{
  "id": "1407.5133",
  "version": "v2",
  "published": "2014-07-18T23:25:22.000Z",
  "updated": "2014-08-26T12:37:47.000Z",
  "title": "Spectral radius, numerical radius, and the product of operators",
  "authors": [
    "Rahim Alizadeh",
    "Mohammad B. Asadi",
    "Che-Man Cheng",
    "Wanli Hong",
    "Chi-Kwong Li"
  ],
  "comment": "9 pages",
  "journal": "J. Math. Anal. Appl., 2014",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\sigma(A)$, $\\rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$\\rho(AB)\\le r(A)r(B) \\quad\\text{ for all bounded linear operators } B$$ if and only if there is a unique $\\mu \\in \\sigma (A)$ satisfying $|\\mu| = \\rho(A)$ and $A = \\frac{\\mu(I + L)}{2}$ for a contraction $L$ with $1\\in\\sigma(L)$. One can get the same conclusion on $A$ if $\\rho(AB) \\le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C \\oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) \\ge 1$ for all unit vector $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-18T23:25:22.000Z",
      "title": "Spectral radius and numerical radius of operators product",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-26T12:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12"
    ],
    "keywords": [
      "spectral radius",
      "numerical radius",
      "operators product",
      "bounded linear operator",
      "hilbert space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5133A"
    }
  }
}