A Toeplitz-like operator with rational matrix symbol having poles on the unit circle:\ Matrix representation and spectral analysis

TL;DR

This paper studies a class of unbounded Toeplitz operators with rational matrix symbols having poles on the unit circle, using state space realization to analyze their matrix representations and spectral properties.

Contribution

It introduces a novel approach connecting rational matrix symbols with poles on the unit circle to their matrix representations and spectral analysis via state space methods.

Findings

Determined the essential spectrum of the operators.

Analyzed the resolvent set for these Toeplitz operators.

Established a connection between the operators and semi-infinite Toeplitz matrices.

Abstract

In this paper we consider a class of unbounded Toeplitz operators with rational matrix symbols that have poles on the unit circle and employ state space realization techniques from linear systems theory, as used in our earlier analysis in [11] of this class of operators, to study the connection with semi-infinite Toeplitz matrices and to determine the essential spectrum and resolvent set.