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

TL;DR

This paper extends trace formulas for perturbations in Schatten classes, broadening the class of functions for which second-order and higher-order trace formulas hold in various operator settings.

Contribution

It introduces a new class of admissible functions ensuring trace class second-order Taylor remainders and generalizes trace formulas for all Schatten p-perturbations with differentiable functions.

Findings

Extended trace formulas to broader function classes.

Proved modified trace formulas for all Schatten p-perturbations.

Established trace formulas for functions in the disk algebra.

Abstract

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, $1<p<\infty$, 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.