Variations on a theme of Beurling

TL;DR

This paper explores the Beurling-Lax-Halmos Theorem within the framework of Hilbert modules, extending its interpretations to classical and multivariate spaces like Hardy, Bergman, and Drury-Arveson spaces.

Contribution

It provides new interpretations and extensions of the Beurling-Lax-Halmos Theorem using Hilbert module language across various function spaces.

Findings

Extended the theorem to multivariate spaces

Connected invariant subspaces with Hilbert module theory

Provided new insights into classical and modern function spaces

Abstract

Interpretations of the Beurling-Lax-Halmos Theorem on invariant subspaces of the unilateral shift are explored using the language of Hilbert modules. Extensions and consequences are considered in both the one and multivariate cases with an emphasis on the classical Hardy, Bergman and Drury-Arveson spaces.