Non-nuclear perturbations of discrete operators and trace formulae

A trace formula is obtained for unbounded discrete operators perturbed by a Hilbert–Schmidt operator; this formula may be called the discrete analogue of M. Krein's formula for nuclear perturbations. A regularized trace formula of Krein's type is also proved for perturbations in the class $S^p$, $2<p\in\mathbb N$, for arbitrary compact and relatively compact perturbations depending on the behaviour at infinity of the distribution function of the spectrum of the unperturbed operator.