论文标题
保存绝对连续的范围
Preservation of absolutely continuous spectrum for contractive operators
论文作者
论文摘要
我们考虑订约操作员$ t $是单一操作员$ u $的跟踪类扰动。我们证明了$ t $，$ t^*$和$ u $ comciend的绝对连续光谱的尺寸功能。特别是，如果$ u $具有纯粹的奇异频谱，那么特征函数$θ$的$ t $是双面内部函数，即$θ（ξ）$是统一的。在$ \ mathbb {t} $上。该结果的一些推论与对操作员$ t $和$ t^*$的渐近稳定性的调查有关（融合$ t^n \ to 0 $ to to $ t to $ and $（t^*）^n \ to 0 $，在强操作员拓扑中）。 证明是基于特征函数的明确计算。
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.
