{
  "id": "2305.17657",
  "version": "v1",
  "published": "2023-05-28T07:47:07.000Z",
  "updated": "2023-05-28T07:47:07.000Z",
  "title": "Power numerical radius inequalities from an extension of Buzano's inequality",
  "authors": [
    "Pintu Bhunia"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Several numerical radius inequalities are studied by developing an extension of the Buzano's inequality. It is shown that if $T$ is a bounded linear operator on a complex Hilbert space, then \\begin{eqnarray*} w^n(T) &\\leq& \\frac{1}{2^{n-1}} w(T^n)+ \\sum_{k=1}^{n-1} \\frac{1}{2^{k}} \\left\\|T^k \\right\\| \\left\\|T \\right\\|^{n-k}, \\end{eqnarray*} for every positive integer $n\\geq 2.$ This is a non-trivial improvement of the classical inequality $w(T)\\leq \\|T\\|.$ The above inequality gives an estimation for the numerical radius of the nilpotent operators, i.e., if $T^n=0$ for some least positive integer $n\\geq 2$, then \\begin{eqnarray*} w(T) &\\leq& \\left(\\sum_{k=1}^{n-1} \\frac{1}{2^{k}} \\left\\|T^k \\right\\| \\left\\|T \\right\\|^{n-k}\\right)^{1/n} \\leq \\left( 1- \\frac{1}{2^{n-1}}\\right)^{1/n} \\|T\\|. \\end{eqnarray*} Also, we deduce a reverse inequality for the numerical radius power inequality $w(T^n)\\leq w^n(T)$. We show that if $\\|T\\|\\leq 1$, then \\begin{eqnarray*} w^n(T) &\\leq& \\frac{1}{2^{n-1}} w(T^n)+ 1- \\frac{1}{2^{n-1}}, \\end{eqnarray*} for every positive integer $n\\geq 2.$ This inequality is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-28T07:47:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30",
      "15A60"
    ],
    "keywords": [
      "power numerical radius inequalities",
      "buzanos inequality",
      "positive integer",
      "complex hilbert space",
      "numerical radius power inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}