A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations

TL;DR

This paper characterizes the invertibility of unbounded Toeplitz-like operators with rational matrix symbols having poles on the unit circle using Riccati equations, providing explicit formulas for their inverses.

Contribution

It introduces a novel approach using state space methods and Riccati equations to determine invertibility, overcoming limitations of previous Wiener-Hopf factorization techniques.

Findings

Invertibility characterized by stabilizing solutions of Riccati equations

Derived pseudo-canonical factorization of the matrix symbol

Explicit formulas for the inverse operator

Abstract

This paper is a continuation of the work on unbounded Toeplitz-like operators $T_\Om$ with rational matrix symbol $\Om$ initiated in Groenewald et. al (Complex Anal. Oper. Theory 15, 1(2021)), where a Wiener-Hopf type factorization of $\Om$ is obtained and used to determine when $T_\Om$ is Fredholm and compute the Fredholm index in case $T_\Om$ is Fredholm. Due to the high level of non-uniqueness and complicated form of the Wiener-Hopf type factorization, it does not appear useful in determining when $T_\Om$ is invertible. In the present paper we use state space methods to characterize invertibility of $T_\Om$ in terms of the existence of a stabilizing solution of an associated nonsymmetric discrete algebraic Riccati equation, which in turn leads to a pseudo-canonical factorization of $\Om$ and concrete formulas of $T_\Om^{-1}$.