{
  "id": "2505.07824",
  "version": "v1",
  "published": "2025-04-15T05:58:45.000Z",
  "updated": "2025-04-15T05:58:45.000Z",
  "title": "Characteristic function of a power partial isometry",
  "authors": [
    "Kritika Babbar",
    "Amit Maji"
  ],
  "comment": "28 pages. Preliminary version",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "The celebrated Sz.-Nagy-Foia\\c{s} model theory says that there is a bijection between the class of purely contractive analytic functions and the class of completely non-unitary (c.n.u. in short) contractions modulo unitary equivalence. In this paper we provide a complete classification of the purely contractive analytic functions such that the associated contraction is a c.n.u. power partial isometry. As an application of our findings, we determine a class of contractive polynomials such that the associated c.n.u. power partial isometry is of the explicit diagonal form $S \\oplus N \\oplus C$, where $S$ and $C^*$ are unilateral shifts and $N$ is nilpotent. Finally, we obtain a characterization of operator-valued symbols for which the corresponding Toeplitz operator on vector-valued Hardy space is a partial isometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-04-15T05:58:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A45",
      "47A48",
      "47B35",
      "46E40",
      "47A20",
      "46E40",
      "30H10"
    ],
    "keywords": [
      "power partial isometry",
      "characteristic function",
      "purely contractive analytic functions",
      "contractions modulo unitary equivalence",
      "model theory says"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}