{
  "id": "1805.00602",
  "version": "v1",
  "published": "2018-05-02T02:16:03.000Z",
  "updated": "2018-05-02T02:16:03.000Z",
  "title": "The generalized numerical range of a set of matrices",
  "authors": [
    "Pan-Shun Lau",
    "Chi-Kwong Li",
    "Yiu-Tung Poon",
    "Nung-Sing Sze"
  ],
  "comment": "22 pages, 8 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a given set of $n\\times n$ matrices $\\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\\mathcal F$, denoted by $W_C({\\mathcal F})$. We obtain basic algebraic and topological properties of $W_C({\\mathcal F})$, and show that there are connections between the geometric properties of $W_C({\\mathcal F})$ and the algebraic properties of $C$ and the matrices in ${\\mathcal F}$. Furthermore, we consider the starshapedness and convexity of the set $W_C({\\mathcal F})$. In particular, we show that if ${\\mathcal F}$ is the convex hull of two matrices such that $W_C(A)$ and $W_C(B)$ are convex, then the set $W_C({\\mathcal F})$ is star-shaped. We also investigate the extensions of the results to the joint $C$-numerical range of an $m$-tuple of matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-02T02:16:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A60"
    ],
    "keywords": [
      "generalized numerical range",
      "algebraic properties",
      "basic algebraic",
      "geometric properties",
      "convex hull"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}