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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the class of perturbations of the semigroup of shifts on by unitary cocycles with the property (where 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 by Markovian cocycles with the property .

**1.**R. T. Powers,*An index theory for -semigroups of endomorphisms of and type 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 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 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47D03,
46L55,
46L57

Retrieve articles in all journals with MSC (2000): 47D03, 46L55, 46L57

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