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

TL;DR

This paper explores how model spaces and inner functions can be used to construct cocyclic perturbations of the semigroup of shifts on the semiaxis, controlling spectral types and Schatten class differences.

Contribution

It introduces a method to construct cocyclic perturbations with prescribed spectral types using model spaces and inner functions, including trace class perturbations.

Findings

Perturbed semigroup elements can have prescribed spectral types.

Perturbations differ from original operators by Schatten-von Neumann class operators.

Special case analysis for trace class perturbations.

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.