{
  "id": "2106.05914",
  "version": "v1",
  "published": "2021-06-10T17:00:09.000Z",
  "updated": "2021-06-10T17:00:09.000Z",
  "title": "Matrix power means and new characterizations of operator monotone functions",
  "authors": [
    "Trung Hoa Dinh",
    "Cong Trinh Le",
    "The Van Nguyen",
    "Bich Khue Vo"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$ P_\\mu(p, A, B) = A^{1/2}\\left(\\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\\right )^{1/p} A^{1/2}\\quad (p \\ge 0). $$ In this paper, for $0\\le p \\le 1 \\le q$, we show that if one of the following inequalities \\begin{align*} f(P_\\mu(p, A, B)) \\le f(P_\\mu(1, A, B)) \\le f(P_\\mu(q, A, B))\\nonumber \\end{align*} holds for any positive definite matrices $A$ and $B$, then the function $f$ is operator monotone on $(0, \\infty).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers $1/2$ and $2$. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-10T17:00:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "47A64",
      "47A56",
      "46E05",
      "15B48"
    ],
    "keywords": [
      "operator monotone functions",
      "non-kubo-ando matrix power means",
      "positive definite matrices",
      "characterizations",
      "inverse problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}