V. V. Kapustin | Searcharxiv

arXiv2012

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 $p>1$.

math.FA↗

arXiv2012

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

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.

math.FA↗