{
  "id": "1304.0159",
  "version": "v1",
  "published": "2013-03-31T06:22:32.000Z",
  "updated": "2013-03-31T06:22:32.000Z",
  "title": "Operator Entropy Inequalities",
  "authors": [
    "A. Morassaei",
    "F. Mirzapour",
    "M. S. Moslehian"
  ],
  "comment": "11 pages; to appear in Colloq. Math",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\\mathbf{A}=(A_1,...,A_n)$ and $\\mathbf{B}=(B_1,...,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\\mathbf{A}|\\mathbf{B}):=\\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\\,, $$ and then give upper and lower bounds for $S_q^f(\\mathbf{A}|\\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219--235] under certain conditions. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-31T06:22:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "15A42",
      "46L05",
      "47A30"
    ],
    "keywords": [
      "inequality",
      "operator entropy inequalities",
      "operator monotone function",
      "linear algebra appl",
      "relative operator entropy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0159M"
    }
  }
}