Functions of dissipative operators under relatively bounded and relatively trace class perturbations

TL;DR

This paper investigates how functions of dissipative operators behave under specific types of perturbations, introducing classes of Lipschitz functions and establishing trace formulas and spectral shift function properties.

Contribution

It introduces and analyzes analytic relatively operator Lipschitz functions and establishes trace formulas for dissipative operators under trace class perturbations.

Findings

Trace formula for relatively trace class perturbations

Spectral shift function integrability result

Characterization of maximal function classes for trace formulas

Abstract

We study the behaviour of functions of dissipative operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of analytic relatively operator Lipschitz functions. An essential role is played by double operator integrals with respect to semispectral measures. We also study the class of analytic 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 analytic relatively operator Lipschitz functions. We also establish the inequality $\int|\boldsymbol{\xi}(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$ for the spectral shift function $\boldsymbol\xi$ in the case of relatively trace class perturbations.