The algebra of multiplication operators on Bergman spaces
type: Item
instance of: scholarly article
title: The algebra of multiplication operators on Bergman spaces
zbMATH Open document ID: 0596.32010
DOI: 10.1007/BF01189287
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Omega\) be a connected domain in \({\mathbb{C}}^ m\), \(m\geq 1\). We will denote by \(H^{\infty}(\Omega)\) the Banach algebra of bounded, holomorphic functions defined in \(\Omega\), and by \(A^ 2(\Omega)\) the Bergman space associated with \(\Omega\). Each function \(\phi \in H^{\infty}(\Omega)\) determines a bounded multiplication operator \(M_{\phi}\) on \(A^ 2(\Omega)\) by the formula \((M_{\phi}f)(\omega)=\phi (\omega)f(\omega),\) \(f\in A^ 2(\Omega)\), \(\omega\in \Omega\). We study the algebra \({\mathcal A}(\Omega)=\{M_{\phi}: \phi \in H^{\infty}(\Omega)\}\) assuming that \(A^ 2(\Omega)\neq \{0\}\) and \(H^{\infty}(\Omega)\) contains nonconstant functions. We show that \({\mathcal A}(\Omega)\) has a rich supply of invariant subspaces, paralleling the known case in which \(\Omega\) is the unit disc in \({\mathbb{C}}\). In particular, \({\mathcal A}(\Omega)\) is weakly closed and reflexive, and the representation \(\phi \to M_{\phi}\) compresses to many other representations of \(H^{\infty}(\Omega)\).
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 3959926
zbMATH Keywords: Banach algebra of bounded, holomorphic functions
zbMATH Keywords: Bergman space
zbMATH Keywords: algebra of multiplication operators
author: H. Bercovici
MaRDI profile type: Publication
cites work: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I
cites work: A Reflexivity Theorem for Weakly Closed Subspaces of Operators
cites work: Q5800023
cites work: Q3896064
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description: scientific article; zbMATH DE number 3959926