{
  "id": "2208.01405",
  "version": "v1",
  "published": "2022-07-31T19:12:12.000Z",
  "updated": "2022-07-31T19:12:12.000Z",
  "title": "The $C$-numerical range and Unitary dilations",
  "authors": [
    "Chi-Kwong Li"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For an $n\\times n$ complex matrix $C$, the $C$-numerical range of a bounded linear operator $T$ acting on a Hilbert space of dimension at least $n$ is the set of complex numbers ${\\rm tr}(CX^*TX)$, where $X$ is a partial isometry satisfying $X^*X = I_n$. It is shown that $${\\bf cl}(W_C(T)) = \\cap \\{{\\bf cl}(W_C(U)): U \\hbox{ is a unitary dilation of } T\\}$$ for any contraction $T$ if and only if $C$ is a rank one normal matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-31T19:12:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A20",
      "15A60"
    ],
    "keywords": [
      "numerical range",
      "unitary dilations",
      "complex matrix",
      "bounded linear operator",
      "complex numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}