AI 中文总结
本文研究自伴算子在厄米特扰动下的奇异谱行为，对几乎所有参数值及迹类扰动的可数集外均得到奇异谱平移结果，还利用高级技术建立广义 disintegration 定理，扩展了有限秩扰动的相关结论。
AI 中文摘要
我们研究自伴算子在厄米特扰动族（甚至是非紧扰动）下的奇异谱行为。在温和假设下，对几乎所有参数值，奇异谱会发生“平移”。此外，若考虑迹类扰动，我们观察到在参数的可数集之外发生“平移”。尽管有限秩扰动已有类似结果，但扩展至迹类扰动远非易事，证明需大量新思想。出人意料的是，部分结果对所有正有界扰动也成立，这源于本文建立的广义 Aleksandrov disintegration 定理。我们使用了涉及 Sz.-Nagy–Foiaş 理论与算子 A₂ 条件的高级技术。
英文摘要
We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the ``shift'' of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the ``shift'' outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy--Foia\c s theory and the operator ${\bf A}_2$ condition.
Comments57 pages
