Representing Kernels of Perturbations of Toeplitz Operators by Backward Shift-Invariant Subspaces

It is well known 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.