{
  "id": "1705.04782",
  "version": "v1",
  "published": "2017-05-13T02:49:43.000Z",
  "updated": "2017-05-13T02:49:43.000Z",
  "title": "A trace formula for functions of contractions and analytic operator Lipschitz functions",
  "authors": [
    "Mark Malamud",
    "Hagen Neidhardt",
    "Vladimir Peller"
  ],
  "comment": "6 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV",
    "math.SP"
  ],
  "abstract": "In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\\in\\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\\Bbb D}$, then $f(T)-f(R)\\in\\boldsymbol{S}_1$. The main result of the note says that there exists a function $\\boldsymbol{\\xi}$ (a spectral shift function) on the unit circle ${\\Bbb T}$ of class $L^1({\\Bbb T})$ such that the following trace formula holds: $\\operatorname{trace}(f(T)-f(R))=\\int_{\\Bbb T} f'(\\zeta)\\boldsymbol{\\xi}(\\zeta)\\,d\\zeta$, whenever $T$ and $R$ are contractions with $T-R\\in\\boldsymbol{S}_1$ and $f$ is an operator Lipschitz function analytic in ${\\Bbb D}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-13T02:49:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "analytic operator lipschitz functions",
      "operator lipschitz function analytic",
      "contractions",
      "trace class difference",
      "spectral shift function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}