Invariant subspaces of perturbed backward shift

TL;DR

This paper characterizes invariant subspaces of the Hardy space under finite-rank perturbations of the backward shift, providing new classifications and representations of nearly invariant subspaces and their relation to perturbed Toeplitz operators.

Contribution

It introduces a detailed classification of invariant and nearly invariant subspaces under finite-rank backward shift perturbations, expanding the understanding of their structure.

Findings

Closed subspaces invariant under finite-rank perturbations are characterized.

Nearly invariant subspaces are represented in a refined manner.

Kernels of certain perturbed Toeplitz operators are examples of nearly invariant subspaces.

Abstract

We represent closed subspaces of the Hardy space that are invariant under finite-rank perturbations of the backward shift. We apply this to classify almost invariant subspaces of the backward shift and represent a more refined version of nearly invariant subspaces. Kernels of certain perturbed Toeplitz operators are examples of the newly introduced nearly invariant subspaces.