Spectral properties for perturbations of unitary operators

Abstract

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 β = U 0 e i K β , with K a compact self-adjoint operator acting on H 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 U β . Moreover, we study the eigenvalue problem for U β 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 β for both cases finite and infinite rank of K, and we give an example where the results can be applied.