{
  "id": "1211.0791",
  "version": "v4",
  "published": "2012-11-05T08:50:34.000Z",
  "updated": "2013-07-31T14:02:48.000Z",
  "title": "Boundary values of resolvents of self-adjoint operators in Krein spaces",
  "authors": [
    "Vladimir Georgescu",
    "Christian Gérard",
    "Dietrich Häfner"
  ],
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if $H$ is a selfadjoint operator on a Krein space $\\cH$, equipped with the Krein scalar product $\\langle \\cdot| \\cdot \\rangle$, $A$ is the generator of a $C_{0}-$group on $\\cH$ and $I\\subset \\rr$ is an interval such that: \\begin{itemize} \\item[]1) $H$ admits a Borel functional calculus on $I$, \\item[]2) the spectral projection $\\one_{I}(H)$ is positive in the Krein sense, \\item[]3) the following {\\em positive commutator estimate} holds: \\[ \\Re \\langle u| [H, \\i A]u\\rangle\\geq c \\langle u| u\\rangle, \\ u \\in {\\rm Ran}\\one_{I}(H), \\ c>0. \\] \\end{itemize} then assuming some smoothness of $H$ with respect to the group $\\e^{\\i t A}$, the following resolvent estimates hold: \\[ \\sup_{z\\in I\\pm \\i]0, \\nu]}\\| \\langle A\\rangle ^{-s}(H-z)^{-1}\\langle A\\rangle^{-s}\\| <\\infty, \\ s>\\12. \\] As an application we consider abstract Klein-Gordon equations \\[ \\p_{t}^{2}\\phi(t)- 2 \\i k \\phi(t)+ h\\phi(t)=0, \\] and obtain resolvent estimates for their generators in {\\em charge spaces} of Cauchy data.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-07-31T14:02:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "krein space",
      "boundary values",
      "self-adjoint operators",
      "selfadjoint operator",
      "paper resolvent estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.0791G"
    }
  }
}