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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
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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  • 1. R. T. Powers, An index theory for $^*$-semigroups of endomorphisms of $\mathcal{B}(\mathcal{H})$ and type $\rm II_1$ factors, Canad. J. Math. 40 (1988), 86-114. MR 89f:46116
  • 2. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 409 (1989), 1-66. MR 90f:47061
  • 3. G. G. Amosov, On cocycle conjugacy of quasifree endomorphism semigroups on the CAR algebra, J. Math. Sci. (New York) 105 (2001), 2496-2503. MR 2003d:46088
  • 4. G. G. Amosov, Approximation modulo $s_2$ of isometric operators and cocycle conjugacy of endomorphisms of the CAR algebra, Fundamental. i Prikl. Matem. 7 (2001), 925-930 (Russian). MR 2002j:46078
  • 5. G. G. Amosov, Cocycle perturbation of quasifree algebraic K-flow leads to required asymptotic dynamics of associated completely positive semigroup, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 237-246. MR 2001m:46148
  • 6. G. G. Amosov, Stationary quantum stochastic processes from the cohomological point of view, Quantum Probability and White Noise Analysis XV, Edited by W. Freudenberg, World Sci. Publ. Co. River Edge, NJ, 2003, 29-40.
  • 7. G. G. Amosov, On the approximation of semigroups of isometries in a Hilbert space, Russian Math. (IzVUZ) 44 (2000), no. 2, 5-10. MR 2001f:47066
  • 8. A. D. Baranov, Isometric embeddings of the spaces $K_{\Theta }$ in the upper half-plane, J. Math. Sci. (New York) 105 (2001), 2319-2329. MR 2002h:46039
  • 9. N. K. Nikolski, Treatise on the shift operator, Springer-Verlag, 1986.
  • 10. N. Dunford and J. T. Schwartz, Linear operators. Part II. Spectral theory, John Wiley & Sons, 1988. MR 90g:47001b
  • 11. P. R. Ahern and D. N. Clark, Radial limits and invariant subspaces, Amer. J. Math., 92 (1970), 332-342. MR 41:7117

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Additional Information

G. G. Amosov
Affiliation: Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudni 141700, Russia

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

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

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