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Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

Eugene Lytvynov Orcid Logo, Irina Rodionova

Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 21, Issue: 02, Start page: 1850011

Swansea University Authors: Eugene Lytvynov Orcid Logo, Irina Rodionova

Abstract

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-spa...

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Published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics
ISSN: 0219-0257 1793-6306
Published: 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa36409
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spelling 2020-07-28T11:47:14.2838420 v2 36409 2017-10-30 Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false dbec195692a77f629e935ca8f4efa502 Irina Rodionova Irina Rodionova true false 2017-10-30 SMA Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$. Journal Article Infinite Dimensional Analysis, Quantum Probability and Related Topics 21 02 1850011 0219-0257 1793-6306 Monotone independence, monotone Levy noise, monotone Levy process, Meixner class of orthogonal polynomials. 18 6 2018 2018-06-18 10.1142/S021902571850011X COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-07-28T11:47:14.2838420 2017-10-30T14:42:27.1149467 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Eugene Lytvynov 0000-0001-9685-7727 1 Irina Rodionova 2 0036409-30102017144510.pdf Monotone_Meixner.pdf 2017-10-30T14:45:10.7770000 Output 307591 application/pdf Accepted Manuscript true 2019-06-18T00:00:00.0000000 true eng
title Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
spellingShingle Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
Eugene Lytvynov
Irina Rodionova
title_short Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
title_full Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
title_fullStr Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
title_full_unstemmed Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
title_sort Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
author_id_str_mv e5b4fef159d90a480b1961cef89a17b7
dbec195692a77f629e935ca8f4efa502
author_id_fullname_str_mv e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
dbec195692a77f629e935ca8f4efa502_***_Irina Rodionova
author Eugene Lytvynov
Irina Rodionova
author2 Eugene Lytvynov
Irina Rodionova
format Journal article
container_title Infinite Dimensional Analysis, Quantum Probability and Related Topics
container_volume 21
container_issue 02
container_start_page 1850011
publishDate 2018
institution Swansea University
issn 0219-0257
1793-6306
doi_str_mv 10.1142/S021902571850011X
college_str Faculty of Science and Engineering
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hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
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department_str 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 $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$.
published_date 2018-06-18T03:44:49Z
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