A Toeplitz-like operator with rational symbol having poles on the unit circle II: the spectrum

TL;DR

This paper analyzes the spectral properties of a class of Toeplitz-like operators with rational symbols having poles on the unit circle, detailing the spectrum, its parts, and examples illustrating these properties.

Contribution

It provides a detailed description of the spectrum, including point, residual, continuous, and essential spectrum, for Toeplitz-like operators with poles on the unit circle.

Findings

The spectrum can be disconnected in the complex plane.

The essential spectrum's structure is characterized.

Examples illustrate the spectral properties.

Abstract

This paper is a continuation of our study of a class of Toeplitz-like operators with a rational symbol which has a pole on the unit circle. A description of the spectrum and its various parts, i.e., point, residual and continuous spectrum, is given, as well as a description of the essential spectrum. In this case, the essential spectrum need not be connected in ${\mathbb C}$. Various examples illustrate the results.