Preservation of absolutely continuous spectrum for contractive operators

TL;DR

This paper proves that contractive operators close to a unitary operator preserve the dimension of the absolutely continuous spectrum, with implications for their asymptotic stability and spectral properties.

Contribution

It establishes that trace class perturbations of a unitary operator maintain the same spectrum dimension functions, revealing spectral stability under such perturbations.

Findings

Dimension functions of the absolutely continuous spectrum coincide for T, T*, and U.

If U has purely singular spectrum, the characteristic function of T is two-sided inner.

Results relate spectral properties to asymptotic stability of the 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 $T$, $T^*$ 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 $T^*$ (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.