{
  "id": "1501.02939",
  "version": "v1",
  "published": "2015-01-13T10:08:22.000Z",
  "updated": "2015-01-13T10:08:22.000Z",
  "title": "Squaring operator Pólya--Szegö and Diaz--Metcalf type inequalities",
  "authors": [
    "Mohammad Sal Moslehian",
    "Xiaohui Fu"
  ],
  "comment": "10 pages, to appear in Linear Algebra Appl. (LAA)",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We square operator P\\'{o}lya--Szeg\\\"{o} and Diaz--Metcalf type inequalities as follows: If operator inequalities $0<m_{1}^{2} \\leq A\\leq M_{1}^{2}$ and $0<m_{2}^{2}\\leq B\\leq M_{2}^{2}$ hold for some positive real numbers $m_{1}\\leq M_{1}$ and $m_{2}\\leq M_{2}$, then for every unital positive linear map $\\Phi$ the following inequalities hold: \\begin{eqnarray*} (\\Phi(A)\\sharp\\Phi(B))^2 &\\leq&\\left(\\frac{M_1M_2 + m_1m_2}{2\\sqrt{M_1M_2m_1m_2}}\\right)^4\\Phi(A\\sharp B)^{2} \\end{eqnarray*} and \\begin{eqnarray*} \\left( \\frac{M_2m_2}{M_1m_1}\\Phi (A) + \\Phi (B) \\right)^2 \\leq \\left( \\frac{(M_1m_1(M_2^2 + m_2^2) + M_2m_2(M_1^2 + m_1^2))^2}{8\\sqrt{M_2M_1m_1m_2} M_1^2m_1^2M_2m_2} \\right)^2\\Phi (A\\sharp B)^2\\,. \\end{eqnarray*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-13T10:08:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A30",
      "47A63",
      "46L05",
      "15A60"
    ],
    "keywords": [
      "diaz-metcalf type inequalities",
      "squaring operator",
      "unital positive linear map",
      "square operator",
      "operator inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150102939S"
    }
  }
}