On necessary and sufficient condition in theory of regularized traces

The present work is devoted to studying the regularized trace formulae for symmetric $L_0$-compact perturbations of a self-adjoint lower semi-bounded operator $L_0$ with a discrete spectrum in a separable Hilbert space. By now, the studies of the regularized trace formulae for the perturbations of abstract self-adjoint discrete operators were mostly aimed on finding a sufficient condition, under which the regularized sum with brackets minus first or several leading terms of the perturbation theory vanished. This condition was formulated in terms of spectral characteristics of an unperturbed operator $L_0$ depending on the belonging of a perturbing operator $V$ to some class. In particular, recently, the traces formulae for model two-dimensional operators in mathematical physics have been intensively studied with a perturbation described by the multiplication operator. Here we study a necessary and sufficient condition for two cases, namely, as the regularized trace with brackets and deduction of the first corrector of the perturbation theory vanishes or is equal to a finite number. We consider a certain summation bracket, which usually arises in the theory of regularized traces of the perturbations of partial differential operators.