TY - JOUR

T1 - On perturbations of the group of shifts on the line by unitary cocycles

AU - Amosov, G. G.

AU - Baranov, A. D.

PY - 2004/11/1

Y1 - 2004/11/1

N2 - It is shown that the class of perturbations of the semigroup of shifts on L2(ℝ+) by unitary cocycles V with the property V t - I ∈ s2, t ≥ 0 (where s2 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 L2 (ℝ) by Markovian cocycles W with the property Wt - I ∈ s2, t ∈ ℝ.

AB - It is shown that the class of perturbations of the semigroup of shifts on L2(ℝ+) by unitary cocycles V with the property V t - I ∈ s2, t ≥ 0 (where s2 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 L2 (ℝ) by Markovian cocycles W with the property Wt - I ∈ s2, t ∈ ℝ.

KW - Cocycle conjugacy

KW - Group of shifts

KW - Unitary cocycles

UR - http://www.scopus.com/inward/record.url?scp=7444245654&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-04-07423-4

DO - 10.1090/S0002-9939-04-07423-4

M3 - Article

AN - SCOPUS:7444245654

VL - 132

SP - 3269

EP - 3273

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 11

ER -