{
  "id": "2105.09715",
  "version": "v1",
  "published": "2021-05-20T13:03:38.000Z",
  "updated": "2021-05-20T13:03:38.000Z",
  "title": "Development of inequality and characterization of equality conditions for the numerical radius",
  "authors": [
    "Pintu Bhunia",
    "Kallol Paul"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $A$ be a bounded linear operator on a complex Hilbert space and $\\Re(A)$ ( $\\Im(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \\begin{eqnarray*} w(A)&\\geq &\\frac{1}{2} \\left \\|A \\right\\| + \\frac{ 1}{2} \\mid \\|\\Re(A)\\|-\\|\\Im(A)\\|\\mid,\\,\\,\\mbox{and}\\\\ w^2(A)&\\geq& \\frac{1}{4} \\left \\|A^*A+AA^* \\right\\| + \\frac{1}{2}\\mid \\|\\Re(A)\\|^2-\\|\\Im(A)\\|^2 \\mid, \\end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\\frac{1}{2}\\sqrt{\\|A^*A+AA^*\\|}$ and prove that $w(A)=\\frac{1}{2}\\sqrt{\\|A^*A+AA^*\\|} $ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\\frac{1}{2}\\sqrt{\\|A^*A+AA^*\\|} $. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T13:03:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30",
      "15A60"
    ],
    "keywords": [
      "numerical radius",
      "equality conditions",
      "development",
      "characterization",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}