On perturbations of the isometric semigroup of shifts on the semiaxis

Abstract

We study perturbations $({\tilde{\tau }}_t{)}_{t\ge 0}$ of the semigroup of shifts $({\tau }_t{)}_{t\ge 0}$ on $L^2({\mathbb{R}}_{+})$ with the property that ${\tilde{\tau }}_t-{\tau }_t$ belongs to a certain Schatten-von Neumann class $\gS_p$ with $p\ge 1$. We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $({\tilde{\tau }}_t{)}_{t\ge 0}$, {\it any singular} spectral type may be achieved by $\gS_1$ perturbations. We provide an explicit construction for a perturbation with a given spectral type based on the theory of model spaces of the Hardy space $H^2$. Also we show that we may obtain {\it any} prescribed spectral type for the unitary component of the perturbed semigroup by a perturbation from the class $\gS_p$ with $p>1$.

Cited by in corpus (1)

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