Preservation of absolutely continuous spectrum for contractive operators

Abstract

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 $T$, $T^{\ast }$ and of $U$ coincide. In particular, if $U$ has a purely singular spectrum then the characteristic function $\theta$ of $T$ is a two-sided inner function, i.e. $\theta (\xi )$ 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 $T^{\ast }$ (convergence $T^n\to 0$ and $(T^{\ast }{)}^n\to 0$, respectively, in the strong operator topology). The proof is based on an explicit computation of the characteristic function.

arXiv comment15 pages. Typos corrected. Some references to relevant work were added