Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line
type: Item
instance of: scholarly article
title: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line
zbMATH Open document ID: 1012.46033
DOI: 10.1016/S0022-247X(02)00052-5
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-09-17T00:00:00Z
review text: The ordinary Fock space is the Hilbert space \(F\) of entire functions \(f(z)= \sum^\infty_{n=0} a_nz^n\), \(g(z)= \sum^\infty_{n=0} c_nz^n\) with inner product \[ (f,g)= \sum^\infty_{n=0} a_n\overline c_nn! \] The present paper studies a generalized Fock space \(F_\alpha\) for \(\alpha> -1/2\) in which the inner product is \[ (f,g)_\alpha= \sum^\infty_{n=0} a_n\overline c_nb_n (\alpha), \] where \(b_n\) is a quotient of gamma functions. The authors show that the Dunkl operator \[ (\Lambda_\alpha f)(z)={d\over dz}f(z) +{2\alpha+ 1\over z} \bigl(f(z)- f(-z)\bigr)/2 \] defines a reproducing kernel \(K_\alpha(w,z)\) in \(F_\alpha\) such that \[ \bigl(f(z), K_\alpha(w,z) \bigr)_\alpha= f(w). \] Among other results, they show that \((f,f)\leq (f,f)_\alpha\) so that \(F_\alpha\) is a subspace of \(F\). They study the multiplication operator \(Q\) on \(F_\alpha\) defined by \((Qf)(z)= zf(z)\) and show that \((\Lambda_\alpha f,g)_\alpha =(f, Qg)_\alpha\), so that \(\Lambda_\alpha\) and \(Q\) are adjoints of one another. This leads to various results on commutator relations and generalized Weyl relations between \(\Lambda_\alpha\) and \(Q\).
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 1801634
zbMATH Keywords: Dunkl operator
zbMATH Keywords: Fock space
zbMATH Keywords: generalized Weyl relations
zbMATH Keywords: chaotic transform
author: Mohamed Sifi
author: Fethi Soltani
reviewed by: James V. Whittaker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1801634