{
  "id": "1008.2943",
  "version": "v2",
  "published": "2010-08-17T17:53:42.000Z",
  "updated": "2011-04-27T10:29:29.000Z",
  "title": "A Characterisation of Anti-Lowner Functions",
  "authors": [
    "Koenraad M. R. Audenaert"
  ],
  "comment": "7 pages; V2: Title changed; mistake corrected in proofs and results about anti-Lowner functions of finite order weakened. The main result about anti-Lowner functions of all orders is unchanged",
  "journal": "Proc. AMS 139, 4217-4223 (2011)",
  "doi": "10.1090/S0002-9939-2011-10935-3",
  "categories": [
    "math.FA"
  ],
  "abstract": "According to a celebrated result by L\\\"owner, a real-valued function $f$ is operator monotone if and only if its L\\\"owner matrix, which is the matrix of divided differences $L_f=(\\frac{f(x_i)-f(x_j)}{x_i-x_j})_{i,j=1}^N$, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,...,x_N$. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions $g$ defined on $(0,+\\infty)$ for which the matrix of divided sums $K_g=(\\frac{g(x_i)+g(x_j)}{x_i+x_j})_{i,j=1}^N$, which we call its anti-L\\\"owner matrix, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,...,x_N\\in(0,+\\infty)$. Such functions, which we call anti-L\\\"owner functions, have applications in the theory of Lyapunov-type equations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-04-27T10:29:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A60"
    ],
    "keywords": [
      "anti-lowner functions",
      "characterisation",
      "real-valued function",
      "positive semidefinite",
      "operator monotone"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.2943A"
    }
  }
}