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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 17/2007

G. Amosov, A. Baranov, V. Kapustin ##
ON
MARKOVIAN PERTURBATIONS OF THE GROUP
OF SHIFTS ON THE REAL LINE

This preprint was accepted December 21, 2007

ABSTRACT:
We study perturbations $(\tilde\tau_t)_{t\in\R}$ of the group of shifts
$(\tau_t)_{t\in\R}$ acting on $L^2(\R)$ by $(\tau_t f)(x)=f(x-t)$,
such that $\tilde\tau_t - \tau_t$
belongs to a certain Schatten--von Neumann class $\S_p$ with $p\ge 1$,
and the perturbations have the Markovian properties in the following sense:
if $f\equiv 0$ on $\R_+$, then, for $t<0$, $\tau_t f=\tilde\tau_t f$.
The problem is to describe possible spectral types of the unitary component
of $\tilde \tau_t$. Previously, we studied this problem for
the isometric semigroup
$(\tilde\tau_t|_{L^2(\R_+)})_{t\ge 0}$ using a special model
in the Hardy space. It was shown that, for $\S_1$-perturbations of
the isometric semigroups, any singular spectral type may be obtained,
whereas for $\S_p$ perturbations the perturbed semigroup
$(\tilde\tau_t|_{L^2(\R_+)})_{t\ge 0}$
may have arbitary spectral type of the unitary component.
For $p>1$ the statement remains true for
the unitary dilations $(\tilde\tau_t)_{t\in \R}$; however, in terms of
our model the difference $\tilde\tau_t - \tau_t$ cannot be in $\S_1$.

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