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

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 $W_t-I\in s_2,\ t\in {\mathbb R}$.