eta)_{l\geq0} ( φ l η ) l ≥ 0
By Lemma 10.1 , we have C ∗ ( x n ) = C ( Y η ) C^*(x_n)=C(Y_\eta) C ∗ ( x n ) = C ( Y η ) μ t η \mu_t^\eta μ t η Y η Y_\eta Y η C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) E K E_K E K Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) x n x_n x n (106) and (107) , hence C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) ω t ≔ μ t η ∘ E K \omega_t\coloneqq\mu_t^\eta\circ E_K ω t : = μ t η ∘ E K C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) S t ( n ) = ( Id ⊗ ω t ) ∘ Δ {\mathcal S}_t^{(n)}=(\operatorname{Id}\otimes\omega_t)\circ\Delta S t ( n ) = ( Id ⊗ ω t ) ∘ Δ h n ∘ S t ( n ) = ( h n ⊗ ω t ) Δ = h n h_n\circ{\mathcal S}_t^{(n)}=(h_n\otimes\omega_t)\Delta=h_n h n ∘ S t ( n ) = ( h n ⊗ ω t ) Δ = h n (106) , (107) and (29) ,
ω t ( u i j ( l ) ) = δ i 0 δ j 0 ∫ φ l η d μ t η = δ i 0 δ j 0 e − t Λ l ( η ) , \omega_t(u_{ij}^{(l)})=\delta_{i0}\delta_{j0}\int\varphi_l^\eta\,d\mu_t^\eta=\delta_{i0}\delta_{j0}e^{-t\Lambda_l(\eta)}, ω t ( u ij ( l ) ) = δ i 0 δ j 0 ∫ φ l η d μ t η = δ i 0 δ j 0 e − t Λ l ( η ) , and ω t ( u i j π ) = 0 \omega_t(u_{ij}^{\pi})=0 ω t ( u ij π ) = 0 π \pi π π l \pi_l π l S t ( n ) ( u i 0 ( l ) ) = ∑ j u i j ( l ) ω t ( u j 0 ( l ) ) = e − t Λ l ( η ) u i 0 ( l ) {\mathcal S}_t^{(n)}(u_{i0}^{(l)})=\sum_j u_{ij}^{(l)}\omega_t(u_{j0}^{(l)})=e^{-t\Lambda_l(\eta)}u_{i0}^{(l)} S t ( n ) ( u i 0 ( l ) ) = ∑ j u ij ( l ) ω t ( u j 0 ( l ) ) = e − t Λ l ( η ) u i 0 ( l ) h K h_K h K ( π K ⊗ h K π K ) Δ = h K π K ( ⋅ ) 1 (\pi_K\otimes h_K\pi_K)\Delta=h_K\pi_K(\cdot)1 ( π K ⊗ h K π K ) Δ = h K π K ( ⋅ ) 1 ( π K ⊗ E K ) Δ = 1 ⊗ E K (\pi_K\otimes E_K)\Delta=1\otimes E_K ( π K ⊗ E K ) Δ = 1 ⊗ E K ( π K ⊗ ω t ) Δ = ω t ( ⋅ ) 1 (\pi_K\otimes\omega_t)\Delta=\omega_t(\cdot)1 ( π K ⊗ ω t ) Δ = ω t ( ⋅ ) 1
( Id ⊗ π K ) Δ ∘ S t ( n ) = ( Id ⊗ ( π K ⊗ ω t ) Δ ) ∘ Δ = S t ( n ) ( ⋅ ) ⊗ 1. (\operatorname{Id}\otimes\pi_K)\Delta\circ{\mathcal S}_t^{(n)}=(\operatorname{Id}\otimes(\pi_K\otimes\omega_t)\Delta)\circ\Delta={\mathcal S}_t^{(n)}(\cdot)\otimes1. ( Id ⊗ π K ) Δ ∘ S t ( n ) = ( Id ⊗ ( π K ⊗ ω t ) Δ ) ∘ Δ = S t ( n ) ( ⋅ ) ⊗ 1. Hence S t ( n ) {\mathcal S}_t^{(n)} S t ( n ) C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 ) C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 ) ( Id ⊗ h K π K ) ∘ Δ (\operatorname{Id}\otimes h_K\pi_K)\circ\Delta ( Id ⊗ h K π K ) ∘ Δ C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 ) Pol ( U q ( n ) ) \operatorname{Pol}(U_q(n)) Pol ( U q ( n )) Pol ( C P q n − 1 ) \operatorname{Pol}(\mathbb{C}\mathbb{P}^{n-1}_q) Pol ( C P q n − 1 ) Pol ( C P q n − 1 ) \operatorname{Pol}(\mathbb{C}\mathbb{P}^{n-1}_q) Pol ( C P q n − 1 ) C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 ) Pol ( C P q n − 1 ) \operatorname{Pol}(\mathbb{C}\mathbb{P}^{n-1}_q) Pol ( C P q n − 1 ) S t ( n ) {\mathcal S}_t^{(n)} S t ( n ) u i 0 ( l ) u_{i0}^{(l)} u i 0 ( l ) S t ( n ) {\mathcal S}_t^{(n)} S t ( n ) C ( C P q n − 1 ) C(\mathbb{C}\mathbb{P}^{n-1}_q) C ( C P q n − 1 ) S t ( n ) ( φ l η ( x n ) ) = e − t Λ l ( η ) φ l η ( x n ) = ( T t η φ l η ) ( x n ) {\mathcal S}_t^{(n)}(\varphi_l^\eta(x_n))=e^{-t\Lambda_l(\eta)}\varphi_l^\eta(x_n)=(T_t^\eta\varphi_l^\eta)(x_n) S t ( n ) ( φ l η ( x n )) = e − t Λ l ( η ) φ l η ( x n ) = ( T t η φ l η ) ( x n ) Proposition 3.3 . Both sides of the last assertion are continuous in F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η )
By Lemma 10.1 , the spectral measure of x n x_n x n h n h_n h n Y n Y_n Y n η k \eta^k η k w k = ( 1 − τ n ) τ n k w_k=(1-\tau_n)\tau_n^k w k = ( 1 − τ n ) τ n k { 0 } \{0\} { 0 } (7) , the shell S k S_k S k O d \mathcal{O}^d O d ( 1 − Q − d ) Q − d k = w k (1-Q^{-d})Q^{-dk}=w_k ( 1 − Q − d ) Q − d k = w k J n J_n J n
unital state-preserving ∗ \ast ∗ Theorem 5.6 . Under J n J_n J n ( S t ( n ) ) (\mathcal{S}^{(n)}_t) ( S t ( n ) ) L η L_\eta L η ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) η = Q − α \eta=Q^{-\alpha} η = Q − α τ n = Q − d \tau_n=Q^{-d} τ n = Q − d Theorem 4.1 applies and gives the equivalence of the forms, with its constants. For n = d + 1 n=d+1 n = d + 1 α = 1 \alpha=1 α = 1 q 2 = Q − 1 q^2=Q^{-1} q 2 = Q − 1
Remark 10.4 Theorem 10.3 shows that the quantum group S U r ( 2 ) \mathrm{SU}_r(2) SU r ( 2 ) r 2 = Q − d r^2=Q^{-d} r 2 = Q − d n = 2 n=2 n = 2 n = 2 , … , d + 1 n=2,\ldots,d+1 n = 2 , … , d + 1 C P q n − 1 \mathbb{C}P^{n-1}_q C P q n − 1 q 2 = Q − d n − 1 q^2=Q^{-\frac{d}{n-1}} q 2 = Q − n − 1 d d n − 1 \frac{d}{n-1} n − 1 d O d \mathcal{O}^d O d n = d + 1 n=d+1 n = d + 1 q = Q − 1 2 q=Q^{-\frac{1}{2}} q = Q − 2 1 d d d O d \mathcal{O}^d O d α = 1 \alpha=1 α = 1 t = Q − 1 t=Q^{-1} t = Q − 1 [30 Section 3] . In the classical limit, Λ l ( q 2 ) ∼ 2 log 1 q l ( l + n − 1 ) \Lambda_l(q^2)\sim2\log\frac{1}{q}l(l+n-1) Λ l ( q 2 ) ∼ 2 log q 1 l ( l + n − 1 ) C P n − 1 \mathbb{C}P^{n-1} C P n − 1 Theorem 7.10 , together with the corresponding case of Theorem 4.6 , holds for any n ≥ 2 n\ge2 n ≥ 2 Theorem 10.5 below.
