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.

31 pages. Changed some terminology and added material on $R_1$ inner functions. v3: Added additional matrial on $R_1$ inner functions and their relation to $A_{\alpha }^p$ inner functions. v4 : Revised terminology for valuation Hilbert modules. Added complete proofs of the Minimum Value Theorem, and the 4th Projection Lemma, plus other additions and corrections. arXiv admin note: text overlap with arXiv:2109.00695, arXiv:2108.12272