The density problem for unbounded Bergman operators
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 \(L^2_a(G)\).
scientific article; zbMATH DE number 1925524