Powers of the Casimir operator and covariant semigroups
We now extend Theorem 7.10 to C P q n − 1 \mathbb{C}P^{n-1}_q C P q n − 1 E R ≔ ( Id ⊗ h K π K ) ∘ Δ E_R\coloneqq(\operatorname{Id}\otimes h_K\pi_K)\circ\Delta E R : = ( Id ⊗ h K π K ) ∘ Δ C ( U q ( n ) ) C(\mathrm{U}_q(n)) C ( U q ( n )) C ( C P q n − 1 ) C(\mathbb{C}P^{n-1}_q) C ( C P q n − 1 ) x ∈ C ( C P q n − 1 ) x\in C(\mathbb{C}P^{n-1}_q) x ∈ C ( C P q n − 1 )
( Id ⊗ E R ) Δ ( x ) = ( Id ⊗ Id ⊗ h K π K ) ( Δ ⊗ Id ) Δ ( x ) = Δ ( E R ( x ) ) = Δ ( x ) . (\operatorname{Id}\otimes E_R)\Delta(x)=(\operatorname{Id}\otimes\operatorname{Id}\otimes h_K\pi_K)(\Delta\otimes\operatorname{Id})\Delta(x)=\Delta(E_R(x))=\Delta(x). ( Id ⊗ E R ) Δ ( x ) = ( Id ⊗ Id ⊗ h K π K ) ( Δ ⊗ Id ) Δ ( x ) = Δ ( E R ( x )) = Δ ( x ) . Hence Δ \Delta Δ C ( C P q n − 1 ) C(\mathbb{C}P^{n-1}_q) C ( C P q n − 1 ) C ( U q ( n ) ) ⊗ min C ( C P q n − 1 ) C(\mathrm{U}_q(n))\otimes_{\min}C(\mathbb{C}P^{n-1}_q) C ( U q ( n )) ⊗ m i n C ( C P q n − 1 ) Id ⊗ E R \operatorname{Id}\otimes E_R Id ⊗ E R S : C ( C P q n − 1 ) → C ( C P q n − 1 ) S:C(\mathbb{C}P^{n-1}_q)\to C(\mathbb{C}P^{n-1}_q) S : C ( C P q n − 1 ) → C ( C P q n − 1 ) ( Id ⊗ S ) ∘ Δ (\operatorname{Id}\otimes S)\circ\Delta ( Id ⊗ S ) ∘ Δ C ( C P q n − 1 ) C(\mathbb{C}P^{n-1}_q) C ( C P q n − 1 )
Theorem 10.5 Let n ≥ 2 n\ge2 n ≥ 2 0 < q < 1 0<q<1 0 < q < 1 η = q 2 \eta=q^2 η = q 2 τ n = q 2 ( n − 1 ) \tau_n=q^{2(n-1)} τ n = q 2 ( n − 1 )
Suppose that τ n = Q − d \tau_n=Q^{-d} τ n = Q − d α > 0 \alpha>0 α > 0 R r a d α R_{\mathrm{rad}}^\alpha R rad α L η ( n − 1 ) α d L_\eta^{\frac{(n-1)\alpha}{d}} L η d ( n − 1 ) α α < ( n + 1 ) d n − 1 \alpha<\frac{(n+1)d}{n-1} α < n − 1 ( n + 1 ) d α ≥ ( n + 1 ) d n − 1 \alpha\ge\frac{(n+1)d}{n-1} α ≥ n − 1 ( n + 1 ) d F ( L η ) F(L_\eta) F ( L η ) R r a d α R_{\mathrm{rad}}^\alpha R rad α
Let ( S t ) t ≥ 0 (S_t)_{t\ge0} ( S t ) t ≥ 0 C ( C P q n − 1 ) C(\mathbb{C}P^{n-1}_q) C ( C P q n − 1 )
Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ on C ( C P q n − 1 ) , t ≥ 0. \Delta\circ S_t=(\operatorname{Id}\otimes S_t)\circ\Delta\quad\text{on }C(\mathbb{C}P^{n-1}_q),\quad t\ge0. Δ ∘ S t = ( Id ⊗ S t ) ∘ Δ on C ( C P q n − 1 ) , t ≥ 0. There exists a function ψ : N 0 → [ 0 , ∞ ) \psi:\mathbb{N}_0\to[0,\infty) ψ : N 0 → [ 0 , ∞ ) S t ( u i 0 ( l ) ) = e − t ψ ( l ) u i 0 ( l ) S_t(u^{(l)}_{i0})=e^{-t\psi(l)}u^{(l)}_{i0} S t ( u i 0 ( l ) ) = e − t ψ ( l ) u i 0 ( l ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 i i i t ≥ 0 t\ge0 t ≥ 0 S t S_t S t C ∗ ( x n ) C^\ast(x_n) C ∗ ( x n ) T t T_t T t C ∗ ( x n ) = C ( Y η ) C^\ast(x_n)=C(Y_\eta) C ∗ ( x n ) = C ( Y η ) T t φ l = e − t ψ ( l ) φ l T_t\varphi_l=e^{-t\psi(l)}\varphi_l T t φ l = e − t ψ ( l ) φ l ψ ( l ) ≤ K τ n , η ψ ( 1 ) Λ l ( η ) \psi(l)\le K_{\tau_n,\eta}\psi(1)\Lambda_l(\eta) ψ ( l ) ≤ K τ n , η ψ ( 1 ) Λ l ( η ) l ≥ 1 l\ge1 l ≥ 1
Suppose that τ n = Q − d \tau_n=Q^{-d} τ n = Q − d α > 0 \alpha>0 α > 0 ( S t ) t ≥ 0 (S_t)_{t\ge0} ( S t ) t ≥ 0 f ↦ ∑ l ≥ 1 ψ ( l ) ∣ f , e l ∣ 2 f\mapsto\sum_{l\ge1}\psi(l)\left|f,e_l\right|^2 f ↦ ∑ l ≥ 1 ψ ( l ) ∣ f , e l ∣ 2 ( T t ) (T_t) ( T t ) ℓ N 0 2 ( w ) \ell^2_{\mathbb{N}_0}(w) ℓ N 0 2 ( w ) R r a d α R_{\mathrm{rad}}^\alpha R rad α α < d n − 1 \alpha<\frac{d}{n-1} α < n − 1 d α ≤ d n − 1 \alpha\le\frac{d}{n-1} α ≤ n − 1 d S t = ∫ [ 0 , ∞ ) S u ( n ) d ν t ( s ) ( u ) S_t=\int_{[0,\infty)}\mathcal{S}^{(n)}_u\,\mathrm{d}\nu_t^{(s)}(u) S t = ∫ [ 0 , ∞ ) S u ( n ) d ν t ( s ) ( u ) s = ( n − 1 ) α d s=\frac{(n-1)\alpha}{d} s = d ( n − 1 ) α ν t ( s ) \nu_t^{(s)} ν t ( s ) L η s L_\eta^s L η s
