Singular traces and perturbation formulae of higher order

TL;DR

This paper extends higher order perturbation formulas for traces of self-adjoint operators with perturbations in weak trace class ideals, generalizing earlier formulas to broader classes of perturbations and measures.

Contribution

It introduces higher order perturbation formulas involving traces and measures for operators with perturbations in weak trace class and quasi-Banach ideals.

Findings

Extended perturbation formulas to weak trace class ideal

Generalized formulas to quasi-Banach ideals

Connected measure representations with operator perturbations

Abstract

Let $H, V$ be self-adjoint operators such that $V$ belongs to the weak trace class ideal. We prove higher order perturbation formula $$\tau\big(f(H+V)-\sum_{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{dt^j} f(H+tV)\big|_{t=0}\big)=\int_{\mathbb{R}} f^{(n)}(t)\,dm_n(t),$$ where $\tau$ is a trace on the weak trace class ideal and $m_n$ is a finite measure that is not necessarily absolutely continuous. This result extends the first and second order perturbation formulas of Dykema and Shripka, who generalised the Krein and Koplienko trace formulas to the weak trace class ideal. We also establish the perturbation formulae when the perturbation $V$ belongs to the quasi-Banach ideal weak-$L_n$ for any $n \geq 1$.