{
  "id": "1111.3112",
  "version": "v1",
  "published": "2011-11-14T06:39:09.000Z",
  "updated": "2011-11-14T06:39:09.000Z",
  "title": "Landau and Gruss type inequalities for inner product type integral transformers in norm ideals",
  "authors": [
    "Danko R. Jocic",
    "Dorde E. Krtinic",
    "Mohammad Sal Moslehian"
  ],
  "comment": "21 pages, to appear in Math. Inequal. Appl. (MIA)",
  "journal": "Math. Inequal. Appl. 16 (2013), no. 1, 109-125",
  "categories": [
    "math.FA",
    "math.CA",
    "math.OA"
  ],
  "abstract": "For a probability measure $\\mu$ and for square integrable fields $(\\mathscr{A}_t)$ and $(\\mathscr{B}_t)$ ($t\\in\\Omega$) of commuting normal operators we prove Landau type inequality \\llu\\int_\\Omega\\mathscr{A}_tX\\mathscr{B}_td\\mu(t)- \\int_\\Omega\\mathscr{A}_t\\,d\\mu(t)X \\int_\\Omega\\mathscr{B}_t\\,d\\mu(t) \\rru \\le \\llu \\sqrt{\\,\\int_\\Omega|\\mathscr{A}_t|^2\\dt-|\\int_\\Omega\\mathscr{A}_t\\dt|^2}X \\sqrt{\\,\\int_\\Omega|\\mathscr{B}_t|^2 \\dt-|\\int_\\Omega\\mathscr{B}_t\\dt|^2} \\rru for all $X\\in\\mathcalb{B}(\\mathcal{H})$ and for all unitarily invariant norms $\\lluo\\cdot\\rruo$. For Schatten $p$-norms similar inequalities are given for arbitrary double square integrable fields. Also, for all bounded self-adjoint fields satisfying $C\\le\\mathscr{A}_t\\le D$ and $E\\le\\mathscr{B}_t\\le F$ for all $t\\in\\Omega $ and some bounded self-adjoint operators $C,D,E$ and $F$, then for all $X\\in\\ccu$ we prove Gr\\\"uss type inequality \\llu\\int_\\Omega\\mathscr{A}_tX\\mathscr{B}_t \\dt- \\int_\\Omega \\mathscr{A}_t\\,d\\mu(t)X \\int_\\Omega\\mathscr{B}_t\\,d\\mu(t) \\rru\\leq \\frac{\\|D-C\\|\\cdot\\|F-E\\|}4\\cdot\\lluo X\\rruo. More general results for arbitrary bounded fields are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-14T06:39:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "46L05",
      "47B10",
      "47A30",
      "47B15"
    ],
    "keywords": [
      "inner product type integral transformers",
      "gruss type inequalities",
      "type inequality",
      "norm ideals",
      "double square integrable fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.3112J"
    }
  }
}