{
  "id": "2309.14698",
  "version": "v1",
  "published": "2023-09-26T06:20:06.000Z",
  "updated": "2023-09-26T06:20:06.000Z",
  "title": "A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations",
  "authors": [
    "G. J. Groenewald",
    "S. ter Horst",
    "J. Jaftha",
    "A. C. M. Ran"
  ],
  "comment": "19 pages",
  "categories": [
    "math.FA"
  ],
  "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 $T_\\Om^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-26T06:20:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47A53",
      "47A68"
    ],
    "keywords": [
      "rational matrix symbol",
      "toeplitz-like operator",
      "unit circle",
      "wiener-hopf type factorization",
      "invertibility"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}