{
  "id": "1507.01418",
  "version": "v1",
  "published": "2015-07-06T12:38:13.000Z",
  "updated": "2015-07-06T12:38:13.000Z",
  "title": "A semigroup approach to the numerical range of operators on Banach spaces",
  "authors": [
    "Martin Adler",
    "Waed Dada",
    "Agnes Radl"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We introduce the numerical spectrum $\\sigma_n(A)\\subset \\mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $\\sigma_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-06T12:38:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A10",
      "47D06"
    ],
    "keywords": [
      "numerical range",
      "banach space",
      "semigroup approach",
      "notions coincide",
      "hilbert spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}