Trace formulas for ${\mathcal{S}}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas

In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class ${\mathcal{S}}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten ${\mathcal{S}}^p$-perturbation, $1<p<\mathrm{\infty }$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case.