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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 15/2007

G. Amosov, A. Baranov, V. Kapustin ##
ON PERTURBATIONS OF THE ISOMETRIC SEMIGROUP
OF SHIFTS ON THE SEMIAXIS

This preprint was accepted December 17, 2007

ABSTRACT:
We study perturbations $(\tilde\tau_t)_{t\ge 0}$ of the semigroup of
shifts $(\tau_t)_{t\ge 0}$ on $L^2(\R_+)$ with the property that
$\tilde\tau_t - \tau_t$ belongs to a certain Schatten--von Neumann class
$\S_p$ with $p\ge 1$. We show that, for the unitary component
in the Wold--Kolmogorov decomposition, any singular
spectral type may be achieved by $\S_1$ perturbations.
We provide an explicit construction based on the theory of model spaces
of the Hardy space, for a perturbation with a given spectral type.
For $p>1$, the unitary component of the perturbed semigroup
may have any prescribed spectral type.

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