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

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 $\int|\boldsymbol{\xi}(t)|(1+|t|)^{-1}\,{\rm d}t<\infty$ for the spectral shift function $\boldsymbol{\xi}$ in the case of relatively trace class perturbations.