Wandering property in the Hardy space
Let \(H^2\) be the usual Hardy space on the open unit disk \(\mathbb{D}\). The authors study the invariant subspaces of the multiplication operator \(T: f\mapsto gf\) whenever \(g\in H^\infty\). Whereas the case of an inner function \(g\) has been dealt with in \textit{T. Lance} and \textit{M. Stessin} [Can. J. Math. 49, No. 1, 100-118 (1997; Zbl 0879.47015)], they now consider the case of an arbitrary bounded analytic function \(g\). For a subset \(A\subseteq H^2\) let \([A]_g\) denote the minimal closed subspace of \(H^2\) containing \(A\) that is invariant under multiplication by \(g\). The following question is investigated: Let \(M\) be an invariant subspace of \(T\). Under which conditions on \(g\) one has \[ [M\ominus gM]_g= M?\tag{\(*\)} \] The following nice result is proved: Theorem. If \(g\in H^\infty\), \(g(0)= 0\), and \((*)\) holds for any subspace \(M\subseteq H^2\) that is invariant under multiplication by \(g\), then there exists a simply connected domain \(\Omega\) and an inner function \(h\) such that \(g= \varphi_\Omega\circ h\), where \(\varphi_\Omega\) is a Riemann mapping of \(\mathbb{D}\) onto \(\Omega\). Conversely, if \(\Omega\) satisfies the additional condition that \(\varphi_\Omega\) is a weak-\(*\) generator of \(H^\infty\), then \((*)\) holds whenever \(g= \varphi_\Omega\circ h\) (\(h\) inner) and \(M\) is invariant under \(T\).
invariant subspaces of the multiplication operator
scientific article; zbMATH DE number 1127774
scientific article