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: B A(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.