{
  "id": "2308.09252",
  "version": "v1",
  "published": "2023-08-18T02:26:47.000Z",
  "updated": "2023-08-18T02:26:47.000Z",
  "title": "Euclidean operator radius and numerical radius inequalities",
  "authors": [
    "Suvendu Jana",
    "Pintu Bhunia",
    "Kallol Paul"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $T$ be a bounded linear operator on a complex Hilbert space $\\mathscr{H}.$ We obtain various lower and upper bounds for the numerical radius of $T$ by developing the Euclidean operator radius bounds of a pair of operators, which are stronger than the existing ones. In particular, we develop an inequality that improves on the inequality $$ w(T) \\geq \\frac12 {\\|T\\|}+\\frac14 {\\left|\\|Re(T)\\|-\\frac12 \\|T\\| \\right|} + \\frac14 { \\left| \\|Im(T)\\|-\\frac12 \\|T\\| \\right|}.$$ Various equality conditions of the existing numerical radius inequalities are also provided. Further, we study the numerical radius inequalities of $2\\times 2$ off-diagonal operator matrices. Applying the numerical radius bounds of operator matrices, we develop the upper bounds of $w(T)$ by using $t$-Aluthge transform. In particular, we improve the well known inequality $$ w(T) \\leq \\frac12 {\\|T\\|}+ \\frac12{ w(\\widetilde{T})}, $$ where $\\widetilde{T}=|T|^{1/2}U|T|^{1/2}$ is the Aluthge transform of $T$ and $T=U|T|$ is the polar decomposition of $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-18T02:26:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "15A60",
      "47A30",
      "47A50"
    ],
    "keywords": [
      "inequality",
      "euclidean operator radius bounds",
      "aluthge transform",
      "upper bounds",
      "complex hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}