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

Arup Chattopadhyay; Clément Coine; Saikat Giri; Chandan Pradhan

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arXiv · 2411.16426

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

Arup Chattopadhyay·Clément Coine·Saikat Giri·Chandan Pradhan

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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.

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Arup Chattopadhyay, Clément Coine, Saikat Giri, Chandan Pradhan. 2024-11-25. Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas. https://arxiv.org/abs/2411.16426

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