A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm properties

TL;DR

This paper investigates the Fredholm properties of unbounded Toeplitz-like operators generated by rational matrix functions with poles on the unit circle, extending scalar cases and providing a Wiener-Hopf factorization approach.

Contribution

It introduces a Wiener-Hopf type factorization for rational matrix functions with poles on the unit circle and analyzes the Fredholm properties of the associated Toeplitz-like operators.

Findings

Derived a Wiener-Hopf factorization for rational matrix functions with poles on T

Established a formula for the index of Toeplitz-like operators using the factorization

Showed that the determinant condition is not sufficient for Fredholmness in this context

Abstract

This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on T is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.