Reducing and Invariant subspaces under two commuting shift operators

AGUILERA, A.; CABRELLI, C.; CARBAJAL, D.; PATERNOSTRO, V.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2023 vol. 528

0022-247X

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-Halmo´s theorems on characterizations of the invariance for the bilateral and unilateral shifts. The motivations for our study were inspired by recent results on Dynamical Sampling in shift-invariant spaces.