Perturbations of Toeplitz operators on vector-valued Hardy spaces

TL;DR

This paper classifies invariant subspaces of finite-rank perturbations of Toeplitz operators on vector-valued Hardy spaces, providing new insights into their structure and related subspace properties.

Contribution

It offers a complete classification of invariant subspaces for these perturbed Toeplitz operators, extending understanding in the vector-valued Hardy space setting.

Findings

Complete classification of invariant subspaces for finite-rank perturbations.

Characterization of invariant and almost invariant subspaces.

Analysis of nearly invariant subspaces with finite defect.

Abstract

In this article, we completely classify invariant subspaces of finite-rank perturbations of a class of Toeplitz operators on vector-valued Hardy spaces. As a consequence, in the vector-valued setting, we characterize invariant and almost invariant subspaces of a class of Toeplitz operators, as well as nearly invariant subspaces associated with certain Blaschke-based operators. We further treat the finite defect case for these nearly invariant subspaces.