Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. Relatively operator Lipschitz functions

TL;DR

This work develops a comprehensive framework for functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. By introducing and characterizing the class of relatively operator Lipschitz (ROL) functions through double operator integrals and Schur multipliers, the authors extend perturbation theory beyond classical trace-class differences. A central result is that the trace formula holds for all $f$ in ${\rm ROL}$ when the perturbation is relatively trace class, with a spectral shift function $\boldsymbol{\xi}$ satisfying $\int |\boldsymbol{\xi}(t)|/(1+t^2)\,dt<\infty$; moreover, ${\rm ROL}$ is shown to be the maximal function class for which this trace formula remains valid. The paper also provides DOI-based representations, necessity results for the multiplier conditions, and a suite of commutator and differentiation formulas, all enabling a robust operator-theoretic treatment of perturbations via double operator integrals and divided differences.

Abstract

We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider study the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also gives us a new approach to the inequality $\int|\boldsymbolξ(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$ for the spectral shift function $\boldsymbolξ$ in the case of relatively trace class perturbations.