For any s > 1 s>1 s > 1 ( e − t L η s ) t ≥ 0 (e^{-tL_{\eta}^{s}})_{t\geq0} ( e − t L η s ) t ≥ 0 ℓ N 0 2 ( w ) \ell^{2}_{N_{0}}(w) ℓ N 0 2 ( w )
Proof: 1. This is Theorem 4.6 and Proposition 4.7 with ϑ = d n − 1 \vartheta=\frac{d}{n-1} ϑ = n − 1 d η = Q − d n − 1 \eta=Q^{-\frac{d}{n-1}} η = Q − n − 1 d α ϑ = ( n − 1 ) α d \frac{\alpha}{\vartheta}=\frac{(n-1)\alpha}{d} ϑ α = d ( n − 1 ) α d + 2 ϑ = ( n + 1 ) d n − 1 d+2\vartheta=\frac{(n+1)d}{n-1} d + 2 ϑ = n − 1 ( n + 1 ) d
Let ω t : = ε ∘ S t \omega_t\mathrel{\mathop:}=\varepsilon\circ S_t ω t : = ε ∘ S t C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 ) Id ⊗ ε \operatorname{Id}\otimes\varepsilon Id ⊗ ε ( Id ⊗ ε ) ∘ Δ = Id (\operatorname{Id}\otimes\varepsilon)\circ\Delta=\operatorname{Id} ( Id ⊗ ε ) ∘ Δ = Id S t = ( Id ⊗ ω t ) ∘ Δ S_t=(\operatorname{Id}\otimes\omega_t)\circ\Delta S t = ( Id ⊗ ω t ) ∘ Δ C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 ) l ∈ N 0 l\in\mathbb{N}_0 l ∈ N 0 j j j u j 0 ( l ) u_{j0}^{(l)} u j 0 ( l ) Pol ( C P q n − 1 ) \operatorname{Pol}(\mathbb{C}P_q^{n-1}) Pol ( C P q n − 1 ) Δ ( u i 0 ( l ) ) = ∑ j u i j ( l ) ⊗ u j 0 ( l ) \Delta(u_{i0}^{(l)})=\sum_j u_{ij}^{(l)}\otimes u_{j0}^{(l)} Δ ( u i 0 ( l ) ) = ∑ j u ij ( l ) ⊗ u j 0 ( l )
S t ( u i 0 ( l ) ) = ∑ j u i j ( l ) ω t ( u j 0 ( l ) ) . S_t(u_{i0}^{(l)})=\sum_j u_{ij}^{(l)}\omega_t(u_{j0}^{(l)}). S t ( u i 0 ( l ) ) = j ∑ u ij ( l ) ω t ( u j 0 ( l ) ) . Since S t ( u i 0 ( l ) ) S_t(u_{i0}^{(l)}) S t ( u i 0 ( l ) ) C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 )
∑ k u i k ( l ) ω t ( u k 0 ( l ) ) ⊗ 1 = ( Id ⊗ π K ) Δ ( S t ( u i 0 ( l ) ) ) = ∑ k u i k ( l ) ⊗ ∑ j π K ( u k j ( l ) ) ω t ( u j 0 ( l ) ) . \sum_k u_{ik}^{(l)}\omega_t(u_{k0}^{(l)})\otimes1=(\operatorname{Id}\otimes\pi_K)\Delta(S_t(u_{i0}^{(l)}))=\sum_k u_{ik}^{(l)}\otimes\sum_j\pi_K(u_{kj}^{(l)})\omega_t(u_{j0}^{(l)}). k ∑ u ik ( l ) ω t ( u k 0 ( l ) ) ⊗ 1 = ( Id ⊗ π K ) Δ ( S t ( u i 0 ( l ) )) = k ∑ u ik ( l ) ⊗ j ∑ π K ( u k j ( l ) ) ω t ( u j 0 ( l ) ) . By the linear independence of the matrix coefficients [[54 Theorem 1.4.3] ], we deduce that ∑ j π K ( u k j ( l ) ) ω t ( u j 0 ( l ) ) = ω t ( u k 0 ( l ) ) 1 \sum_j\pi_K(u_{kj}^{(l)})\omega_t(u_{j0}^{(l)})=\omega_t(u_{k0}^{(l)})1 ∑ j π K ( u k j ( l ) ) ω t ( u j 0 ( l ) ) = ω t ( u k 0 ( l ) ) 1 k k k v : = ∑ j ω t ( u j 0 ( l ) ) e j v\mathrel{\mathop:}=\sum_j\omega_t(u_{j0}^{(l)})e_j v : = ∑ j ω t ( u j 0 ( l ) ) e j ( Id ⊗ π K ) ( u ( l ) ) ( v ⊗ 1 ) = v ⊗ 1 (\operatorname{Id}\otimes\pi_K)(u^{(l)})(v\otimes1)=v\otimes1 ( Id ⊗ π K ) ( u ( l ) ) ( v ⊗ 1 ) = v ⊗ 1 v v v K n K_n K n K n K_n K n π l \pi_l π l C e 0 \mathbb{C}e_0 C e 0 ω t ( u j 0 ( l ) ) = 0 \omega_t(u_{j0}^{(l)})=0 ω t ( u j 0 ( l ) ) = 0 j ≠ 0 j\ne0 j = 0
