{
  "id": "2404.01982",
  "version": "v1",
  "published": "2024-04-02T14:20:03.000Z",
  "updated": "2024-04-02T14:20:03.000Z",
  "title": "Norm Inequalities for Hilbert space operators with Applications",
  "authors": [
    "Pintu Bhunia"
  ],
  "comment": "23 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \\begin{eqnarray*} \\|A\\|_{p} &\\leq &\\left(\\textit{rank} \\, A\\right)^{1/{2p}} \\|A\\|_{2p} \\,\\, \\leq \\,\\, \\left(\\textit{rank} \\, A\\right)^{{(2p-1)}/{2p^2}} \\|A\\|_{2p^2}, \\quad \\textit{for all $p\\geq 1 $} \\end{eqnarray*} where $\\|\\cdot\\|_p$ is the Schatten $p$-norm. If $\\{ \\lambda_n(A) \\}$ is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator $A$, then we show that \\begin{eqnarray*} \\sum_{n} \\left|\\lambda_n(A)\\right|^{p} &\\leq& \\frac12 \\| A\\|_{ p}^{ p} + \\frac12 \\| A^2\\|_{p/2}^{p/2}, \\quad \\textit{for all $p\\geq 2$} \\end{eqnarray*} which improves the classical Weyl's inequality $\\sum_{n} \\left|\\lambda_n(A)\\right|^{p} \\leq \\| A\\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA 1949]. For an $n\\times n$ matrix $A$, we show that the function $p\\to n^{-{1}/{p}}\\|A\\|_p$ is monotone increasing on $p\\geq 1,$ complementing the well known decreasing nature of $p\\to \\|A\\|_p.$ \\indent As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph $G$, namely, $\\mathcal{E}(G) \\leq \\sqrt{2m\\left(\\textit{rank Adj(G)} \\right)},$ where $m$ is the number of edges, improving on a bound by McClelland in $1971$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-02T14:20:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A60",
      "47A30",
      "47A12",
      "26C10",
      "05C50"
    ],
    "keywords": [
      "hilbert space operators",
      "application",
      "upper bound",
      "unitarily invariant norm inequalities",
      "finite rank operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}