{
  "id": "math/0503076",
  "version": "v1",
  "published": "2005-03-04T11:24:16.000Z",
  "updated": "2005-03-04T11:24:16.000Z",
  "title": "On the intrinsic and the spatial numerical range",
  "authors": [
    "Miguel Martin",
    "Javier Meri",
    "Rafael Paya"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a bounded linear operator $T:X\\longrightarrow Y$ such that $\\bar{co} W(T)\\neq V(T)$. We also show that, up to renormig, for every non-reflexive Banach space $Y$, one can find a closed subspace $X$ and a bounded linear operator $T\\in L(X,Y)$ such that $\\bar{co} W(T)\\neq V(T)$. Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobas property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-04T11:24:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "47A12"
    ],
    "keywords": [
      "spatial numerical range",
      "intrinsic numerical range",
      "closed convex hull",
      "bounded linear operator",
      "uniform smoothness"
    ],
    "publication": {
      "journal": "Journal of Mathematical Analysis and Applications",
      "year": 2006,
      "month": "Jun",
      "volume": 318,
      "number": 1,
      "pages": 175
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006JMAA..318..175M"
    }
  }
}