S t ( u i 0 ( l ) ) = ω t ( φ l ( n ) ) u i 0 ( l ) . S_t(u_{i0}^{(l)})=\omega_t(\varphi_l^{(n)})u_{i0}^{(l)}. S t ( u i 0 ( l ) ) = ω t ( φ l ( n ) ) u i 0 ( l ) . The number ω t ( φ l ( n ) ) \omega_t(\varphi_l^{(n)}) ω t ( φ l ( n ) ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] φ l ( n ) = φ l n ( x n ) \varphi_l^{(n)}=\varphi_l^n(x_n) φ l ( n ) = φ l n ( x n ) 1 1 1 Lemma 10.1 . It is multiplicative in t t t t t t 1 1 1 t = 0 t=0 t = 0 e − t ψ ( l ) e^{-t\psi(l)} e − t ψ ( l ) ψ ( l ) ∈ [ 0 , ∞ ) \psi(l)\in[0,\infty) ψ ( l ) ∈ [ 0 , ∞ ) (107) , the polynomials in x n x_n x n φ l ( n ) \varphi_l^{(n)} φ l ( n ) C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) S t S_t S t C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) T t φ l n = e − t ψ ( l ) φ l n T_t\varphi_l^n=e^{-t\psi(l)}\varphi_l^n T t φ l n = e − t ψ ( l ) φ l n C ∗ ( x n ) = C ( Y η ) C^*(x_n)=C(Y_\eta) C ∗ ( x n ) = C ( Y η ) Lemma 10.1 . The maps T t T_t T t ∣ φ l n ∣ ≤ 1 |\varphi_l^n|\leq1 ∣ φ l n ∣ ≤ 1 Y η Y_\eta Y η Lemma 10.1 . Hence Proposition 7.9 with τ = τ n \tau=\tau_n τ = τ n
Put ϑ = d n − 1 \vartheta=\frac{d}{n-1} ϑ = n − 1 d s = α ϑ s=\frac{\alpha}{\vartheta} s = ϑ α e l η ∈ dom E R e_l^\eta\in\operatorname{dom}\mathcal{E}_R e l η ∈ dom E R l ≥ 1 l\geq1 l ≥ 1 Theorem 4.6 , there exist constants c , c ′ > 0 c,c'>0 c , c ′ > 0 ψ ( l ) ≥ c E R ( e l η ) ≥ c ′ Λ l ( η ) s \psi(l)\geq c\mathcal{E}_R(e_l^\eta)\geq c'\Lambda_l(\eta)^s ψ ( l ) ≥ c E R ( e l η ) ≥ c ′ Λ l ( η ) s l ≥ 1 l\geq1 l ≥ 1 Λ l ( η ) s − 1 \Lambda_l(\eta)^{s-1} Λ l ( η ) s − 1 l l l Λ l ( η ) → ∞ \Lambda_l(\eta)\to\infty Λ l ( η ) → ∞ s ≤ 1 s\leq1 s ≤ 1 α ≤ d n − 1 \alpha\leq\frac{d}{n-1} α ≤ n − 1 d
Conversely, suppose that s ≤ 1 s\leq1 s ≤ 1 u ↦ F u ( n ) ( x ) u\mapsto\mathcal{F}_u^{(n)}(x) u ↦ F u ( n ) ( x ) x ∈ C ( C P q n − 1 ) x\in C(\mathbb{C}P_q^{n-1}) x ∈ C ( C P q n − 1 ) S t ( x ) : = ∫ F u ( n ) ( x ) d ν t ( s ) ( u ) S_t(x)\mathrel{\mathop:}=\int\mathcal{F}_u^{(n)}(x)\,d\nu_t^{(s)}(u) S t ( x ) : = ∫ F u ( n ) ( x ) d ν t ( s ) ( u ) S t S_t S t C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 ) S t ( u i 0 ( l ) ) = e − t Λ l ( η ) s u i 0 ( l ) S_t(u_{i0}^{(l)})=e^{-t\Lambda_l(\eta)^s}u_{i0}^{(l)} S t ( u i 0 ( l ) ) = e − t Λ l ( η ) s u i 0 ( l ) C ∗ ( x n ) = C ( Y η ) C^*(x_n)=C(Y_\eta) C ∗ ( x n ) = C ( Y η ) ∫ T u n d ν t ( s ) ( u ) = e − t L η s \int T_u^n\,d\nu_t^{(s)}(u)=e^{-tL_\eta^s} ∫ T u n d ν t ( s ) ( u ) = e − t L η s F u ( n ) = ( Id ⊗ ω u ) ∘ Δ \mathcal{F}_u^{(n)}=(\operatorname{Id}\otimes\omega_u)\circ\Delta F u ( n ) = ( Id ⊗ ω u ) ∘ Δ Theorem 7.10 , and integrating with respect to ν t ( s ) \nu_t^{(s)} ν t ( s ) S t S_t S t ψ ( l ) = Λ l ( η ) s \psi(l)=\Lambda_l(\eta)^s ψ ( l ) = Λ l ( η ) s L η s L_\eta^s L η s
Suppose that the operators e − t L η s e^{-tL_\eta^s} e − t L η s Theorem 7.10 , they restrict to positive unital maps on C ( Y η ) C(Y_\eta) C ( Y η ) φ l n \varphi_l^n φ l n e − t Λ l ( η ) s φ l n e^{-t\Lambda_l(\eta)^s}\varphi_l^n e − t Λ l ( η ) s φ l n Lemma 10.1 , Proposition 7.9 with τ = τ n \tau=\tau_n τ = τ n ψ ( l ) = Λ l ( η ) s \psi(l)=\Lambda_l(\eta)^s ψ ( l ) = Λ l ( η ) s Λ l ( η ) s − 1 \Lambda_l(\eta)^{s-1} Λ l ( η ) s − 1 l l l s > 1 s>1 s > 1
Remark 10.6 Suppose that τ n = Q q − d \tau_n=Q_q^{-d} τ n = Q q − d Theorem 10.5 describes the orders α \alpha α C P q n − 1 \mathbb{C}\mathrm{P}_q^{n-1} C P q n − 1 α ≤ d n − 1 \alpha\leq\frac{d}{n-1} α ≤ n − 1 d C ( C P q n − 1 ) C(\mathbb{C}\mathrm{P}_q^{n-1}) C ( C P q n − 1 ) d n − 1 < α < ( n + 1 ) d n − 1 \frac{d}{n-1}<\alpha<\frac{(n+1)d}{n-1} n − 1 d < α < n − 1 ( n + 1 ) d α ≥ ( n + 1 ) d n − 1 \alpha\geq\frac{(n+1)d}{n-1} α ≥ n − 1 ( n + 1 ) d s = ( n − 1 ) α d s=\frac{(n-1)\alpha}{d} s = d ( n − 1 ) α s ≤ 1 s\leq1 s ≤ 1 1 < s < n + 1 1<s<n+1 1 < s < n + 1 s ≥ n + 1 s\geq n+1 s ≥ n + 1 d d d n = 2 n=2 n = 2 Theorems 4.6 and 7.10 . For n = d + 1 n=d+1 n = d + 1 q 2 = Q − 1 q^2=Q^{-1} q 2 = Q − 1 s = α s=\alpha s = α 0 < α < d + 2 0<\alpha<d+2 0 < α < d + 2 L Q − 1 α L_{Q^{-1}}^\alpha L Q − 1 α C P q d \mathbb{C}\mathrm{P}_q^d C P q d 0 < α ≤ 1 0<\alpha\leq1 0 < α ≤ 1 C ( C P q d ) C(\mathbb{C}\mathrm{P}_q^d) C ( C P q d )
