Paper detail

On perturbations of the isometric semigroup of shifts on the semiaxis

We study perturbations $(\tildeτ_t)_{t\ge 0}$ of the semigroup of shifts $(τ_t)_{t\ge 0}$ on $L^2(\R_+)$ with the property that $\tildeτ_t - τ_t$ 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 $(\tildeτ_t)_{t\ge 0}$, {\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 $p>1$.