{
  "id": "1910.09944",
  "version": "v1",
  "published": "2019-10-22T13:04:31.000Z",
  "updated": "2019-10-22T13:04:31.000Z",
  "title": "The nuclearity of Gelfand-Shilov spaces and kernel theorems",
  "authors": [
    "Andreas Debrouwere",
    "Lenny Neyt",
    "Jasson Vindas"
  ],
  "comment": "24 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the nuclearity of the Gelfand-Shilov spaces $\\mathcal{S}^{(\\mathfrak{M})}_{(\\mathscr{W})}$ and $\\mathcal{S}^{\\{\\mathfrak{M}\\}}_{\\{\\mathscr{W}\\}}$, defined via a weight (multi-)sequence system $\\mathfrak{M}$ and a weight function system $\\mathscr{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for K\\\"{o}the sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\\mathcal{S}^{(M)}_{(A)}$ and $\\mathcal{S}^{\\{M\\}}_{\\{A\\}}$ (defined via weight sequences $M$ and $A$) and the Beurling-Bj\\\"orck spaces $\\mathcal{S}^{(\\omega)}_{(\\eta)}$ and $\\mathcal{S}^{\\{\\omega\\}}_{\\{\\eta\\}}$ (defined via weight functions $\\omega$ and $\\eta$). Our results cover anisotropic cases as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-22T13:04:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A11",
      "46E10",
      "46A45"
    ],
    "keywords": [
      "gelfand-shilov spaces",
      "kernel theorems",
      "nuclearity",
      "results cover anisotropic cases",
      "weight function system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}