de Branges spaces contained in some Banach spaces of analytic functions
\textit{L. de Branges} has proved in Theorem 15 of [`Square summable power series', to appear]\ an invariant subspace theorem which generalizes not only Beurling's famous theorem but also its generalizations due to Lax and Halmos. The scalar version of the theorem says: Theorem A. Let \(M\) be a Hilbert space contractively contained in the Hardy space \(H^2\) of the unit disk \(D\) such that \(S(M) \subset M\) (where \(S\) is the operator of multiplication by the coordinate function \(z\)) and \(S\) acts as an isometry on \(M\). Then there exists a unique \(b\) in the unit ball of \(H^\infty\) such that \(M= b(z) H^2\). Further, \(|bf |_M= |f|_{H^2}\). In this note we characterize those Hilbert spaces \(M\) which are algebraically contained in various Banach spaces of analytic functions on the unit disk \(D\). We drop the contractivity requirement on \(M\) (no continuity assumptions are made on the inclusion relation). Thus even in the particular case of \(M\subset H^2\), we obtain an extension of de Branges theorem by having characterized the class of all Hilbert spaces which are vector subspaces of \(H^2\) and on which \(S\) acts as an isometry.
no continuity assumptions are made on the inclusion relation
Banach spaces of analytic functions on the unit disk
scientific article; zbMATH DE number 856547