{
  "id": "1808.06898",
  "version": "v1",
  "published": "2018-08-21T13:52:59.000Z",
  "updated": "2018-08-21T13:52:59.000Z",
  "title": "The C-Numerical Range for Schatten-Class Operators",
  "authors": [
    "Gunther Dirr",
    "Frederik vom Ende"
  ],
  "comment": "10 pages; follow-up paper of arXiv:1712.01023",
  "categories": [
    "math.FA"
  ],
  "abstract": "We generalize the $C$-numerical range $W_C(T)$ from trace-class to conjugate Schatten-class operators and show that its closure is always star-shaped with star-center $\\lbrace 0\\rbrace$. Equivalently, the closure of the image of the unitary orbit of $T\\in\\mathcal B^p(\\mathcal H)$ under any continous linear functional $L\\in(\\mathcal B^p)'(\\mathcal H)$ is star-shaped with star-center $\\lbrace 0\\rbrace$ if $p\\in(1,\\infty]$ and star-center $\\operatorname{tr}(T)W_e(L)$ if $p=1$, where $W_e(L)$ denotes the essential numerical range of $L$. Moreover, the closure of $W_C(T)$ is convex if either $C$ or $T$ is normal with collinear eigenvalues. 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 the $C$-spectrum. This closure coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-21T13:52:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47B10",
      "15A60"
    ],
    "keywords": [
      "c-numerical range",
      "conjugate schatten-class operators",
      "star-center",
      "compact normal operators",
      "closure coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}