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

Abstract

It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^{\ast }$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^{\ast }$-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.

17 pages. Extensive revisions and corrections. Minor change to the title