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An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions / Eugene, Lytvynov; Irina, Rodionova

Russian Mathematical Surveys, Volume: 70, Issue: 5, Start page: 857

Swansea University Authors: Eugene, Lytvynov, Irina, Rodionova

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DOI (Published version): 10.1070/RM2015v070n05ABEH004965

Abstract

Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R...

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Published in: Russian Mathematical Surveys
Published: 2015
URI: https://cronfa.swan.ac.uk/Record/cronfa22143
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spelling 2019-05-23T08:25:07.6961701 v2 22143 2015-06-22 An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false dbec195692a77f629e935ca8f4efa502 Irina Rodionova Irina Rodionova true false 2015-06-22 SMA Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ 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 variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon L\'evy white noise of the Meixner type is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. Furthermore, we get the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x \eta\partial_x^\dag\partial_x\partial_x+\partial_x$.Here $\partial_x$ and $\partial_x^\dag$ are annihilation and creation operators at point $x$. Journal Article Russian Mathematical Surveys 70 5 857 31 12 2015 2015-12-31 10.1070/RM2015v070n05ABEH004965 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2019-05-23T08:25:07.6961701 2015-06-22T17:07:41.3424203 College of Science Mathematics Marek Bozejko 1 Eugene Lytvynov 0000-0001-9685-7727 2 Irina Rodionova 3
title An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
spellingShingle An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
Eugene, Lytvynov
Irina, Rodionova
title_short An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
title_full An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
title_fullStr An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
title_full_unstemmed An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
title_sort An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions
author_id_str_mv e5b4fef159d90a480b1961cef89a17b7
dbec195692a77f629e935ca8f4efa502
author_id_fullname_str_mv e5b4fef159d90a480b1961cef89a17b7_***_Eugene, Lytvynov
dbec195692a77f629e935ca8f4efa502_***_Irina, Rodionova
author Eugene, Lytvynov
Irina, Rodionova
format Journal article
container_title Russian Mathematical Surveys
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container_issue 5
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publishDate 2015
institution Swansea University
doi_str_mv 10.1070/RM2015v070n05ABEH004965
college_str College of Science
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department_str Mathematics{{{_:::_}}}College of Science{{{_:::_}}}Mathematics
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description Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ 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 variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon L\'evy white noise of the Meixner type is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. Furthermore, we get the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x \eta\partial_x^\dag\partial_x\partial_x+\partial_x$.Here $\partial_x$ and $\partial_x^\dag$ are annihilation and creation operators at point $x$.
published_date 2015-12-31T03:37:44Z
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