Perturbations of operator semigroups generated by operator-valued measures

The report will present a new method for constructing perturbations based on a generalization of the integral equation of variation of parameters, which develops the construction of A.S. Holevo. Instead of the Lebesgue measure on $ {\ mathbb R} $, integration is carried out over an operator-valued measure. Moreover, in order for the operator-valued measure to correctly generalize the absolutely continuous one in the original equation, the covariance condition with respect to the perturbed semigroup is imposed on it. In this paper, we study a perturbation of an abstract $ C_0 $ -semigroup in a Banach space using an arbitrary covariant operator-valued measure. Under technical constraints on the measure, it is proved that a solution to an integral equation with a measure exists, is unique, and is a $ C_0 $ -semigroup. An example of a perturbation of a semigroup of shifts on a half-line is described, showing that the domain of the generator can change. An example of a perturbation of a quantum dynamical semigroup in the Fock space by an unbounded covariant measure is also presented, the idea of ​​which is taken from the investigated restricted case.