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

### Alexei B. Aleksandrov

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

Michigan State University, East Lansing, USA

## 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 ${\boldsymbol S}_m$, where $m$ 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 $m$th operator Taylor polynomial. In the case $m=1$ this corresponds to the Livshits–Krein trace formula, while in the case $m=2$ this corresponds to the Koplienko trace formula. In the case of an arbitrary positive integer $m$, the trace formula for the $m$th 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 $B_{\infty1}^m(\Bbb R)$.

## Cite this article

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

DOI 10.4171/JST/1