Operator algebras related to bounded positive operator. operator with simple spectrum

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1 Operator algebras related to bounded positive operator with simple spectrum Institute of Mathematics, University of Bia lystok joint work with A. Odzijewicz, A. Tereszkiewicz 13 June 2007 Varna

2 Coherent state map and corresponding algebra H a bounded positive operator with simple spectrum in Hilbert space H 0 cyclic vector, i.e. {E( ) 0 } B(R) is linearly dense in H 0 0 = 1 is an isomorphism for I : L 2 (R, dµ) f R f(λ)e(dλ) 0 H µ( ) = 0 E( ) 0. I H I acts in L 2 (R, dµ) as the multiplication by argument

3 By Gram-Schmidt orthonormalization one obtains orthonormal polynomials P n in L 2 (R, dµ) The orthonormal basis in H n := P n (H) 0 = I(P n ) H n = b n 1 n 1 + a n n + b n n + 1 b n > 0, b 1 = 0 and a n, b n R Jacobi matrix a 0 b 0 0 b 0 a 1 b 1 J =. 0 b 1 a

4 Moments of measure µ σ k := Resolvent R λ := (H λ1) 1 R λ k µ(dλ) > 0 σ k = 0 H k 0, k N {0} 0 R λ 0 = R µ(dx) x λ = σ k λ k+1, k=0 where the second equality is valid for λ > H.

5 Coherent state map a complex analytic map K : D H from disc D with linearly dense image, expressed by K(z) = c n z n n, n=0 0 < c n R Annihilation operator AK(z) = zk(z). where c 1 = 0. Creation operator A n := c n 1 c n n 1, A n = c n c n+1 n + 1.

6 Generalized exponential function E : D C E( vw) := K(v) K(w), We consider two coherent state maps for z < H 1 2 for z < H 1 2. K 1 (z) := K 2 (z) := σn z n n, n=0 n=0 1 σn z n n

7 E 1 (z) = 1 z 0 R 1 0 = z σ n z n n=0 Decomposition of unity K 2 (z) K 2 (z) ν(dz) = 1, ν(dz) := 1 2π dϕ f µ(dr), where z = re iϕ and f µ is pullback of (2) by f(x) := x 2. E 2 (vw) = E 2 (vz)e 2 (zw) ν(dz),

8 T - Toeplitz algebra, i.e. C -algebra generated by shift operator S n = n 1, n N S 0 = 0. Proposition i) Let us assume that A 1 is bounded. Then the C -algebra A 1 generated by A 1 coincides with T if and only if the sequence } n N is convergent. { σ n 1 σ n ii) Let us assume that A 2 is bounded. Then the C -algebra A 2 generated by A 2 coincides with T if and only if the sequence } n N is convergent. { σ n 1 σ n iii) If both A 1 and A 2 are bounded then A 1 = A 2.

9 Sketch of proof i) lim inf n σ n lim inf σ n 1 n n σn lim sup n n σn lim sup n σ n σ n 1 lim n n σn = lim n σ n σ n 1 S = (A 1 A 1) 1 2 A1 A 1 A 1 A 1 ( H ) 1 1 is compact. Thus A 1 = (A 1 A 1 ) 1 2 S T. iii) A 1 bounded A 2 A 2 bounded from below A 1 A 1 = (A 2 A 2) 1 A 1 = (A 2 A 2) 1 A 2 A 2 = (A 1 A 1) 1 A 1

10 We will restrict our considerations to the case A 1 = A 2 = T. Q := q n n n T, 0 < q < 1 n=0 Structural function R : spec Q spec A 1 A 1 (continuous) Structural relations: R(q n ) := σ n 1 σ n for n N {0}, σ n 1 R(0) := lim = H 1, n σ n A 1A 1 = R(Q) A 1 A 1 = R(qQ) qqa 1 = A 1 Q qa 1Q = QA 1

11 N n := n n [A 1, N] = A 1 [A 1, N] = A 1 A k 1 A k 1 = R(Q)...R(q k Q), A k 1A k 1 = R(qQ)...R(q k+1 Q) σ k = 1 R(q)...R(q k ) = 1 0 A k 1 A k 1 0 E 1 (z) := E 2 (z) := n=0 z n R(q)...R(q k ), R(q)...R(q k ) z n. n=0

12 Example (little q-jacobi polynomials) R(x) = (1 x)(1 b 1q 1 x)... (1 b r 1 q 1 x) (1 a 1 q 1 x)... (1 a r q 1 (1 χ x) {1} (x)), χ {1} is a characteristic function of the set {1} a i, b i < 1. Generalized exponential functions basic hypergeometric series ( ) a1... a E 1 (z) = r Φ r r 1 b 1...b r 1 q; z = for z < 1 n=0 E 2 (z) = r+1 Φ r ( q q b1...b r 1 a 1...a r (a 1 ; q) n... (a r ; q) n (q; q) n (b 1 ; q) n... (b r 1 ; q) n z n ) q; z

