On perturbations of the isometric semigroup of shifts on the semiaxis
Abstract
We study perturbations
(
τ
~
t
)
t≥0
of the semigroup of shifts
(
τ
t
)
t≥0
on $L^2(\R_+)$ with the property that
τ
~
t
−
τ
t
belongs to a certain Schatten-von Neumann class $\gS_p$ with
p≥1
. We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup
(
τ
~
t
)
t≥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
.