{
  "id": "2307.06697",
  "version": "v1",
  "published": "2023-07-13T11:51:31.000Z",
  "updated": "2023-07-13T11:51:31.000Z",
  "title": "A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics",
  "authors": [
    "G. J. Groenewald",
    "S. ter Horst",
    "J. J. Jaftha",
    "A. C. M. Ran"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In a recent paper (Groenewald et al.\\ {\\em Complex Anal.\\ Oper.\\ Theory} \\textbf{15:1} (2021)) we considered an unbounded Toeplitz-like operator $T_\\Omega$ generated by a rational matrix function $\\Omega$ that has poles on the unit circle $\\mathbb{T}$ of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator $T_\\Omega$, including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of $T_\\Omega$ from this factorization, and hence of the co-kernel, even when $T_\\Omega$ is Fredholm. In the current paper we provide a formula for the dimension of the kernel of $T_\\Omega$ under an additional assumption on the Wiener-Hopf type factorization. In the case that $\\Omega$ is a $2 \\times 2$ matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the $2 \\times 2$ case is partially extended to the case of matrix functions of arbitrary size.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T11:51:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47A53",
      "47A68"
    ],
    "keywords": [
      "wiener-hopf type factorization",
      "rational matrix symbol",
      "unit circle",
      "toeplitz-like operator",
      "fredholm characteristics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}