{
  "id": "1710.09555",
  "version": "v1",
  "published": "2017-10-26T06:14:21.000Z",
  "updated": "2017-10-26T06:14:21.000Z",
  "title": "Convexity and Star-shapedness of Matricial Range",
  "authors": [
    "Pan-Shun Lau",
    "Chi-Kwong Li",
    "Yiu-Tung Poon",
    "Nung-Sing Sze"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let ${\\bf A} = (A_1, \\dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${\\mathcal H}$. Their joint $(p,q)$-matricial range $\\Lambda_{p,q}({\\bf A})$ is the collection of $(B_1, \\dots, B_m) \\in {\\bf M}_q^m$, where $I_p\\otimes B_j$ is a compression of $A_j$ on a $pq$-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of $p,q,m$. In this paper, it is shown that $\\Lambda_{p,q}({\\bf A})$ is star-shaped if the dimension of ${\\mathcal H}$ is sufficiently large. If $\\dim{\\mathcal H}$ is infinite, we extend the definition of $\\Lambda_{p,q}({\\bf A})$ to $\\Lambda_{\\infty,q}({\\bf A})$ consisting of $(B_1, \\dots, B_m) \\in {\\bf M}_q^m$ such that $I_\\infty \\otimes B_j$ is a compression of $A_j$ on a closed subspace of ${\\mathcal H}$, and consider the joint essential $(p,q)$-matricial range $$\\Lambda^{ess}_{p,q}({\\bf A}) = \\bigcap \\{ {\\bf cl}(\\Lambda_{p,q}(A_1+F_1, \\dots, A_m+F_m)): F_1, \\dots, F_m \\hbox{ are compact operators}\\}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-26T06:14:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A13",
      "47A55",
      "15A60"
    ],
    "keywords": [
      "matricial range",
      "star-shapedness",
      "joint essential",
      "definition covers",
      "dimensional subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}