The frontier between C P q n − 1 \mathbb{C}\mathrm{P}_q^{n-1} C P q n − 1 S q 2 n − 1 S_q^{2n-1} S q 2 n − 1
We now show that the phenomenon of Section 8 occurs in the whole family. Recall that K n K_n K n π K ( x i j ) \pi_K(x_{ij}) π K ( x ij ) ( i , j ) (i,j) ( i , j ) U q ( n − 1 ) U_q(n-1) U q ( n − 1 ) i , j ≤ n − 1 i,j\leq n-1 i , j ≤ n − 1 π K ( x n n ) = 1 ⊗ z \pi_K(x_{nn})=1\otimes z π K ( x nn ) = 1 ⊗ z z z z T \mathbb{T} T π K ( x i n ) = π K ( x n i ) = 0 \pi_K(x_{in})=\pi_K(x_{ni})=0 π K ( x in ) = π K ( x ni ) = 0 i ≤ n − 1 i\leq n-1 i ≤ n − 1 π K ′ ≔ ( Id ⊗ ϵ 1 ) ∘ π K : C ( U q ( n ) ) → C ( U q ( n − 1 ) ) \pi_{K'}\coloneqq(\operatorname{Id}\otimes\epsilon_1)\circ\pi_K:C(U_q(n))\to C(U_q(n-1)) π K ′ : = ( Id ⊗ ϵ 1 ) ∘ π K : C ( U q ( n )) → C ( U q ( n − 1 )) ϵ 1 \epsilon_1 ϵ 1 1 ∈ T 1\in\mathbb{T} 1 ∈ T U q ( n − 1 ) U_q(n-1) U q ( n − 1 ) K n K_n K n
C ( S q 2 n − 1 ) ≔ { x ∈ C ( U q ( n ) ) : ( Id ⊗ π K ′ ) Δ ( x ) = x ⊗ 1 } . C(S_q^{2n-1})\coloneqq\{x\in C(U_q(n)):(\operatorname{Id}\otimes\pi_{K'})\Delta(x)=x\otimes1\}. C ( S q 2 n − 1 ) : = { x ∈ C ( U q ( n )) : ( Id ⊗ π K ′ ) Δ ( x ) = x ⊗ 1 } . It contains C ( C P q n − 1 ) C(\mathbb{C}\mathrm{P}_q^{n-1}) C ( C P q n − 1 ) Δ ( x n n ) = ∑ j x n j ⊗ x j n \Delta(x_{nn})=\sum_j x_{nj}\otimes x_{jn} Δ ( x nn ) = ∑ j x nj ⊗ x j n ( Id ⊗ π K ′ ) Δ ( x n n ) = x n n ⊗ 1 (\operatorname{Id}\otimes\pi_{K'})\Delta(x_{nn})=x_{nn}\otimes1 ( Id ⊗ π K ′ ) Δ ( x nn ) = x nn ⊗ 1 ( Id ⊗ π K ) Δ ( x n n ) = x n n ⊗ ( 1 ⊗ z ) (\operatorname{Id}\otimes\pi_K)\Delta(x_{nn})=x_{nn}\otimes(1\otimes z) ( Id ⊗ π K ) Δ ( x nn ) = x nn ⊗ ( 1 ⊗ z ) x n n x_{nn} x nn C ( S q 2 n − 1 ) C(S_q^{2n-1}) C ( S q 2 n − 1 ) C ( C P q n − 1 ) C(\mathbb{C}\mathrm{P}_q^{n-1}) C ( C P q n − 1 )
Lemma 10.7 We have
x n n x n n ∗ = 1 − x n , x n n ∗ x n n = 1 − η x n , x n x n n = η x n n x n . x_{nn}x_{nn}^{*}=1-x_n,\qquad x_{nn}^{*}x_{nn}=1-\eta x_n,\qquad x_nx_{nn}=\eta x_{nn}x_n. x nn x nn ∗ = 1 − x n , x nn ∗ x nn = 1 − η x n , x n x nn = η x nn x n . Consequently, ∥ x n n ∥ = 1 \lVert x_{nn}\rVert=1 ∥ x nn ∥ = 1 k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 p k ≔ 1 { η k } ( x n ) p_k\coloneqq\mathbf{1}_{\{\eta^k\}}(x_n) p k : = 1 { η k } ( x n ) C ∗ ( x n ) C^{*}(x_n) C ∗ ( x n ) x n n ∗ p 0 = 0 x_{nn}^{*}p_0=0 x nn ∗ p 0 = 0 x n n ∗ p k = p k − 1 x n n ∗ x_{nn}^{*}p_k=p_{k-1}x_{nn}^{*} x nn ∗ p k = p k − 1 x nn ∗ k ≥ 1 k\geq1 k ≥ 1 ( x n n p k ) ∗ ( x n n p k ) = ( 1 − η k ) p k (x_{nn}p_k)^{*}(x_{nn}p_k)=(1-\eta^k)p_k ( x nn p k ) ∗ ( x nn p k ) = ( 1 − η k ) p k
