{
  "id": "2407.06724",
  "version": "v1",
  "published": "2024-07-09T09:57:06.000Z",
  "updated": "2024-07-09T09:57:06.000Z",
  "title": "Sharper bounds for the numerical radius of $n \\times n$ operator matrices II",
  "authors": [
    "Pintu Bhunia"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A=[A_{ij}]$ be an $n\\times n$ operator matrix where each $A_{ij}$ is a bounded linear operator on a complex Hilbert space $\\mathcal{H}$. With other numerical radius bounds via contraction operators, we show that $w(A) \\leq w(\\tilde{A}),$ where $\\tilde{A}=[a_{ij}]$ is an $n\\times n$ complex matrix with \\begin{eqnarray*} a_{ij}=\\begin{cases} w(A_{ii}) \\quad \\text{if } i=j\\\\ \\underset{0\\leq t \\leq 1}{\\min} \\left\\| |A_{ij}|^{2t} + |A_{ji}^*|^{2t} \\right\\|^{1/2} \\left\\| |A_{ij}^*|^{2(1-t)}+ |A_{ji}|^{2(1-t)} \\right\\|^{1/2} \\quad \\text{if } i< j 0 \\quad \\text{if } i> j. \\end{cases} \\end{eqnarray*} This bound refines the well known bound $w(A) \\leq w(\\hat{A}),$ where $\\hat{A}=[\\hat{a}_{ij}]$ is an $n\\times n$ matrix with $\\hat{a}_{ij}= w(A_{ii}) $ \\text{if } $i=j$ and $\\hat{a}_{ij}= \\|A_{ij}\\| $ \\text{if } $i\\neq j$ [Linear Algebra Appl. 468 (2015), 18--26]. We deduce that if $A$, $B$ are bounded linear operators on $\\mathcal{H},$ then \\begin{eqnarray*} w\\left(\\begin{bmatrix} 0&A\\\\ B&0 \\end{bmatrix}\\right) \\leq \\frac12 \\left\\| |A|^{2t} + |B^*|^{2t} \\right\\|^{1/2} \\left\\| |A^*|^{2(1-t)}+ |B|^{2(1-t)} \\right\\|^{1/2} \\quad \\text{for all } t\\in [0,1]. \\end{eqnarray*} Further by applying the numerical radius bounds of operator matrices, we deduce some numerical radius bounds for a single operator, the product of two operators, the commutator of operators. We show that if $A$ is a bounded linear operator on $\\mathcal{H},$ then $w(A) \\leq \\frac12 \\|A\\|^t \\left\\| |A|^{1-t}+|A^*|^{1-t} \\right\\| \\quad \\text{for all } t\\in [0,1],$ which refines as well as generalizes the existing ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T09:57:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30",
      "15A60",
      "47A63"
    ],
    "keywords": [
      "operator matrix",
      "numerical radius bounds",
      "bounded linear operator",
      "sharper bounds",
      "complex hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}