An Abstract Beurling's Theorem for Several Complex Variables I

TL;DR

This paper generalizes Beurling's theorem to several complex variables by developing an abstract framework for valuation Hilbert modules, leading to a complete description of invariant subspaces in multivariable Hardy spaces.

Contribution

It introduces an abstract Beurling's theorem for valuation Hilbert modules, extending classical results to multiple complex variables and providing a comprehensive description of invariant subspaces.

Findings

Complete description of invariant subspaces of $H^2$ in the polydisk

Extension of Beurling's theorem to valuation Hilbert modules

Framework applicable to various Hilbert spaces of analytic functions

Abstract

How to extend Beurling's theorem on the shift invariant subspaces of Hardy class $H^2$ of the unit disk to several complex variables has been an open problem at least since 1964. In this paper, we prove a generalization of Beurling's theorem to valuation Hilbert modules over valuation algebras. We shall apply our abstract Beurling's theorem to obtain a complete description of the closed invariant subspaces of $H^2$ of the polydisk. As we shall show in a subsequent paper, further consequences of our abstract Beurling's theorem are complete extensions of Beurling's theorem to many more Hilbert spaces of analytic functions in several complex variables.