Proof. With the notation of [57 (4.6) and (4.26)] , we have z n = x n n z_n=x_{nn} z n = x nn w n = x n n ∗ w_n=x_{nn}^{*} w n = x nn ∗ ζ n − 1 = x n \zeta_{n-1}=x_n ζ n − 1 = x n ζ n = 1 \zeta_n=1 ζ n = 1 [57 (4.28.a)] with k = n k=n k = n z n w n = 1 − ζ n − 1 z_nw_n=1-\zeta_{n-1} z n w n = 1 − ζ n − 1 w n z n = 1 − q − 2 ζ n − 1 w_nz_n=1-q^{-2}\zeta_{n-1} w n z n = 1 − q − 2 ζ n − 1 [57 (4.28.b)] with k = n k=n k = n i = n − 1 i=n-1 i = n − 1 z n ζ n − 1 = ζ n − 1 z n z_n\zeta_{n-1}=\zeta_{n-1}z_n z n ζ n − 1 = ζ n − 1 z n Lemma 10.1 , the spectrum of x n x_n x n Y η ⊂ [ 0 , 1 ] Y_\eta\subset[0,1] Y η ⊂ [ 0 , 1 ] 0 0 0 ∥ x n n ∥ 2 = ∥ 1 − x n ∥ = 1 \lVert x_{nn}\rVert^2=\lVert1-x_n\rVert=1 ∥ x nn ∥ 2 = ∥ 1 − x n ∥ = 1 x n n ∗ x n = η x n x n n ∗ x_{nn}^{*}x_n=\eta x_nx_{nn}^{*} x nn ∗ x n = η x n x nn ∗ x n n ∗ P ( x n ) = P ( η x n ) x n n ∗ x_{nn}^{*}P(x_n)=P(\eta x_n)x_{nn}^{*} x nn ∗ P ( x n ) = P ( η x n ) x nn ∗ P P P η Y η ⊂ Y η \eta Y_\eta\subset Y_\eta η Y η ⊂ Y η Y η Y_\eta Y η x n n ∗ F ( x n ) = F ( η x n ) x n n ∗ x_{nn}^{*}F(x_n)=F(\eta x_n)x_{nn}^{*} x nn ∗ F ( x n ) = F ( η x n ) x nn ∗ F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η ) F ( η x n ) F(\eta x_n) F ( η x n ) x ↦ F ( η x ) x\mapsto F(\eta x) x ↦ F ( η x ) F = 1 { η k } F=\mathbf{1}_{\{\eta^k\}} F = 1 { η k } Y η Y_\eta Y η η k \eta^k η k x ∈ Y η x\in Y_\eta x ∈ Y η F ( η x ) = 1 { η k − 1 } ( x ) F(\eta x)=\mathbf{1}_{\{\eta^{k-1}\}}(x) F ( η x ) = 1 { η k − 1 } ( x ) k ≥ 1 k\geq1 k ≥ 1 F ( η x ) = 0 F(\eta x)=0 F ( η x ) = 0 k = 0 k=0 k = 0 ( x n n p k ) ∗ ( x n n p k ) = p k x n n ∗ x n n p k = p k ( 1 − x n ) p k = ( 1 − η k ) p k (x_{nn}p_k)^{*}(x_{nn}p_k)=p_kx_{nn}^{*}x_{nn}p_k=p_k(1-x_n)p_k=(1-\eta^k)p_k ( x nn p k ) ∗ ( x nn p k ) = p k x nn ∗ x nn p k = p k ( 1 − x n ) p k = ( 1 − η k ) p k
For n = 2 n=2 n = 2 Lemma 10.7 shows that x 22 ∗ x_{22}^{*} x 22 ∗ x 2 x_2 x 2 (38) of b b b A A A r 2 = q 2 r^2=q^2 r 2 = q 2 x n n ∗ x_{nn}^{*} x nn ∗ x n x_n x n η k \eta^k η k ( 1 − τ n ) τ n k (1-\tau_n)\tau_n^k ( 1 − τ n ) τ n k
Theorem 10.8 Let n ≥ 2 n\geq2 n ≥ 2 0 < q < 1 0<q<1 0 < q < 1 η = q 2 \eta=q^2 η = q 2 τ n = q 2 ( n − 1 ) \tau_n=q^{2(n-1)} τ n = q 2 ( n − 1 ) A A A C ∗ C^{*} C ∗ C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) x n x_n x n x n n x_{nn} x nn C ( S q 2 n − 1 ) C(S_q^{2n-1}) C ( S q 2 n − 1 ) C ( U q ( n ) ) C(U_q(n)) C ( U q ( n ))
Let S : A → A S:\mathcal{A}\to\mathcal{A} S : A → A K K K N 0 \mathbb{N}_0 N 0 K ( m , ⋅ ) K(m,\cdot) K ( m , ⋅ ) ℓ 1 \ell^1 ℓ 1 μ \mu μ m → ∞ m\to\infty m → ∞ F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η ) S ( F ( x n n ) ) = G ( x n n ) S(F(x_{nn}))=G(x_{nn}) S ( F ( x nn )) = G ( x nn ) G ∈ C ( Y η ) G\in C(Y_\eta) G ∈ C ( Y η ) G ( η m ) = ∑ k K ( m , k ) F ( η k ) G(\eta^m)=\sum_k K(m,k)F(\eta^k) G ( η m ) = ∑ k K ( m , k ) F ( η k ) m ∈ N 0 m\in\mathbb{N}_0 m ∈ N 0
∥ S ( x n n ) − x n n ∥ ≥ 1 − F ( μ ) ≥ 1 − 1 − ( sup k μ ( k ) ) 2 . \left\|S(x_{nn})-x_{nn}\right\|\geq1-F(\mu)\geq1-\sqrt{1-\left(\sup_k\mu(k)\right)^2}. ∥ S ( x nn ) − x nn ∥ ≥ 1 − F ( μ ) ≥ 1 − 1 − ( k sup μ ( k ) ) 2 . where F ( μ ) F(\mu) F ( μ ) (98) .
Suppose that τ n = Q − d \tau_n=Q^{-d} τ n = Q − d α > 0 \alpha>0 α > 0 J n J_n J n ( S t ) t > 0 (S_t)_{t>0} ( S t ) t > 0 A \mathcal{A} A C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) J n e − t R α rad J n − 1 J_ne^{-tR_\alpha^{\operatorname{rad}}}J_n^{-1} J n e − t R α rad J n − 1
∥ S ρ − j ( x n n ) − x n n ∥ ≥ 1 − 1 − δ T 2 > 0 , j ≥ 1 , \left\|S_{\rho-j}(x_{nn})-x_{nn}\right\|\geq1-\sqrt{1-\delta_T^2}>0,\qquad j\geq1, ∥ S ρ − j ( x nn ) − x nn ∥ ≥ 1 − 1 − δ T 2 > 0 , j ≥ 1 , where δ T \delta_T δ T Lemma 2.6 .