13 Example (little q-jacobi polynomials) Moments σ n = q-pochhammer symbol (a 1 ; q) n... (a r ; q) n (q; q) n (b 1 ; q) n... (b r 1 ; q) n, (α; q) n := (1 α)(1 αq)... (1 αq n 1 )

14 Example (little q-jacobi polynomials, case r = 1) 1 < a < q 1 Structural relations R(x) = 1 x 1 ax 1 Q = (1 aq)a 1A 1 1 qq = (1 aqq)a 1 A 1 Moments are σ k = (aq; q) k (q; q) k

15 Example (little q-jacobi polynomials, case r = 1) Coefficients of Jacobi matrix a n = qn (1 aq n+1 )(1 q n ) (1 q 2n )(1 q 2n+1 ) + aqn (1 q n )(1 a 1 q n 1 ) (1 q 2n 1 )(1 q 2n, ) aq b n = 2n+1 (1 q n+1 )(1 aq n+1 )(1 a 1 q n )(1 q n ) (1 q 2n )(1 q 2n+1 ) 2 (1 q 2n+2. ) Measure µ is discrete µ(dλ) = (aq; q) (a 1, q) (q; q) (q; q) n=0 (qλ; q) n a n λ (a 1 λ; q) n δ(λ q n )dλ. The polynomials orthonormal with respect to this measure are a subclass of little q-jacobi polynomials.

16 Example (little q-jacobi polynomials, case r = 1) Exponential functions Reproducing property E 2 (vw) = 1 2π E 1 (z) = 1 Φ 0 ( aq ( q E 2 (z) = 2 Φ 1 a 1 0 (aq; q) (a 1, q) (q; q) (q; q) q ) q; z ) q; z 2π dr dϕe 2 (vre iϕ )E 2 (re iϕ w) 0 n=0 (qr; q) n a n r (a 1 r; q) n δ(r q 2n )

17 Example (little q-jacobi polynomials, case r = 2 and a 1 = q, b 1 < a 2, 0 < a 2 < 1) Structural relations R(x) = (1 b 1q 1 x) (1 a 2 q 1 x) (1 χ {1}(x)) (1 a 2 q 1 Q)A 1A 1 = (1 b 1 q 1 Q)(1 0 0 ) (1 a 2 Q)A 1 A 1 = 1 b 1 Q, Moments σ n σ n = (a 2; q) n (b 1 ; q) n

18 Example (little q-jacobi polynomials, case r = 2 and a 1 = q, b 1 < a 2, 0 < a 2 < 1) Coefficients of Jacobi matrix a n = qn (1 a 2 q n )(1 b 1 q n 1 ) (1 b 1 q 2n 1 )(1 b 1 q 2n ) + a 2q n 1 (1 q n )(1 q n 1 b 1 /a 2 ) (1 b 1 q 2n 2 )(1 b 1 q 2n 1 ) a 2 q b n = 2n (1 q n+1 )(1 q n b 1 /a 2 )(1 a 2 q n )(1 b 1 q n 1 ) (1 b 1 q 2n 1 )(1 b 1 q 2n ) 2 (1 b 1 q 2n+1 ) Measure µ(dλ) = (a 2; q) (b 1 /a 2 ; q) (b 1 ; q) (q; q) n=0 (qλ; q) a n 2 (λb 1 /a 2 ; q) δ(λ q n )dλ

19 Example (little q-jacobi polynomials, case r = 2 and a 1 = q, b 1 < a 2, 0 < a 2 < 1) Exponential functions Reproducing property E 2 (vw) = 1 2π E 1 (z) = 2 Φ 1 ( a2 b 1 E 2 (z) = 2 Φ 1 ( b1 1 (a 2; q) (b 1 /a 2 ; q) (b 1 ; q) (q; q) 0 a 2 q q ) q; z ) q; z 2π dr dϕe 2 (vre iϕ )E 2 (re iϕ w) 0 n=0 (rq; q) a n 2 (rb 1 /a 2 ; q) δ(r q 2n )

20 Example (little q-jacobi polynomials, case r = 2 and a 1 = q, b 1 < a 2, 0 < a 2 < 1) Orthogonal polynomials corresponding to this case are the little q-jacobi polynomials. b 1 = q previous case r = 1 b 1 = q 2, a 2 = q the little q-legendre polynomials b 1 = 0, 0 < a 2 < 1 the little q-laguerre/wall polynomials

