Spectral properties for perturbations of unitary operators

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