Trace formulae for perturbations of class ${\mathbf{S}}_{\mathbf{m}}$

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 ${\mathbf{S}}_m$, 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 $B_{\infty 1}^m(\mathbb{R})$.