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

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.