Lowering and Raising Operators for the Free Meixner Class of Orthogonal Polynomials

Lytvynov, Eugene; Rodionova, Irina

doi:10.1142/s0219025709003720

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…Результаты, рассмотренные выше, имеют некоммутативные аналоги в свободной теории вероятностей [16], [17] (см. также [4], [5], [10], [12], [13], [43] и приведенные там ссылки). По поводу дальнейших связей между классическими распределениями из класса Мейкснера и свободной теорией вероятностей см.…”

Section: Doi: 104213/rm9668unclassified

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Божейко¹,

Bożejko²,

Литвинов³

et al. 2015

Uspekhi Mat. Nauk

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Section: Doi: 104213/rm9668unclassified

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Божейко¹,

Bożejko²,

Литвинов³

et al. 2015

Uspekhi Mat. Nauk

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“…t , cf. [14]. In the infinite-dimensional case, we define operators of free differentiation by setting, for each t ∈ T ,…”

Section: A Globality Of the Annihilation Operatorsmentioning

confidence: 99%

Meixner Class of Non-commutative Generalized Stochastic Processes with Freely Independent Values II. The Generating Function

Bożejko

Lytvynov

2010

Commun. Math. Phys.

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Let T be an underlying space with a non-atomic measure σ on it. In [Comm. Math. Phys. 292 (2009), 99-129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, ω = (ω(t)) t∈T , was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables?We construct a class of operator-valued functions Z = (Z(t)) t∈T such that Z(t) commutes with ω(s) for any s, t ∈ T . Then a generating function can be understood as G(Z, ω) = ∞ n=0 T n P (n) (ω(t 1 ), . . . , ω(t n ))Z(t 1 ) · · · Z(t n )σ(dt 1 ) · · · σ(dt n ), where P (n) (ω(t 1 ), . . . , ω(t n )) is (the kernel of the) n-th orthogonal polynomial. We derive an explicit form of G(Z, ω), which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators ∂ t , t ∈ T . In contrast to the classical case, we prove that the operators ∂ t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality. Construction of integral of an operator-valued functionwith respect to an operator-valued measure Let G be a real separable Hilbert space, and let L (G) denote the Banach space of all bounded linear operators in G. We will call a mapping Z : T → L (G) simple if it has

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An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Bożejko¹,

Lytvynov²,

Rodionova³

2015

Russ. Math. Surv.

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Let ν be a finite measure on R whose Laplace transform is analytic in a neighborhood of zero. An anyon Lévy white noise on (R d , dx) is a certain family of noncommuting operators ω, ϕ in the anyon Fock space over L 2 (R d × R, dx ⊗ ν). Here ϕ = ϕ(x) runs over a space of test functions on R d , while ω = ω(x) is interpreted as an operator-valued distribution on R d . Let L 2 (τ ) be the noncommutative L 2 -space generated by the algebra of polynomials in variables ω, ϕ , where τ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in L 2 (τ ) of the form P n (ω), f (n) , where f (n) is a test function on (R d ) n . Using these orthogonal polynomials, we derive a unitary isomorphism U between L 2 (τ ) and an extended anyon Fock space over dx)) if and only if the measure ν is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism U , we realize the operators ω, ϕ as a Jacobi (i.e., tridiagonal) field in F(L 2 (R d , dx)). We derive a Meixner-type class of anyon Lévy white noise for which the respective Jacobi field in F(L 2 (R d , dx)) has a relatively simple structure. Each anyon Lévy white noise of the Meixner type is characterized by two parameters: λ ∈ R and η ≥ 0. Furthermore, we get the representation ω(x) = ∂ † x + λ∂ † x ∂ x + η∂ † x ∂ x ∂ x + ∂ x . Here ∂ x and ∂ † x are annihilation and creation operators at point x.

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Lowering and Raising Operators for the Free Meixner Class of Orthogonal Polynomials

Abstract: We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

Cited by 3 publications

References 17 publications

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Meixner Class of Non-commutative Generalized Stochastic Processes with Freely Independent Values II. The Generating Function

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

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