{
  "id": "0905.4855",
  "version": "v1",
  "published": "2009-05-29T13:11:49.000Z",
  "updated": "2009-05-29T13:11:49.000Z",
  "title": "Lipschitz functions of perturbed operators",
  "authors": [
    "Fyodor Nazarov",
    "Vladimir Peller"
  ],
  "comment": "6 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV",
    "math.SP"
  ],
  "abstract": "We prove that if $f$ is a Lipschitz function on $\\R$, $A$ and $B$ are self-adjoint operators such that ${\\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\\boldsymbol{S}_{1,\\be}$, i.e., $s_j(A-B)\\le{\\rm const} (1+j)^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class $\\boldsymbol{S}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\\in\\boldsymbol{S}_\\Omega$, i.e., $\\sum_{j=0}^ns_j(f(A)-f(B))\\le\\const\\log(2+n)$. We also obtain more general results about the behavior of double operator integrals of the form $Q=\\iint(f(x)-f(y))(x-y)^{-1}dE_1(x)TdE_2(y)$, where $E_1$ and $E_2$ are spectral measures. We show that if $T\\in\\boldsymbol{S}_1$, then $Q\\in\\boldsymbol{S}_\\Omega$ and if $\\rank T=1$, then $Q\\in\\boldsymbol{S}_{1,\\be}$. Finally, if $T$ belongs to the Matsaev ideal $\\boldsymbol{S}_\\omega$, then $Q$ is a compact operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-05-29T13:11:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55",
      "47B49",
      "47A60"
    ],
    "keywords": [
      "lipschitz function",
      "perturbed operators",
      "self-adjoint operators",
      "compact operator",
      "spectral measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.4855N"
    }
  }
}