Traces of operators with a relatively compact perturbation

This is a continuation of the authors' series of papers on the theory of regularized traces of abstract discrete operators. We prove a theorem in which the perturbing operator B is subordinate to the operator A (0) in the sense that BA(0)(-delta) is a compact operator belonging to some Schatten-von Neumann class of finite order. Apart from covering new classes of operators, and in contrast to our preceding papers, we give a unified statement of the theorem regardless of whether the resolvent of the unperturbed operator belongs to the trace class. Two examples are given in which the result is applied to ordinary differential operators as well as to partial differential operators.