An extension of the Koplienko-Neidhardt trace formulae

Koplienko [Ko] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class ${\text{bS}}_2$. A similar formula in the case of unitary operators was obtained by Neidhardt [N]. In this paper we improve their results and obtain sharp conditions under which the Koplienko–Neidhardt trace formulae hold.