On rank one perturbations of semigroup of shifts on half-axis

We study a special case of perturbations of the semigroup of shifts on the half-axis changing the domain of its generator. We consider a rank one perturbation of generator defined by an exponential. We show that such perturbation of the generator always produces the generator of some $C_0$-semigroup, the action of which is described explicitly. The criterion of isometricity and contractivity of the perturbed semigroup is obtained. For the contractive case, we show that the considered generator perturbation produces a rank one perturbation of the cogenerator. The studied special case is used to build a model of perturbation for the semigroup of shifts defined by an integral equation with respect to some operator-valued measure. In the case when the domain of the generator remains unchanged, this integral equation is reduced to a well-known equation of the perturbation theory, where the integration is made with respect to the usual Lebesgue measure. If the domain is changed, the perturbation satisfies an integral equation with a nontrivial measure that having no density with respect to the Lebesgue measure. We study completely the problem of constructing an operator-valued measure that defines the integral equation relating the perturbed semigroup with the original one. The measure, when it exists, is obtained explicitly and we show that it is defined non-uniquely. We study the possibility of choosing an operator-valued measure with values in the set of self-adjoint and positive operators.