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
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
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.. 2.......
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.
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.
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
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),
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.
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
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
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
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
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 )
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
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.
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 )
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
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
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 )
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
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 + 1 + α)(n + 1 + β)(n + 1 + 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
Example (Classical Jacobi polynomials) Structural function Structural relations R(q x ) = α + β + 1 + x (1 χ β + x {1} (x)) (β + N)A 1A 1 = (α + β + 1 + N)(1 0 0 ) (β + 1 + N)A 1 A 1 = α + β + 2 + N
Example (Classical Jacobi polynomials) Exponential functions E 1 (z) = 2 F 1 ( β + 1 1 α + β + 2 E 2 (z) = 2 F 1 ( α + β + 2 1 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
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
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.
Evolution of basis d dt n t = B t n t, B t := ( ) d dt U t U t.
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
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
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
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),
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]
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.
Naum Il ich Akhiezer. The classical moment problem and some related questions in analysis. Oliver & Boyd, Edinburgh, 1965. 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, http://aw.twi.tudelft.nl/ koekoek/askey.html, 1998. Anatol Odzijewicz. Quantum algebras and q-special functions related to coherent states maps of the disc. Commun. Math. Phys., 192:183 215, 1998. Anatol Odzijewicz and Tudor S. Ratiu. The Banach Poisson geometry of the multi-diagonal Toda-like lattices. math.sg/0310318, Tomasz 2006. Goliński