Representing kernels of perturbations of Toeplitz operators by backward shift-invariant subspaces

TL;DR

This paper investigates the structure of kernels of finite-rank perturbations of Toeplitz operators, showing they are nearly backward shift-invariant with finite defect and representing them via shift-invariant subspaces.

Contribution

It extends the understanding of Toeplitz operator kernels by characterizing perturbations as nearly shift-invariant and applying recent theorems for their representation.

Findings

Kernels of finite-rank perturbations are nearly shift-invariant with finite defect.

Representation of kernels using backward shift-invariant subspaces.

Identification of kernels in several important cases.

Abstract

It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.