{
  "id": "1008.1638",
  "version": "v1",
  "published": "2010-08-10T06:08:52.000Z",
  "updated": "2010-08-10T06:08:52.000Z",
  "title": "Functions of normal operators under perturbations",
  "authors": [
    "Alexei Aleksandrov",
    "Vladimir Peller",
    "Denis Potapov",
    "Fedor Sukochev"
  ],
  "comment": "32 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV",
    "math.SP"
  ],
  "abstract": "In \\cite{Pe1}, \\cite{Pe2}, \\cite{AP1}, \\cite{AP2}, and \\cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the H\\\"older class $\\L_\\a(\\R^2)$, $0<\\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\\|f(N_1)-f(N_2)\\|\\le\\const\\|f\\|_{\\L_\\a}\\|N_1-N_2\\|^\\a$. We obtain a more general result for functions in the space $\\L_\\o(\\R^2)=\\big\\{f:~|f(\\z_1)-f(\\z_2)|\\le\\const\\o(|\\z_1-\\z_2|)\\big\\}$ for an arbitrary modulus of continuity $\\o$. We prove that if $f$ belongs to the Besov class $B_{\\be1}^1(\\R^2)$, then it is operator Lipschitz, i.e., $\\|f(N_1)-f(N_2)\\|\\le\\const\\|f\\|_{B_{\\be1}^1}\\|N_1-N_2\\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\\in\\L_\\a(\\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\\bS_p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-10T06:08:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normal operators",
      "perturbations",
      "schatten-von neuman class",
      "operator lipschitz",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.1638A"
    }
  }
}