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

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.