Spectral properties for perturbations of unitary operators

Consider a unitary operator U-0 acting on a complex separable Hilbert space H. In this paper we study spectral properties for perturbations of U-0 of the type, U-beta = U(0)e(iK beta), with K a compact self-adjoint operator acting on H and beta 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 U-beta. Moreover, we study the eigenvalue problem for U-beta when the unperturbed operator U-0 does not have any. A typical example of this situation corresponds to the case when U-0 is purely absolutely continuous. Conditions on the eigenvalues of K are given to produce eigenvalues for U-beta for both cases finite and infinite rank of K, and we give an example where the results can be applied. (C) 2011 Elsevier Inc. All rights reserved.