{
  "id": "2412.02326",
  "version": "v1",
  "published": "2024-12-03T09:40:42.000Z",
  "updated": "2024-12-03T09:40:42.000Z",
  "title": "A sharp bound for the functional calculus of $ρ$-contractions",
  "authors": [
    "Felix L. Schwenninger",
    "Jens de Vries"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.FA",
    "cs.NA",
    "math.NA",
    "math.OA"
  ],
  "abstract": "Let $A$ be a $\\rho$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $\\rho$-contractions we prove that \\begin{align*} \\big\\|f(A)\\big\\|\\leq \\frac{\\rho}{2}\\big(1-|f(0)|^{2}\\big)+\\sqrt{\\frac{\\rho^{2}}{4}\\big(1-|f(0)|^{2}\\big){}^{2}+|f(0)|^{2}}. \\end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $\\rho=2$, is consistent with a result by Drury.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-03T09:40:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A60",
      "15A60",
      "47A20"
    ],
    "keywords": [
      "functional calculus",
      "sharp bound",
      "contraction",
      "closed unit disk",
      "rational function mapping"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}