Trace formulae for perturbations of class $\bs{\bS_m}$

TL;DR

This paper develops comprehensive trace formulae for self-adjoint operator perturbations within Schatten class $S_m$, extending previous formulas and including new results for operator differences and functions in Besov spaces.

Contribution

It introduces the most general trace formulae for perturbations in Schatten class $S_m$, improving previous results and covering a wider class of functions and operator differences.

Findings

Established general trace formulae for $S_m$ perturbations.

Extended trace formula for operator Taylor polynomials to Besov space functions.

Derived trace formulae for $m$th order operator differences.

Abstract

We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained. This formula includes the Livshits--Krein trace formula in the case $m=1$ and the Koplienko trace formula in the case $m=2$. We establish most general trace formulae in the case of perturbation of Schatten--von Neumann class $\bS_m$. We also improve the trace formula obtained in \cite{PSS} for operator Taylor polynomials and prove it for arbitrary functions in he Besov space $B_{\be1}^m(\R)$. We consider several other special cases of our general trace formulae. In particular, we establish a trace formula for $m$th order operator differences.