For any 0 < t ≤ Λ 1 ( η ) − 1 0<t\leq\Lambda_1(\eta)^{-1} 0 < t ≤ Λ 1 ( η ) − 1 S : A → A S:\mathcal{A}\to\mathcal{A} S : A → A S ( F ( x n n ) ) = ( T t η F ) ( x n n ) S(F(x_{nn}))=(T_t^\eta F)(x_{nn}) S ( F ( x nn )) = ( T t η F ) ( x nn ) F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η )
∥ S ( x n n ) − x n n ∥ ≥ 1 − 1 − δ C ( τ n , η ) 2 , \left\|S(x_{nn})-x_{nn}\right\|\geq1-\sqrt{1-\delta_C(\tau_n,\eta)^2}, ∥ S ( x nn ) − x nn ∥ ≥ 1 − 1 − δ C ( τ n , η ) 2 , where δ C ( τ n , η ) \delta_C(\tau_n,\eta) δ C ( τ n , η )
Let ( ω t ) t > 0 (\omega_t)_{t>0} ( ω t ) t > 0 C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) ω t ( x j n ) → δ j n \omega_t(x_{jn})\to\delta_{jn} ω t ( x j n ) → δ j n t → 0 + t\to0^+ t → 0 + 1 ≤ j ≤ n 1\leq j\leq n 1 ≤ j ≤ n t 1 > 0 t_1>0 t 1 > 0 0 < t ≤ t 1 0<t\leq t_1 0 < t ≤ t 1 ( Id ⊗ ω t ) ∘ Δ (\operatorname{Id}\otimes\omega_t)\circ\Delta ( Id ⊗ ω t ) ∘ Δ F ( x n n ) ↦ ( T t η F ) ( x n n ) F(x_{nn})\mapsto(T_t^\eta F)(x_{nn}) F ( x nn ) ↦ ( T t η F ) ( x nn ) C ∗ ( x n ) C^*(x_n) C ∗ ( x n )
Proof : 1. We realize C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) H H H Lemma 10.1 , η m \eta^m η m x n x_n x n p m ≠ 0 p_m\ne0 p m = 0 m ∈ N 0 m\in\mathbb{N}_0 m ∈ N 0 m ≥ 1 m\geq1 m ≥ 1 ξ ∈ p m H \xi\in p_mH ξ ∈ p m H Lemma 10.7 , the vector ξ ′ ≔ ( 1 − η m ) − 1 / 2 x n n ∗ ξ \xi'\coloneqq(1-\eta^m)^{-1/2}x_{nn}^*\xi ξ ′ : = ( 1 − η m ) − 1/2 x nn ∗ ξ p m − 1 H p_{m-1}H p m − 1 H ⟨ x n n ∗ ξ , ξ ′ ⟩ = ( 1 − η m ) 1 / 2 \langle x_{nn}^*\xi,\xi'\rangle=(1-\eta^m)^{1/2} ⟨ x nn ∗ ξ , ξ ′ ⟩ = ( 1 − η m ) 1/2 K \mathcal{K} K Π \Pi Π A \mathcal{A} A K \mathcal{K} K V : H → K V:H\to\mathcal{K} V : H → K S ( x ) = V ∗ Π ( x ) V S(x)=V^*\Pi(x)V S ( x ) = V ∗ Π ( x ) V x ∈ A x\in\mathcal{A} x ∈ A ζ ≔ V ξ \zeta\coloneqq V\xi ζ : = V ξ ζ ′ ≔ V ξ ′ \zeta'\coloneqq V\xi' ζ ′ : = V ξ ′ F ∈ C ( Y η ) F\in C(Y_\eta) F ∈ C ( Y η ) i ∈ N 0 i\in\mathbb{N}_0 i ∈ N 0 F ( x n ) F(x_n) F ( x n ) p i H p_iH p i H F ( η i ) F(\eta^i) F ( η i ) S S S F = 1 { η k } F=1_{\{\eta^k\}} F = 1 { η k }
∥ Π ( p k ) ζ ∥ 2 = ⟨ S ( p k ) ξ , ξ ⟩ = K ( m , k ) , ∥ Π ( p k ) ζ ′ ∥ 2 = K ( m − 1 , k ) , k ∈ N 0 . \|\Pi(p_k)\zeta\|^2=\langle S(p_k)\xi,\xi\rangle=K(m,k),\qquad\|\Pi(p_k)\zeta'\|^2=K(m-1,k),\qquad k\in\mathbb{N}_0. ∥Π ( p k ) ζ ∥ 2 = ⟨ S ( p k ) ξ , ξ ⟩ = K ( m , k ) , ∥Π ( p k ) ζ ′ ∥ 2 = K ( m − 1 , k ) , k ∈ N 0 . Since ∑ k K ( m , k ) = 1 = ∥ ζ ∥ 2 \sum_kK(m,k)=1=\|\zeta\|^2 ∑ k K ( m , k ) = 1 = ∥ ζ ∥ 2 Π ( p k ) \Pi(p_k) Π ( p k ) ζ = ∑ k Π ( p k ) ζ \zeta=\sum_k\Pi(p_k)\zeta ζ = ∑ k Π ( p k ) ζ ζ ′ = ∑ k Π ( p k ) ζ ′ \zeta'=\sum_k\Pi(p_k)\zeta' ζ ′ = ∑ k Π ( p k ) ζ ′ Lemma 10.7 , we have x n n ∗ p 0 = 0 x_{nn}^*p_0=0 x nn ∗ p 0 = 0 x n n ∗ p k = p k − 1 x n n ∗ p k x_{nn}^*p_k=p_{k-1}x_{nn}^*p_k x nn ∗ p k = p k − 1 x nn ∗ p k k ≥ 1 k\geq1 k ≥ 1
⟨ S ( x n n ∗ ) ξ , ξ ′ ⟩ = ⟨ Π ( x n n ∗ ) ζ , ζ ′ ⟩ = ∑ k ≥ 1 ⟨ Π ( x n n ∗ p k ) Π ( p k ) ζ , Π ( p k − 1 ) ζ ′ ⟩ , \langle S(x_{nn}^*)\xi,\xi'\rangle=\langle\Pi(x_{nn}^*)\zeta,\zeta'\rangle=\sum_{k\geq1}\langle\Pi(x_{nn}^*p_k)\Pi(p_k)\zeta,\Pi(p_{k-1})\zeta'\rangle, ⟨ S ( x nn ∗ ) ξ , ξ ′ ⟩ = ⟨ Π ( x nn ∗ ) ζ , ζ ′ ⟩ = k ≥ 1 ∑ ⟨ Π ( x nn ∗ p k ) Π ( p k ) ζ , Π ( p k − 1 ) ζ ′ ⟩ , and, since ∥ x n n ∗ p k ∥ ≤ 1 \|x_{nn}^*p_k\|\leq1 ∥ x nn ∗ p k ∥ ≤ 1 ∣ ⟨ S ( x n n ∗ ) ξ , ξ ′ ⟩ ∣ ≤ F m ≔ ∑ k ≥ 1 K ( m , k ) K ( m − 1 , k − 1 ) \left|\langle S(x_{nn}^*)\xi,\xi'\rangle\right|\leq F_m\coloneqq\sum_{k\geq1}\sqrt{K(m,k)K(m-1,k-1)} ∣ ⟨ S ( x nn ∗ ) ξ , ξ ′ ⟩ ∣ ≤ F m : = ∑ k ≥ 1 K ( m , k ) K ( m − 1 , k − 1 ) Lemma 8.1 , F m → F ( μ ) F_m\to F(\mu) F m → F ( μ ) m → ∞ m\to\infty m → ∞ S S S S ( x n n ∗ ) = S ( x n n ) ∗ S(x_{nn}^*)=S(x_{nn})^* S ( x nn ∗ ) = S ( x nn ) ∗
∥ S ( x n n ) − x n n ∥ = ∥ S ( x n n ∗ ) − x n n ∗ ∥ ≥ ∣ ⟨ ( S ( x n n ∗ ) − x n n ∗ ) ξ , ξ ′ ⟩ ∣ ≥ ( 1 − η m ) 1 / 2 − F m . \|S(x_{nn})-x_{nn}\|=\|S(x_{nn}^*)-x_{nn}^*\|\geq\left|\langle(S(x_{nn}^*)-x_{nn}^*)\xi,\xi'\rangle\right|\geq(1-\eta^m)^{1/2}-F_m. ∥ S ( x nn ) − x nn ∥ = ∥ S ( x nn ∗ ) − x nn ∗ ∥ ≥ ∣ ⟨( S ( x nn ∗ ) − x nn ∗ ) ξ , ξ ′ ⟩ ∣ ≥ ( 1 − η m ) 1/2 − F m . Letting m → ∞ m \to\infty m → ∞ Lemma 8.2 gives the second one.
