{
  "id": "2112.13656",
  "version": "v2",
  "published": "2021-12-17T16:45:29.000Z",
  "updated": "2022-02-10T15:35:40.000Z",
  "title": "Unitarily invariant Norms on Operators",
  "authors": [
    "Jor-Ting Chan",
    "Chi-Kwong Li"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $f$ be a symmetric norm on ${\\mathbb R}^n$ and let ${\\mathcal B}({\\mathcal H})$ be the set of all bounded linear operators on a Hilbert space ${\\mathcal H}$ of dimension at least $n$. Define a norm on ${\\mathcal B}({\\mathcal H})$ by $\\|A\\|_f = f(s_1(A), \\dots, s_n(A))$, where $s_k(A) = \\inf\\{\\|A-X\\|: X\\in {\\mathcal B}({\\mathcal H}) \\hbox{ has rank less than } k\\}$ is the $k$th singular value of $A$. Basic properties of the norm $\\|\\cdot\\|_f$ are obtained including some norm inequalities and characterization of the equality case. Geometric properties of the unit ball of the norm are obtained; the results are used to determine the structure of maps $L$ satisfying $\\|L(A)-L(B)\\|_f=\\|A - B\\|_f$ for any $A, B \\in {\\mathcal B}({\\mathcal H})$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-10T15:35:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A04",
      "15A60",
      "47B48"
    ],
    "keywords": [
      "unitarily invariant norms",
      "th singular value",
      "symmetric norm",
      "geometric properties",
      "equality case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}