Linear operators on Fock spaces
type: Item
instance of: scholarly article
title: Linear operators on Fock spaces
zbMATH Open document ID: 1384.30011
author: Senhua Zhu
author: Zengjian Lou
author: Kehe Zhu
published in: Integral Equations and Operator Theory
publication date: 2017-11-10T00:00:00Z
review text: The aim of this paper is to extend several results about properties of some linear operators on the Fock space \(F_\alpha^2\) to linear operators on Fock spaces \(F_\alpha^p\) for \( 0<p<\infty \). The main results of the paper concern the conditions for boundedness and compactness of such operators on \(F_\alpha^p\). The obtained conditions for a linear operator \(T\) on \(F_\alpha^p\) are obtained imposing some boundedness conditions on the operator \(T_a=W_aTW_a\), where \(W_a\) is the Weil operator on \(H(\mathbb C)\) -- the linear space of all entire functions on \(\mathbb C\). Since the vector space \(\mathcal K\) of all finite linear combinations of normalized reproducing kernels is dense in the Fock space \(F_\alpha^p\) for each \(0<p<\infty\), a linear mapping \(T:{\mathcal K}\rightarrow H(\mathbb C)\) is called a bounded operator on \(F_\alpha^p\) if there exists a positive constant \(C\) such that \(\|Tf\|_{p,\alpha}\leq C\|f\|_{p,\alpha}\). Beside specific results on boundedness and compactness obtained in the paper, one should mention here Theorem 3.4 which says that, if \(T:{\mathcal K}\rightarrow H(\mathbb C)\) is linear and has the property that \(\|T_a1\|\rightarrow 0\) as \(a\rightarrow\infty\), and if there exists a positive constant \(C\) such that \(\|T^*_a1\|_{1,\alpha}\leq C\) for all \(a\in\mathbb C\), then for any \(1\leq p<\infty\) the operator \(T\) is bounded on \(F_\alpha^p\) and has the property that \(\|Tf_n\|_{p,\alpha}\rightarrow 0\) as \(a\rightarrow\infty\), whenever \(\{f_n\}\) is a bounded sequence in \(F_\alpha^p\) that converges to zero on compact sets. In particular, \(T\) is compact on \(F_\alpha^p\) for \(1<p<\infty\).
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 47B32
zbMATH DE Number: 6805237
zbMATH Keywords: Fock spaces
zbMATH Keywords: boundedness and compactness of linear operators
zbMATH Keywords: reproducing kernels
MaRDI profile type: Publication
full work available at URL: s00020-017-2381-y
OpenAlex ID: W2619761592
cites work: Compact operators via the Berezin transform
cites work: Compactness characterization of operators in the Toeplitz algebra of the Fock space \(F^p_\alpha\)
cites work: Toeplitz operators with BMO symbols on the Segal-Bargmann space
cites work: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains
cites work: The complex interpolation method preserves compactness of linear operators
cites work: Compact Toeplitz operators on the Segal-Bargmann space
cites work: Hankel forms and the Fock space
cites work: Compact operators on Bergman spaces
cites work: A reproducing kernel thesis for operators on Bergman-type function spaces
cites work: Boundedness and compactness of operators on the Fock space
cites work: Theory of Bergman Spaces in the Unit Ball of C^n
cites work: Toeplitz operators and Hankel operators
cites work: Analysis on Fock Spaces
cites work: Q3447431
cites work: Toeplitz operators with BMO symbols and the Berezin transform
DOI: 10.1007/S00020-017-2381-Y
description: scientific article; zbMATH DE number 6805237