{
  "id": "0908.3623",
  "version": "v1",
  "published": "2009-08-25T14:13:52.000Z",
  "updated": "2009-08-25T14:13:52.000Z",
  "title": "Functions of operators under perturbations of class $\\bS_p$",
  "authors": [
    "A. B. Aleksandrov",
    "V. V. Peller"
  ],
  "comment": "49 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV",
    "math.SP"
  ],
  "abstract": "This is a continuation of our paper \\cite{AP2}. We prove that for functions $f$ in the H\\\"older class $\\L_\\a(\\R)$ and $1<p<\\be$, the operator $f(A)-f(B)$ belongs to $\\bS_{p/\\a}$, whenever $A$ and $B$ are self-adjoint operators with $A-B\\in\\bS_p$. We also obtain sharp estimates for the Schatten--von Neumann norms $\\big\\|f(A)-f(B)\\big\\|_{\\bS_{p/\\a}}$ in terms of $\\|A-B\\|_{\\bS_p}$ and establish similar results for other operator ideals. We also estimate Schatten--von Neumann norms of higher order differences $\\sum\\limits_{j=0}^m(-1)^{m-j}(m\\j)f\\big(A+jK\\big)$. We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on $f$ for $f(A)-f(B)$ to belong to $\\bS_q$ under the assumption that $A-B\\in\\bS_p$. We also obtain Schatten--von Neumann estimates for quasicommutators $f(A)Q-Qf(B)$, and introduce a spectral shift function and find a trace formula for operators of the form $f(A-K)-2f(A)+f(A+K)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-25T14:13:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55",
      "47B10",
      "47B35",
      "47B49",
      "47A60",
      "46E15",
      "46E35"
    ],
    "keywords": [
      "perturbations",
      "estimate schatten-von neumann norms",
      "schatten-von neumann estimates",
      "higher order differences",
      "spectral shift function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.3623A"
    }
  }
}