{
  "id": "1608.06094",
  "version": "v1",
  "published": "2016-08-22T09:21:04.000Z",
  "updated": "2016-08-22T09:21:04.000Z",
  "title": "The star-shapedness of a generalized numerical range",
  "authors": [
    "Pan-Shun Lau",
    "Tuen-Wai Ng",
    "Nam-Kiu Tsing"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathcal{H}_n$ be the set of all $n\\times n$ Hermitian matrices and $\\mathcal{H}^m_n$ be the set of all $m$-tuples of $n\\times n$ Hermitian matrices. For $A=(A_1,...,A_m)\\in \\mathcal{H}^m_n$ and for any linear map $L:\\mathcal{H}^m_n\\to\\mathbb{R}^\\ell$, we define the $L$-numerical range of $A$ by \\[ W_L(A):=\\{L(U^*A_1U,...,U^*A_mU): U\\in \\mathbb{C}^{n\\times n}, U^*U=I_n\\}. \\] In this paper, we prove that if $\\ell\\leq 3$, $n\\geq \\ell$ and $A_1,...,A_m$ are simultaneously unitarily diagonalizable, then $W_L(A)$ is star-shaped with star center at $L\\left(\\frac{\\mathrm{tr} A_1}{n}I_n,...,\\frac{\\mathrm{tr} A_m}{n}I_n\\right)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-22T09:21:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A04",
      "15A60"
    ],
    "keywords": [
      "generalized numerical range",
      "star-shapedness",
      "hermitian matrices",
      "star center"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}