{
  "id": "2308.06936",
  "version": "v1",
  "published": "2023-08-14T04:47:11.000Z",
  "updated": "2023-08-14T04:47:11.000Z",
  "title": "The Commutant of Multiplication by z on the Closure of Rational Functions in $L^t(μ)$",
  "authors": [
    "Liming Yang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2212.10811",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a compact set $K\\subset \\mathbb C,$ a finite positive Borel measure $\\mu$ on $K,$ and $1 \\le t < \\i,$ let $\\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \\mu)$ be the closure of $\\text{Rat}(K)$ in $L^t(\\mu).$ For a bounded Borel subset $\\mathcal D\\subset \\mathbb C,$ let $\\area_{\\mathcal D}$ denote the area (Lebesgue) measure restricted to $\\mathcal D$ and let $H^\\i (\\mathcal D)$ be the weak-star closed sub-algebra of $L^\\i(\\area_{\\mathcal D})$ spanned by $f,$ bounded and analytic on $\\mathbb C\\setminus E_f$ for some compact subset $E_f \\subset \\mathbb C\\setminus \\mathcal D.$ We show that if $R^t(K, \\mu)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\\mathcal R \\subset K$ whose closure contains the support of $\\mu$ and there exists an isometric isomorphism and a weak-star homeomorphism $\\rho$ from $R^t(K, \\mu) \\cap L^\\infty(\\mu)$ onto $H^\\infty(\\mathcal R)$ such that $\\rho(r) = r$ for all $r\\in\\text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for $\\rtkmu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-14T04:47:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational functions",
      "borel subset",
      "multiplication",
      "structural decomposition theorems",
      "finite positive borel measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}