{
  "id": "1401.3072",
  "version": "v2",
  "published": "2014-01-14T05:31:05.000Z",
  "updated": "2014-05-02T11:57:30.000Z",
  "title": "On an invariance property of the space of smooth vectors",
  "authors": [
    "Karl-Hermann Neeb",
    "Hadi Salmasian",
    "Christoph Zellner"
  ],
  "comment": "Accepted by Kyoto Journal of Math",
  "categories": [
    "math.RT",
    "math.FA"
  ],
  "abstract": "Let $(\\pi, \\mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\\gamma \\: \\mathbb R \\to \\mathrm{Aut}(G)$ define a continuous action of $\\mathbb R$ on $G$. Suppose that $\\pi^\\#(g,t) = \\pi(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \\rtimes_\\gamma \\mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\\mathcal H^\\infty$ of smooth vectors of $\\pi$ under the operators $U_f = \\int_\\mathbb R f(t)U_t\\, dt$ for $f \\in L^1(\\mathbb R)$, resp., $f \\in \\mathcal S(\\mathbb R)$. Using this theorem we show that, for suitably defined spectral subspaces $\\mathfrak g_{\\mathbb C}(E)$, $E \\subseteq \\mathbb R$, in the complexified Lie algebra $\\mathfrak g_{\\mathbb C}$, and $\\mathcal H^\\infty(F)$, $F\\subseteq \\mathbb R$, for $U$ in $\\mathcal H^\\infty$, we have \\[ \\mathsf{d}\\pi(\\mathfrak g_{\\mathbb C}(E)) \\mathcal H^\\infty(F) \\subseteq \\mathcal H^\\infty(E + F).\\]",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-02T11:57:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E65",
      "22E45",
      "17B65"
    ],
    "keywords": [
      "smooth vectors",
      "invariance property",
      "continuous unitary representation",
      "first main theorem",
      "semidirect product group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.3072N"
    }
  }
}