Paper detail

Preservation of absolutely continuous spectrum for contractive operators

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