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 , where 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 th operator Taylor polynomial. In the case this corresponds to the Livshits–Krein trace formula, while in the case this corresponds to the Koplienko trace formula. In the case of an arbitrary positive integer , the trace formula for the th operator Taylor polynomial was obtained recently in . Our results allow us to essentially enlarge the class of functions, for which the trace formula obtained in  holds. Namely, we prove that it holds for functions in the Besov space .
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Alexei B. Aleksandrov, Vladimir V. Peller, Trace formulae for perturbations of class . J. Spectr. Theory 1 (2011), no. 1, pp. 1–26DOI 10.4171/JST/1