By Lemma 2.6 , the rows of the transition matrix P t P_t P t e − t R α q e^{-tR_{\alpha}^{q}} e − t R α q ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 μ t \mu_t μ t J n J_n J n k ↦ f ( k ) k \mapsto f(k) k ↦ f ( k ) F ( x n ) F(x_n) F ( x n ) F ( η k ) = f ( k ) F(\eta^k) = f(k) F ( η k ) = f ( k ) S t S_t S t K = P t K = P_t K = P t μ = μ t \mu= \mu_t μ = μ t sup k μ ρ − j ( k ) ≥ δ T \sup_k \mu_{\rho-j}(k) \ge\delta_T sup k μ ρ − j ( k ) ≥ δ T Lemma 2.6 .
By Lemma 10.1 , we have ∣ φ l η ∣ ≤ 1 |\varphi_l^\eta| \le1 ∣ φ l η ∣ ≤ 1 Y η Y_\eta Y η l l l Proposition 3.3 , the operator T t η T_t^\eta T t η ℓ N 0 1 \ell^1_{\mathbb{N}_0} ℓ N 0 1 μ t η \mu_t^\eta μ t η 0 < τ < 1 0 < \tau< 1 0 < τ < 1 τ = τ n \tau= \tau_n τ = τ n
The map S t : = ( Id ⊗ ω t ) ∘ Δ S_t \mathrel{\mathop:}= (\operatorname{Id} \otimes\omega_t) \circ\Delta S t : = ( Id ⊗ ω t ) ∘ Δ C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) S t ( x n n ) = ∑ j x n j ω t ( x j n ) S_t(x_{nn}) = \sum_j x_{nj}\omega_t(x_{jn}) S t ( x nn ) = ∑ j x nj ω t ( x j n ) ∥ x n j ∥ ≤ 1 \|x_{nj}\| \le1 ∥ x nj ∥ ≤ 1 ∥ S t ( x n n ) − x n n ∥ ≤ ∑ j ∣ ω t ( x j n ) − δ j n ∣ → 0 \|S_t(x_{nn}) - x_{nn}\| \le\sum_j |\omega_t(x_{jn}) - \delta_{jn}| \to0 ∥ S t ( x nn ) − x nn ∥ ≤ ∑ j ∣ ω t ( x j n ) − δ j n ∣ → 0 t → 0 + t \to0^+ t → 0 + 0 < t 1 ≤ Λ 1 ( η ) − 1 0 < t_1 \le\Lambda_1(\eta)^{-1} 0 < t 1 ≤ Λ 1 ( η ) − 1 1 − 1 − δ C ( τ n , η ) 2 1-\sqrt{1-\delta_C(\tau_n,\eta)^2} 1 − 1 − δ C ( τ n , η ) 2 0 < t ≤ t 1 0 < t \le t_1 0 < t ≤ t 1 A = C ( U q ( n ) ) \mathcal{A} = C(U_q(n)) A = C ( U q ( n )) S t S_t S t F ( x n ) ↦ ( T t η F ) ( x n ) F(x_n) \mapsto(T_t^\eta F)(x_n) F ( x n ) ↦ ( T t η F ) ( x n ) C ∗ ( x n ) C^*(x_n) C ∗ ( x n ) t t t
Remark 10.9 Theorem 10.8 shows that the natural domain of the radial Taibleson and Casimir dynamics is C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 n ≥ 2 n \ge2 n ≥ 2 C ( C P q n − 1 ) C(\mathbb{C}P_q^{n-1}) C ( C P q n − 1 ) ( S t ( n ) ) (\mathcal{S}_t^{(n)}) ( S t ( n ) ) ( Id ⊗ ω t ) ∘ Δ (\operatorname{Id} \otimes\omega_t) \circ\Delta ( Id ⊗ ω t ) ∘ Δ ω t = μ t η ∘ E K \omega_t = \mu_t^\eta\circ E_K ω t = μ t η ∘ E K ω t ( x n n ) = 0 \omega_t(x_{nn}) = 0 ω t ( x nn ) = 0 t > 0 t > 0 t > 0 (106) , since x n n x_{nn} x nn π l \pi_l π l C ( S q 2 n − 1 ) C(S_q^{2n-1}) C ( S q 2 n − 1 ) x n n x_{nn} x nn C ( U q ( n ) ) C(U_q(n)) C ( U q ( n )) U q ( n ) U_q(n) U q ( n ) [21 ] , has the Casimir dynamics of C P q n − 1 \mathbb{C}P_q^{n-1} C P q n − 1 x n n ∗ x_{nn}^* x nn ∗
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, 6 rue Didier Daurat, 81000 Albi, France
Personal webpage cedric.arhancet@protonmail.com ORCID: 0000-0002-5179-6972
