Spectral properties for perturbations of unitary operators

Consider a unitary operator U0U0 acting on a complex separable Hilbert space HH. In this paper we study spectral properties for perturbations of U0U0 of the type,Uβ=U0eiKβ,Uβ=U0eiKβ, with K   a compact self-adjoint operator acting on HH and β a real parameter. We apply the commutator theory developed for unitary operators in Astaburuaga et al. (2006) [1] to prove the absence of singular continuous spectrum for UβUβ. Moreover, we study the eigenvalue problem for UβUβ when the unperturbed operator U0U0 does not have any. A typical example of this situation corresponds to the case when U0U0 is purely absolutely continuous. Conditions on the eigenvalues of K   are given to produce eigenvalues for UβUβ for both cases finite and infinite rank of K, and we give an example where the results can be applied.