On local non-tangential growth of the resolvent of a banded Toeplitz operator

TL;DR

This paper investigates the local growth behavior of the resolvent of banded Hardy--Toeplitz operators near boundary points of their spectrum, revealing inverse linear growth in specific non-tangential regions.

Contribution

It establishes the inverse linear growth rate of the resolvent near boundary points, except for a finite set, for banded Toeplitz operators with Laurent polynomial symbols.

Findings

Resolvent growth is inverse linear near most boundary points.

Growth behavior is analyzed in non-tangential domains.

Finite exceptional set where growth behavior differs.

Abstract

We study the growth of the resolvent of a Hardy--Toeplitz operator $T_b$ with a Laurent polynomial symbol (\emph{i.e., } the matrix $T_b$ is banded), at the neighborhood of a point $w_0\in\partial(\sigma(T_b))$ on the boundary of its spectrum. We show that such growth is inverse linear in some non-tangential domains at the vertex $w_0$, provided that $w_0$ does not belong to a certain finite set on the complex plane.