{
  "id": "2010.12750",
  "version": "v1",
  "published": "2020-10-24T02:59:58.000Z",
  "updated": "2020-10-24T02:59:58.000Z",
  "title": "Refinements of norm and numerical radius inequalities",
  "authors": [
    "Pintu Bhunia",
    "Kallol Paul"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ \\frac{1}{4}\\|A^*A+AA^*\\| \\leq \\frac{1}{8}\\bigg( \\|A+A^*\\|^2+\\|A-A^*\\|^2 +c^2(A+A^*)+c^2(A-A^*)\\bigg) \\leq w^2(A)$$ and \\begin{eqnarray*} \\frac{1}{2}\\|A^*A+AA^*\\| - \\frac{1}{4}\\bigg\\|(A+A^*)^2 (A-A^*)^2 \\bigg\\|^{1/2} \\leq w^2(A) \\leq \\frac{1}{2}\\|A^*A+AA^*\\|, \\end{eqnarray*} %$$ \\frac{1}{4}\\|A^*A+AA^*\\| \\leq \\frac{1}{2}w^2(A) + \\frac{1}{8}\\bigg\\|(A+A^*)^2 (A-A^*)^2 \\bigg\\|^{1/2}\\leq w^2(A),$$ where $\\|.\\|$, $w(.)$ and $c(.)$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then \\begin{eqnarray*} \\|AD^*\\| \\leq \\left\\| \\int_0^1 \\left( (1-t) \\left(\\frac{ |A|^2+|D|^2}{2}\\right) +t\\|AD^*\\|I \\right)^2dt \\right\\|^{1/2} \\leq \\frac{1}{2}\\left\\| |A|^2+|D|^2 \\right\\|, \\end{eqnarray*} where $|A|=(A^*A)^{1/2}$ and $|D|=(D^*D)^{1/2}$. This is a refinement of well known inequality obtained by Bhatia and Kittaneh.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-24T02:59:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30"
    ],
    "keywords": [
      "numerical radius inequalities",
      "inequality",
      "complex hilbert space",
      "bounded linear operator",
      "refinement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}