{
  "id": "0806.1428",
  "version": "v1",
  "published": "2008-06-09T11:29:34.000Z",
  "updated": "2008-06-09T11:29:34.000Z",
  "title": "Domains of uniqueness for $C_0$-semigroups on the dual of a Banach space",
  "authors": [
    "Ludovic Dan Lemle"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $({\\cal X},\\|\\:.\\:\\|)$ be a Banach space. In general, for a $C_0$-semigroup \\semi on $({\\cal X},\\|\\:.\\:\\|)$, its adjoint semigroup \\semia is no longer strongly continuous on the dual space $({\\cal X}^{*},\\|\\:.\\:\\|^{*})$. Consider on ${\\cal X}^{*}$ the topology of uniform convergence on compact subsets of $({\\cal X},\\|\\:.\\:\\|)$ denoted by ${\\cal C}({\\cal X}^{*},{\\cal X})$, for which the usual semigroups in literature becomes $C_0$-semigroups. The main purpose of this paper is to prove that only a core can be the domain of uniqueness for a $C_0$-semigroup on $({\\cal X}^{*},{\\cal C}({\\cal X}^{*},{\\cal X}))$. As application, we show that the generalized Schr\\\"odinger operator ${\\cal A}^Vf={1/2}\\Delta f+b\\cdot\\nabla f-Vf$, $f\\in C_0^\\infty(\\R^d)$, is $L^\\infty(\\R^d,dx)$-unique. Moreover, we prove the $L^1(\\R^d,dx)$-uniqueness of weak solution for the Fokker-Planck equation associated with ${\\cal A}^V$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-09T11:29:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "banach space",
      "uniqueness",
      "fokker-planck equation",
      "compact subsets",
      "adjoint semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1428L"
    }
  }
}