Doubly commuting submodules of the Hardy module over polydiscs

TL;DR

This paper extends Beurling's Theorem to vector-valued settings over polydiscs and provides conditions for the completion problem in bounded holomorphic functions on the polydisc.

Contribution

It introduces a vector-valued version of Beurling's Theorem for the polydisc and characterizes the completion problem in $H^(\u2202^n)$.

Findings

Established a vector-valued Beurling's Theorem for the polydisc

Provided necessary and sufficient conditions for the completion problem in $H^(\u2202^n)$

Extended classical results to a multivariable, vector-valued context

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 $H^\infty(\mathbb{D}^n)$.