The density problem for unbounded Bergman operators
type: Item
instance of: scholarly article
title: The density problem for unbounded Bergman operators
zbMATH Open document ID: 1037.32006
author: Sherwin Kouchekian
published in: Integral Equations and Operator Theory
publication date: 2003-01-01T00:00:00Z
review text: Let \(G\) be a domain in the complex plane, \(L^2_a(G)\) the subspace of all holomorphic functions in \(L^2(G)\), and \(S: f(z)\mapsto zf(z)\) the operator of multiplication by the independent variable, defined on dom\(\,S:=\{f\in L^2_a(G):\,zf\in L^2_a(G)\}\). The author gives a number of results concerning the density of dom\(\,S\) in \(L^2_a(G)\). In~particular, if \(L^2_a(G)\neq\{0\}\) and \(D(a,r)\) is a disc whose closure is contained in~\(G\), then dom\(\,S\) is dense in \(L^2_a(G)\) if and only if \(0\) is not a bounded point evaluation for \(L^2_a(\mathbb D\setminus K)\), where \(K=\{0\} \cup\{\frac r{z-a},\,z\in\mathbb C\setminus G\}\). This implies, for instance, that \(S\) is densely defined whenever the connected component of \(\mathbb C\cup\{\infty\}\setminus G\) containing \(\infty\) does not reduce to the singleton \(\{\infty\}\), or whenever Area\((\mathbb C\setminus G)=+\infty\). An equivalent characterization in terms of Wiener capacity is also given. On~the other hand, if \(K\subset\mathbb C\) is compact and \(G=\mathbb C\setminus K\), then dom\(\,S\) is not dense in \(L^2_a(G)\).
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1925524
zbMATH Keywords: unbounded subnormal operators
zbMATH Keywords: bounded point evaluations
zbMATH Keywords: logarithmic capacity
zbMATH Keywords: Wiener capacity
reviewed by: Miroslav Engliš
MaRDI profile type: Publication
full work available at URL: s000200300008
OpenAlex ID: W1995559164
DOI: 10.1007/S000200300008
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description: scientific article; zbMATH DE number 1925524