Preservation of absolutely continuous spectrum for contractive operators

Abstract:

Contractive operators $T$ that are trace class perturbations of a unitary operator $U$ are treated. It is proved 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 $\theta$ of $T$ is a two-sided inner function, i.e., $\theta (\xi )$ is unitary a.e. on $\mathbb {T}$. Some corollaries to this result are related to investigations of the asymptotic stability of the operators $T$ and $T^*$ (the 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.