{
  "id": "1401.1804",
  "version": "v2",
  "published": "2014-01-08T20:37:31.000Z",
  "updated": "2014-05-29T17:41:13.000Z",
  "title": "Chebyshev type inequalities for Hilbert space operators",
  "authors": [
    "Mohammad Sal Moslehian",
    "Mojtaba Bakherad"
  ],
  "comment": "18 pages, to appear in J. Math. Anal. Appl. (JMAA)",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\\mu$, $\\alpha: T\\rightarrow [0, +\\infty)$ is a measurable function and $(A_t)_{t\\in T}, (B_t)_{t\\in T}$ are suitable continuous fields of operators in ${\\mathscr A}$ having the synchronous Hadamard property, then \\begin{align*} \\int_{T} \\alpha(s) d\\mu(s)\\int_{T}\\alpha(t)(A_t\\circ B_t) d\\mu(t)\\geq\\left(\\int_{T}\\alpha(t) A_t d\\mu(t)\\right)\\circ\\left(\\int_{T}\\alpha(s) B_s d\\mu(s)\\right). \\end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\\times n$ matrices. Several applications are given as well.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-29T17:41:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "47A60",
      "46L05"
    ],
    "keywords": [
      "hilbert space operators",
      "chebyshev type inequalities",
      "inequality",
      "main version deals",
      "compact hausdorff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.1804S"
    }
  }
}