{
  "id": "1203.5498",
  "version": "v4",
  "published": "2012-03-25T14:09:04.000Z",
  "updated": "2013-09-05T14:57:23.000Z",
  "title": "Regularity of Semigroups via the Asymptotic Behaviour at Zero",
  "authors": [
    "Stephan Fackler"
  ],
  "comment": "14 pages; final version",
  "journal": "Semigroup Forum 87 (2013), no. 1, 1-17",
  "doi": "10.1007/s00233-013-9466-y",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) on a uniformly convex space X is holomorphic iff limsup_{t \\downarrow 0} ||T(t)-Id|| < 2. We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent {\\theta} in (0,1). As application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to R-sectoriality: We obtain a characterization of maximal regularity by the behaviour of the semigroup at zero and show extrapolation results.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-09-05T14:57:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47D06",
      "47D09",
      "46B70"
    ],
    "keywords": [
      "asymptotic behaviour",
      "arbitrary banach spaces",
      "general extrapolation result",
      "maximal regularity",
      "zero-two law"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.5498F"
    }
  }
}