{
  "id": "2108.05414",
  "version": "v1",
  "published": "2021-08-11T19:07:50.000Z",
  "updated": "2021-08-11T19:07:50.000Z",
  "title": "Commuting normal operators and joint numerical range",
  "authors": [
    "Jor-Ting Chan",
    "Chi-Kwong Li",
    "Yiu-Tung Poon"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let ${\\mathcal H}$ be a complex Hilbert space and let ${\\mathcal B}({\\mathcal H})$ be the algebra of all bounded linear operators on ${\\mathcal H}$. For a positive integer $k$ less than the dimension of ${\\mathcal H}$ and ${\\mathbf A} = (A_1, \\dots, A_m)\\in {\\mathcal B}({\\mathcal H})^m$, the joint $k$-numerical range $W_k({\\mathbf A})$ is the set of $(\\alpha_1, \\dots, \\alpha_m) \\in{\\mathbb C}^m$ such that $\\alpha_i = \\sum_{j = 1}^k \\langle A_ix_j, x_j\\rangle$ for an orthonormal set $\\{x_1, \\ldots, x_k\\}$ in ${\\mathcal H}$. Relations between the geometric properties of $W_k({\\mathbf A})$ and the algebraic and analytic properties of $A_1, \\dots, A_m$ are studied. It is shown that there is $k\\in {\\mathbb N}$ such that $W_k({\\mathbf A})$ is a polyhedral set, i.e., the convex hull of a finite set, if and only if $A_1, \\dots, A_k$ have a common reducing subspace ${\\mathbf V}$ of finite dimension such that the compression of $A_1, \\dots, A_m$ on the subspace ${\\mathbf V}$ are diagonal operators $D_1, \\dots, D_m$ and $W_k({\\mathbf A}) = W_k(D_1, \\dots, D_m)$. Characterization is also given to ${\\bf A}$ such that the closure of $W_k({\\mathbf A})$ is polyhedral. The conditions are related to the joint essential numerical range of ${\\mathbf A}$. These results are used to study ${\\bf A}$ such that (a) $\\{A_1, \\dots, A_m\\}$ is a commuting family of normal operators, or (b) $W_k(A_1, \\dots, A_m)$ is polyhedral for every positive integer $k$. It is shown that conditions (a) and (b) are equivalent for finite rank operators but it is no longer true for compact operators. Characterizations are given for compact operators $A_1, \\dots, A_m$ satisfying (a) and (b), respectively. Results are also obtained for general non-compact operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-11T19:07:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "15A60"
    ],
    "keywords": [
      "commuting normal operators",
      "joint numerical range",
      "positive integer",
      "general non-compact operators",
      "finite rank operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}