Reducing and Invariant subspaces under two commuting shift operators

TL;DR

This paper characterizes subspaces that are invariant or reducing under two commuting shift operators in Hardy spaces, extending classical theorems and motivated by recent dynamical sampling research.

Contribution

It provides new characterizations of reducing and invariant subspaces for two commuting shifts, generalizing Helson and Beurling-Lax-Halmos theorems.

Findings

Conditions for invariance and reduction under bilateral and unilateral shifts

Extension of classical shift theorems to Hardy spaces with multiplicity

Connections to recent dynamical sampling results

Abstract

In this article, we characterize reducing and invariant subspaces of the space of square integrable functions defined in the unit circle and having values in some Hardy space with multiplicity. We consider subspaces that reduce the bilateral shift and at the same time are invariant under the unilateral shift acting locally. We also study subspaces that reduce both operators. The conditions obtained are of the type of the ones in Helson and Beurling-Lax-Halmos theorems on characterizations of the invariance for the bilateral and unilateral shift. The motivations for our study were inspired by recent results on Dynamical Sampling in shift-invariant spaces.