On perturbations of the isometric semigroup of shifts on the semiaxis

Perturbations $(\widetilde\tau_t)_{t\ge0}$ of the semigroup of shifts $(\tau_t)_{t\ge 0}$ on $L^2(\mathbb R_+)$ are studied under the assumption that $\widetilde\tau_t-\tau_t$ belongs to a certain Schatten–von Neumann class $\mathfrak S_p$ with $p\ge1$. It is shown that, for the unitary component in the Wold–Kolmogorov decomposition of the cogenerator of the semigroup $(\widetilde\tau_t)_{t\ge0}$, any singular spectral type may be achieved by $\mathfrak S_1$-perturbations. An explicit construction is provided for a perturbation with a given spectral type, based on the theory of model spaces of the Hardy space $H^2$. Also, it is shown that an arbitrary prescribed spectral type may be obtained for the unitary component of the perturbed semigroup by a perturbation of class $\mathfrak S_p$ with $p>1$.