The algebra of multiplication operators on Bergman spaces
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 \(H^{\infty}(\Omega)\).
Banach algebra of bounded, holomorphic functions
scientific article; zbMATH DE number 3959926
scientific article