Preservation of absolutely continuous spectrum for contractive operators

Abstract

Contractive operators T T that are trace class perturbations of a unitary operator U U are treated. It is proved that the dimension functions of the absolutely continuous spectrum of T T , T ∗ T^* , and of U U coincide. In particular, if U U has a purely singular spectrum, then the characteristic function θ \theta of T T is a two-sided inner function, i.e., θ ( ξ ) \theta (\xi ) is unitary a.e. on T \mathbb {T} . Some corollaries to this result are related to investigations of the asymptotic stability of the operators T T and T ∗ T^* (the convergence T n → 0 T^n\to 0 and ( T ∗ ) n → 0 (T^*)^n\to 0 , respectively, in the strong operator topology). The proof is based on an explicit computation of the characteristic function.