{
  "id": "1712.01023",
  "version": "v1",
  "published": "2017-12-04T11:50:11.000Z",
  "updated": "2017-12-04T11:50:11.000Z",
  "title": "The C-Numerical Range in Infinite Dimensions",
  "authors": [
    "Frederik vom Ende",
    "Gunther Dirr"
  ],
  "comment": "21 pages, no figures, comments welcome",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In infinite dimensions, on the level of trace-class operators rather than matrices $C$, we show that the closure of the $C$-numerical range $\\overline{W_C(T)}$ is star-shaped with respect to the set $\\operatorname{tr}(C)W_e(T)$, where $W_e(T)$ denotes the essential numerical range of the bounded operator $T$. Further, we will see that in the case of compact normal operators, the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the closure of the convex hull of said $C$-spectrum. In addition, if the eigenvalues of $C$ are collinear, then the latter coincides with the closure of the $C$-numerical range for any compact normal operator $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-04T11:50:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "15A60"
    ],
    "keywords": [
      "infinite dimensions",
      "c-numerical range",
      "compact normal operator",
      "convex hull",
      "trace-class operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}