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 Uspekhi Mat. Nauk, 2015, Volume 70, Issue 5(425), Pages 75–120 (Mi umn9668)

This article is cited in 7 scientific papers (total in 7 papers)

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

M. Bożejkoa, E. W. Lytvynovb, I. V. Rodionovab

a Institute of Mathematics, Wrocław University, Wrocław, Poland
b Swansea University, Swansea, UK

Abstract: Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighbourhood of zero. An anyon Lévy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ on the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$, where $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in the variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. Noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$ are constructed, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$, and are then used to derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space $\mathbf F(L^2(\mathbb R^d,dx))$ over $L^2(\mathbb R^d,dx)$. The usual anyon Fock space $\mathscr F(L^2(\mathbb R^d,dx))$ over $L^2(\mathbb R^d,dx)$ is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathscr F(L^2(\mathbb R^d,dx))$ holds if and only if the measure $\nu$ is concentrated at a single point, that is, in the Gaussian or Poisson case. With use of the unitary isomorphism $U$, the operators $\langle \omega,\varphi\rangle$ are realized as a Jacobi (that is, tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. A Meixner-type class of anyon Lévy white noise is derived for which the corresponding Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon Lévy white noise of Meixner type is characterized by two parameters, $\lambda\in\mathbb R$ and $\eta\geqslant0$. In conclusion, the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x +\eta\partial_x^\dag\partial_x\partial_x+\partial_x$ is obtained, where $\partial_x$ and $\partial_x^\dag$ are the annihilation and creation operators at the point $x$.
Bibliography: 57 titles.

Keywords: anyon commutation relations, anyon Fock space, gamma process, Jacobi field, Lévy white noise, Meixner class of orthogonal polynomials.

 Funding Agency Grant Number National Science Centre (Narodowe Centrum Nauki) Dec-2012/05/B/ST1/00626Dec-2011/02/A/ST1/00119 Universität Bielefeld SFB 701 The work of the first and second authors was carried out with the financial support of the Polish National Science Center (grant no. Dec-2012/05/B/ST1/00626) and of the Collaborative Research Centre at Bielefeld University (SFB 701), in the framework of the programme "Spectral structures and topological methods in mathematics". The first author was also partially supported by the MAESTRO programme (grant no. Dec-2011/02/A/ST1/00119).

DOI: https://doi.org/10.4213/rm9668

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English version:
Russian Mathematical Surveys, 2015, 70:5, 857–899

Bibliographic databases:

UDC: 517.98
MSC: Primary 46L53, 60G51, 60H40; Secondary 33C45
Received: 01.12.2014

Citation: M. Bożejko, E. W. Lytvynov, I. V. Rodionova, “An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions”, Uspekhi Mat. Nauk, 70:5(425) (2015), 75–120; Russian Math. Surveys, 70:5 (2015), 857–899

Citation in format AMSBIB
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This publication is cited in the following articles:
1. E. Lytvynov, “Gauge-invariant quasi-free states on the algebra of the anyon commutation relations”, Comm. Math. Phys., 351:2 (2017), 653–687
2. M. M. Frei, N. A. Kachanovsky, “Some remarks on operators of stochastic differentiation in the Lévy white noise analysis”, Methods Funct. Anal. Topology, 23:4 (2017), 320–345
3. L. Accardi, W. Ayed, “Free white noise flows”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20:3 (2017), 1750014, 33 pp.
4. E. Lytvynov, I. Rodionova, “Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 21:2 (2018), 1850011, 21 pp.
5. N. A. Kachanovsky, “On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis”, Carpathian Math. Publ., 10:1 (2018), 114–132
6. A. Boussayoud, A. Abderrezzak, S. Araci, “A new symmetric endomorphism operator for some generalizations of certain generating functions”, Notes Number Theory Discret. Math., 24:4 (2018), 45–58
7. Kachanovsky N.A., Kachanovska T.O., “Interconnection Between Wick Multiplication and Integration on Spaces of Nonregular Generalized Functions in the Levy White Noise Analysis”, Carpathian Math. Publ., 11:1 (2019), 70–88
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