{
  "id": "2203.07821",
  "version": "v1",
  "published": "2022-03-15T12:23:06.000Z",
  "updated": "2022-03-15T12:23:06.000Z",
  "title": "Wiener-Hopf factorization indices of rational matrix functions with respect to the unit circle in terms of realization",
  "authors": [
    "G. J. Groenewald",
    "M. A. Kaashoek",
    "A. C. M. Ran"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "As in the paper [G. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators. \\emph{Indag. Math.} 28 (2017) 694--710] our aim is to obtain explicitly the Wiener-Hopf indices of a rational $m\\times m$ matrix function $R(z)$ that has no poles and no zeros on the unit circle $\\mathbb{T}$ but, in contrast with that paper, the function $R(z)$ is not required to be unitary on the unit circle. On the other hand, using a Douglas-Shapiro-Shields type of factorization, we show that $R(z)$ factors as $R(z)=\\Xi(z)\\Psi(z)$, where $\\Xi(z)$ and $\\Psi(z)$ are rational $m\\times m$ matrix functions, $\\Xi(z)$ is unitary on the unit circle and $\\Psi(z)$ is an invertible outer function. Furthermore, the fact that $\\Xi(z)$ is unitary on the unit circle allows us to factor as $\\Xi(z) =V(z)W^*(z)$ where $V(z)$ and $W(z)$ are rational bi-inner $m\\times m$ matrix functions. The latter allows us to solve the Wiener-Hopf indices problem. To derive explicit formulas for the functions $V(z)$ and $W(z)$ requires additional realization properties of the function $\\Xi(z)$ which are given in the last two sections.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-15T12:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47A35"
    ],
    "keywords": [
      "unit circle",
      "wiener-hopf factorization indices",
      "rational matrix functions",
      "additional realization properties",
      "wiener-hopf indices problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}