de Branges spaces contained in some Banach spaces of analytic functions
type: Item
instance of: scholarly article
title: de Branges spaces contained in some Banach spaces of analytic functions
zbMATH Open document ID: 0844.47017
author: D. C. Singh
author: Sanjeev Agrawal
published in: Illinois Journal of Mathematics
publication date: 1996-08-29T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 856547
zbMATH Keywords: no continuity assumptions are made on the inclusion relation
zbMATH Keywords: invariant subspace theorem
zbMATH Keywords: Hardy space
zbMATH Keywords: Banach spaces of analytic functions on the unit disk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 856547