21 Example (Classical Jacobi polynomials) Coefficients of Jacobi matrix (α, β > 1) β 2 α 2 a n = (2n + α + β)(2n + α + β + 2) (n + 1)(n α)(n β)(n α + β) b n = 2 (2n + α + β + 1)(2n + α + β + 2) 2 (2n + α + β + 3) Measure µ(dλ) = (1 λ) α λ β Γ(α + β + 2) Γ(α + 1)Γ(β + 1) χ [0,1](λ)dλ Moments σ n = (β + 1) n (α + β + 2) n

22 Example (Classical Jacobi polynomials) Structural function Structural relations R(q x ) = α + β x (1 χ β + x {1} (x)) (β + N)A 1A 1 = (α + β N)(1 0 0 ) (β N)A 1 A 1 = α + β N

23 Example (Classical Jacobi polynomials) Exponential functions E 1 (z) = 2 F 1 ( β α + β + 2 E 2 (z) = 2 F 1 ( α + β β + 1 Reproduction property of E 2 holds for measure ) z ) z ν(dz) = 1 2π (1 r2 ) α r 2β Γ(α + β + 2) Γ(α + 1)Γ(β + 1) χ [0,1](r)dϕdr

24 Example (Classical Jacobi polynomials) Subcases: α = β = λ 1 2 Gegenbauer/ultraspherical polynomials α = β = 1 2 Chebychev I kind α = β = 1 2 Chebychev II kind α = β = 0 Legendre/spherical

25 One-parameter subgroup R t U t Aut H such that matrix J H = (J t ) nm m t t n n,m=0 is three-diagonal n t := U t n.

26 Evolution of basis d dt n t = B t n t, B t := ( ) d dt U t U t.

27 Instead of considering the evolution of the basis, one can consider the evolution of H H t := U t HU t = n,m=0 (J t ) nm m n d dt H t = [H t, B t ] with condition that 0 is cyclic for all H t and J t is three-diagonal

28 Let H t depend on infinite number of times t = (t 1, t 2,...) Toda lattice equations t k H t = [H t, B k t ] B k t := H k t P 0 (H k t ) 2P + (H k t ) P 0 (H k t ) is diagonal operator P 0 (H k t ) := (Jt k ) nn n n n=0 P + (H k t ) is upper-triangular operator n 1 P + (H k t ) := (Jt k ) mn m n n=1 m=0

29 Moments σ k (t) satisfy the equations t l σ k (t) = 2(σ k+l (t) σ k (t)σ l (t)) Since t k σ l (t) = t l σ k (t) then there exists a function τ(t) = τ(t 1, t 2,...) such that Evolution equation in terms of τ σ k (t) = 1 log τ(t) 2 t k τ(t) = 2 τ(t) t k t l t k+l

30 Equation on measure µ t t k µ t (dλ) = Solution of evolution equations τ(t) = τ(0) σ k (t) = µ t (dλ) = R e2 P ( ) λ k γ k µ t (dγ) µ t (dλ) R R e 2 P l=1 t lλ l µ 0 (dλ) e 2 P l=1 t lλ l R e2 P l=1 t lγ l µ 0 (dγ) µ 0(dλ) 1 l=1 t lγ l µ 0 (dγ) R λ k e 2 P l=1 t lλ l µ 0 (dλ),

31 Evolution equation on structural function ( 1 R t (Q) = 2R t (Q) t l R t (Q)R t (qq)... R t (q l 1 Q) ) 1 R t (qq)r t (q 2 Q)... R t (q l Q) Hierarchy of equations on annihilation and creation operators t l A 1 t = [(A l 1 ta l 1 t) 1, A 1 t ] t l A 2 t = [(A l 2 ta l 2 t) 1, A 2 t ] t l A 1 t = [(A l 1 ta l 1 t) 1, A 1 t] t l A 2 t = [(A l 2 ta l 2 t) 1, A 2 t]

32 Proposition Provided that lim σ n 1(0) σ n(0) exists, lim σ n 1(t) σ n(t) exists for all t. Thus if A 1 = T for t = 0 it is also true for any t.

33 Naum Il ich Akhiezer. The classical moment problem and some related questions in analysis. Oliver & Boyd, Edinburgh, Roelof Koekoek and René F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report DU 98-17, Delft University of Technology, koekoek/askey.html, Anatol Odzijewicz. Quantum algebras and q-special functions related to coherent states maps of the disc. Commun. Math. Phys., 192: , Anatol Odzijewicz and Tudor S. Ratiu. The Banach Poisson geometry of the multi-diagonal Toda-like lattices. math.sg/ , Tomasz Goliński