Hilbert-Schmidt Hankel operators over complete Reinhardt domains
type: Item
instance of: scholarly article
title: Hilbert-Schmidt Hankel operators over complete Reinhardt domains
zbMATH Open document ID: 1318.47047
author: Trieu Le
published in: Integral Equations and Operator Theory
publication date: 2014-10-06T00:00:00Z
review text: Given a bounded domain in \({\mathbb C}^n\), the Bergman space \(A^2(\Omega)\) over \(\Omega\) is the Hilbert space consisting of all holomorphic functions in \(L^2(\Omega)\), the Lebesgue space with respect to the Lebesgue volume measure over \(\Omega\). Given \(\varphi\in L^2(\Omega)\), the Hankel operator \(H_\varphi: A^2(\Omega)\to L^2(\Omega)\ominus A^2(\Omega)\) is defined by \(H_\varphi h = (I-P)(\varphi h)\) for all \(h\in A^2(\Omega)\) for which \(\varphi h \in L^2(\Omega)\). In this paper, the author considers operators of the form \(H_{\overline f}\), where \(f\) is holomorphic and investigates when they are Hilbert-Schmidt. In the case of dimension \(1\), it was proved by \textit{J. Arazy} et al. [J. Reine Angew. Math. 406, 179--199 (1990; Zbl 0686.47023)] that \(H_{\overline f}\) is Hilbert-Schmidt if and only if \(f'\in A^2(\Omega)\). In higher dimensions \((n \geq 2)\), the situation becomes quite different. In fact, in the setting of the unit ball of \({\mathbb C}^n\), it was proved by \textit{K. Zhu} [Proc. Am. Math. Soc. 109, No. 3, 721--730 (1990; Zbl 0731.47028)] that \(H_{\overline f}\) is Hilbert-Schmidt if and only if \(f\) is constant. In this paper, the author extends Zhu's result to the complete Reinhardt domains by means of a quite elementary approach. The author also indicates how his approach can be applied to certain general settings.
reviewed by: Boo Rim Choe
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 32A36
zbMATH DE Number: 6351693
zbMATH Keywords: Bergman space
zbMATH Keywords: Hankel operator
zbMATH Keywords: Hilbert-Schmidt operator
zbMATH Keywords: \(\overline\partial\)-Neumann operator
MaRDI profile type: Publication
full work available at URL: s00020-013-2103-z
OpenAlex ID: W2041089811
cites work: Hankel Operators on Weighted Bergman Spaces
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cites work: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains
cites work: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complex ellipsoids
cites work: Q4520911
cites work: The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces
cites work: Q4840755
cites work: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains
cites work: Q3633063
cites work: On the Schatten class membership of Hankel operators on the unit ball
cites work: Hilbert-Schmidt Hankel Operators on the Bergman Space
cites work: Schatten Class Hankel Operators on the Bergman Space of the Unit Ball
DOI: 10.1007/S00020-013-2103-Z
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description: scientific article; zbMATH DE number 6351693