Trace formulae for perturbations of class $\text{bs}{\text{bS}}_m$

We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class ${\text{bS}}_m$, 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 ${\text{bS}}_m$. We also improve the trace formula obtained in \cite{PSS} for operator Taylor polynomials and prove it for arbitrary functions in he Besov space $B_{\text{be}1}^m(\mathbb{R})$. We consider several other special cases of our general trace formulae. In particular, we establish a trace formula for $m$th order operator differences.