Beurling's Theorem for Valuation Hilbert Modules and Several Complex Variables

TL;DR

This paper extends Beurling's theorem to Valuation Hilbert Modules and applies it to characterize invariant subspaces in various multivariable analytic function spaces.

Contribution

It introduces Valuation Hilbert Modules and proves a Beurling-type theorem for them, enabling new descriptions of invariant subspaces in several complex variables.

Findings

Complete descriptions of invariant subspaces in $H^2$ of the polydisk and ball.

Extension of Beurling's theorem to Valuation Hilbert Modules.

Applications to weighted $A^2$ spaces on complex manifolds.

Abstract

We develop a theory of Valuation Hilbert Modules and prove a version of Beurling's theorem for these. Then we apply our version of Beurling's theorem to obtain complete descriptions of the closed invariant subspaces of a number of Hilbert spaces of analytic functions in several complex variables, including $H^2$ of the polydisk the ball, and bounded symmetric domains, and weighted $A^2$ spaces on complex analytic manifolds.