JournalsjstVol. 1, No. 1pp. 1–26

Trace formulae for perturbations of class Sm\boldsymbol{{\boldsymbol S}_m}

  • Alexei B. Aleksandrov

    Steklov Institute of Mathematics, St-Petersburg, Russian Federation
  • Vladimir V. Peller

    Michigan State University, East Lansing, USA
Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$ cover
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Abstract

We offer a new approach to trace formulae for functions of perturbed self-adjoint operators. We establish most general trace formulae in the case of perturbation of Schatten–von Neumann class Sm{\boldsymbol S}_m, where mm is a positive integer. We consider several special cases of our general trace formulae. In particular, as a special case we obtain the trace formula for the mmth operator Taylor polynomial. In the case m=1m=1 this corresponds to the Livshits–Krein trace formula, while in the case m=2m=2 this corresponds to the Koplienko trace formula. In the case of an arbitrary positive integer mm, the trace formula for the mmth operator Taylor polynomial was obtained recently in [25]. Our results allow us to essentially enlarge the class of functions, for which the trace formula obtained in [25] holds. Namely, we prove that it holds for functions in the Besov space B1m(R)B_{\infty1}^m(\Bbb R).

Cite this article

Alexei B. Aleksandrov, Vladimir V. Peller, Trace formulae for perturbations of class Sm\boldsymbol{{\boldsymbol S}_m}. J. Spectr. Theory 1 (2011), no. 1, pp. 1–26

DOI 10.4171/JST/1