An Abstract Beurling's Theorem for Several Complex Variables I

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.