Representations of \(H^ \infty(G)\) and invariant subspaces
The author considers the algebra of bounded, complex functions, analytic on \(G\), a bounded, open subset of the complex plane. Representations of this algebra in the algebra of linear, bounded operators acting on a Banach space are considered. The Banach space is supposed to satisfy certain geometrical conditions. In all cases these conditions hold for reflexive Banach spaces. The author proves that if the image under this representation of the identity function \(z\), is an operator with spectrum dominating for \(G\), then, under some supplementary conditions, the image of the algebra of bounded, analytic functions under the given representation is shown to have a rich invariant subspace lattice. Among other things, the author proves this last operator algebra is reflexive.
scientific article; zbMATH DE number 537314
scientific article