A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain

TL;DR

This paper extends the Beurling theorem to generalized Hardy spaces on multiply connected domains, characterizing invariant subspaces, cyclic vectors, and operator commutativity in this broader context.

Contribution

It proves a Beurling-Helson-Lowdenslager type theorem for Banach spaces of functions on multiply connected domains, including Hardy spaces with various norms.

Findings

Invariant subspace characterization established

Cyclic vectors identified as outer functions

Analytic multiplication operators shown to be maximal abelian and reflexive

Abstract

The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in the complex plane. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.