{
  "id": "1510.08132",
  "version": "v1",
  "published": "2015-10-27T23:41:35.000Z",
  "updated": "2015-10-27T23:41:35.000Z",
  "title": "On mapping theorems for numerical range",
  "authors": [
    "Hubert Klaja",
    "Javad Mashreghi",
    "Thomas Ransford"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\\le\\|f\\|_\\infty$. We give a new and elementary proof of this result using finite Blaschke products. A well-known result relating numerical radius and norm says $\\|T\\| \\leq 2w(T)$. We obtain a local improvement of this estimate, namely, if $w(T)\\le1$ then \\[ \\|Tx\\|^2\\le 2+2\\sqrt{1-|\\langle Tx,x\\rangle|^2} \\qquad(x\\in H,~\\|x\\|\\le1). \\] Using this refinement, we give a simplified proof of Drury's teardrop theorem, which extends the Berger-Stampfli theorem to the case $f(0)\\ne0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-27T23:41:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical range",
      "mapping theorems",
      "finite blaschke products",
      "well-known result relating numerical radius",
      "drurys teardrop theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}