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On perturbations of the group of shifts on the line by unitary cocycles


Authors: G. G. Amosov and A. D. Baranov
Journal: Proc. Amer. Math. Soc. 132 (2004), 3269-3273
MSC (2000): Primary 47D03; Secondary 46L55, 46L57
DOI: https://doi.org/10.1090/S0002-9939-04-07423-4
Published electronically: June 2, 2004
MathSciNet review: 2073301
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Abstract: It is shown that the class of perturbations of the semigroup of shifts on $L^2({\mathbb R}_+)$ by unitary cocycles $V$ with the property $V_t-I\in s_2, t\geq 0$ (where $s_2$ is the Hilbert-Schmidt class) contains strongly continuous semigroups of isometric operators, whose unitary parts possess spectral decompositions with the measure being singular with respect to the Lebesgue measure. Thus, we describe also the subclass of strongly continuous groups of unitary operators that are perturbations of the group of shifts on $L^2({\mathbb R})$ by Markovian cocycles $W$ with the property $W_t-I\in s_2, t\in {\mathbb R}$.


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Additional Information

G. G. Amosov
Affiliation: Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudni 141700, Russia
Email: amosov@fizteh.ru

A. D. Baranov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetski pr., St. Petersburg, 198504, Russia
Address at time of publication: Laboratoire de Mathématiques Pures, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence, France
Email: antonbaranov@netscape.net

DOI: https://doi.org/10.1090/S0002-9939-04-07423-4
Keywords: Group of shifts, unitary cocycles, cocycle conjugacy
Received by editor(s): March 31, 2003
Received by editor(s) in revised form: July 8, 2003
Published electronically: June 2, 2004
Additional Notes: The first author is partially supported by INTAS-00-738
The second author is partially supported by RFBR Grant No. 03-01-00377
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society