Applications of model spaces to construction of cocyclic perturbations of a semigroup of shifts on a semiaxis

Abstract

We describe a construction of cocyclic perturbations of the semigroup of shifts on the half-line by means of 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.