Doubly commuting submodules of the Hardy module over polydiscs

Abstract

In this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in $H^\infty(\mathbb{D}^n)$.

14 pages, revised, to appear in Studia Mathematica