Formula of the regularized trace for perturbation in the Schatten–von Neumann of discrete operators

In the paper we study a formula of the regularized trace for a perturbation in Schatten–von Neumann class ($\sigma_p$, $p\in\mathbb N$) of discrete self-adjoint operators. We prove that the regularized vanishes after deducting $(p-1)$ terms of perturbation theory if there are no dilating gaps in the spectrum of the unperturbed operator.