Traces of operators with relatively compact perturbations

Regularized trace formulae are proved for abstract operators with a perturbing operator $B$ subordinate to the non-perturbed operator $A_0$ in the following sense: $BA_0^{-1}$ is a compact operator in some Schatten–von Neumann class of finite order. Two essentially different cases can be distinguished here: the resolvent of $A_0$ is either of trace class or not. Five theorems describing various cases of operator subordination and the structure of the spectrum of the unperturbed operator are proved.