Fock space associated to Coxeter groups of type B

Marek Bożejko , Wiktor Ejsmont , Takahiro Hasebe

Abstract

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an (α,q)(α,q)-Fock space, which satisfy the commutation relation View the MathML sourcebα,q(x)bα,q⁎(y)−qbα,q⁎(y)bα,q(x)=〈x,y〉I+α〈x‾,y〉q2N, where x,yx,y are elements of a complex Hilbert space with a self-adjoint involution View the MathML sourcex↦x¯ and N is the number operator with respect to the grading on the (α,q)(α,q)-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators View the MathML sourcebα,q(x)+bα,q⁎(x) with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution is associated to the orthogonal polynomials called the q-Meixner–Pollaczek polynomials, yielding the q -Hermite polynomials when α=0α=0 and free Meixner polynomials when q=0q=0.
Author Marek Bożejko
Marek Bożejko,,
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, Wiktor Ejsmont (MISaF / IZM / KMiC)
Wiktor Ejsmont,,
- Katedra Matematyki i Cybernetyki
, Takahiro Hasebe
Takahiro Hasebe,,
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Journal seriesJournal of Functional Analysis, ISSN 0022-1236, (A 40 pkt)
Issue year2015
Vol269
No6
Pages1769-1795
Publication size in sheets1.3
Keywords in English Noncommutative probability; q-Gaussian process; Fock spaces
DOIDOI:10.1016/j.jfa.2015.06.026
URL http://www.sciencedirect.com/science/journal/00221236/269/6?sdc=1
Languageen angielski
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Bozejko_Ejsmont_Hasebe_Fock_space_associated2015.pdf 487,43 KB
Score (nominal)40
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