Relatively Operator Lipschitz Functions of Dissipative Operators

In this note we study the behaviour of functions of maximal dissipative operators under relatively bounded and relatively trace class perturbations. We introduce the class of analytic relatively operator Lipschitz functions.We obtain a formula for the derivative in the strong operator topology in the parameter of functions of one-parametric families of dissipative operators.We also establish a trace formula for the difference of a function of a perturbed operator and the function of the initial operator. It turns out that the corresponding spectral shift function is integrable with weight (1+|x|). Moreover, the maximal class of functions, for which the trace formula holds for all pairs of maximal dissipative operators under relatively trace class perturbations coincides with the class of analytic relatively operator Lipschitz functions.