On applications of the model spaces to the construction of cocyclic perturbations of the semigroup of shifts on the semiaxis

Abstract

We describe a construction of cocyclic perturbations of the semigroup of shifts on the semiaxis by means of the theory of model spaces. It is shown that one can choose an inner function that determines the model space so that the elements of the perturbed semigroup have a prescribed spectral type and differ from the elements of the initial semigroup by operators from the Schatten-von Neumann class $\mathfrak{S}_p$, $p>1$. The case of the trace class $\mathfrak{S}_1$ perturbations is considered separately.