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An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions
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...
Published in:  Russian Mathematical Surveys 

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2015

URI:  https://cronfa.swan.ac.uk/Record/cronfa22143 
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20190523T08:25:07.6961701 v2 22143 20150622 An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions e5b4fef159d90a480b1961cef89a17b7 0000000196857727 Eugene Lytvynov Eugene Lytvynov true false dbec195692a77f629e935ca8f4efa502 Irina Rodionova Irina Rodionova true false 20150622 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 operatorvalued 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 Meixnertype 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 20151231 10.1070/RM2015v070n05ABEH004965 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 20190523T08:25:07.6961701 20150622T17:07:41.3424203 Faculty of Science and Engineering School of Mathematics and Computer Science  Mathematics Marek Bozejko 1 Eugene Lytvynov 0000000196857727 2 Irina Rodionova 3 
title 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions 
spellingShingle 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions Eugene Lytvynov Irina Rodionova 
title_short 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions 
title_full 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions 
title_fullStr 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions 
title_full_unstemmed 
An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions 
title_sort 
An extended anyon Fock space and noncommutative Meixnertype 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 
author2 
Marek Bozejko Eugene Lytvynov Irina Rodionova 
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Journal article 
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Russian Mathematical Surveys 
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70 
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2015 
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Swansea University 
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10.1070/RM2015v070n05ABEH004965 
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Faculty of Science and Engineering 
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Faculty of Science and Engineering 
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Faculty of Science and Engineering 
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School of Mathematics and Computer Science  Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer 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 operatorvalued 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 Meixnertype 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 
20151231T03:27:01Z 
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1756416412688056320 
score 
10.926618 