{
  "id": "2105.09718",
  "version": "v1",
  "published": "2021-05-20T13:12:45.000Z",
  "updated": "2021-05-20T13:12:45.000Z",
  "title": "Numerical radius inequalities of $2 \\times 2$ operator matrices",
  "authors": [
    "Pintu Bhunia",
    "Kallol Paul"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Several upper and lower bounds for the numerical radius of $2 \\times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then \\begin{eqnarray*} && \\frac{1}{2}\\max \\left \\{ \\|B\\|, \\|C\\| \\right \\}+\\frac{1}{4} \\left | \\|B+C^*\\|-\\|B-C^*\\| \\right | &&\\leq w \\left(\\left[\\begin{array}{cc} 0 & B C& 0 \\end{array}\\right]\\right)\\\\ &&\\leq \\frac{1}{2} \\max \\left\\{\\|B\\|,\\|C\\|\\right \\}+\\frac{1}{2}\\max \\left \\{r^{\\frac{1}{2}}(|B||C^*|),r^{\\frac{1}{2}}(|B^*||C|)\\right\\}, \\end{eqnarray*} where $w(.)$, $r(.)$ and $\\|.\\|$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $\\left[\\begin{array}{cc} 0 & B C& 0 \\end{array}\\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\\max \\Big \\{\\|B+C\\|^2 , \\|B-C\\|^2 \\Big\\}\\leq \\left \\|B^2+C^2 \\right \\|+2w(|B||C|). $",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T13:12:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30"
    ],
    "keywords": [
      "operator matrix",
      "numerical radius inequalities",
      "bounded linear operator",
      "complex hilbert space",
      "operator norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}