有界自伴算子的Weyl定理

中文摘要 AI 辅助

设$T$是可分无穷维右四元数Hilbert空间上的有界自伴算子，令$\sigdS(T)$为有限型右特征值之集。我们证明Weyl定理对$T$成立：$$\sigeS(T)=\sigS(T)\setminus\sigdS(T),$$这补全了在一般情形下得到的包含关系。证明通过一个单一的三重判据进行：$\sigS(T)$中的一点位于$\sigeS(T)$之外，且同样位于$\sigdS(T)$之中，当且仅当该点是具有有限秩谱原子的孤立点。经由Riesz投影的自然途径不能直接使用，因为Riesz投影未必是正交的；我们证明对于自伴算子它实际上是正交的，孤立谱部分的Riesz投影与相应的谱投影一致，从而代数重数与几何重数相等。一个$2\times2$四元数矩阵表明，没有自伴性时最后的恒等式不成立；而一个具有真正球状$S$-谱的正规算子则展示了在正规情形下需要改变什么。

英文摘要

Let $T$ be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let $\sigdS(T)$ be the set of right eigenvalues of finite type. We prove that Weyl's theorem holds for $T$: $$\sigeS(T)=\sigS(T)\setminus\sigdS(T),$$ which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of $\sigS(T)$ lies outside $\sigeS(T)$, and equally lies in $\sigdS(T)$, exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A $2\times2$ quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical $S$-spectrum shows what has to change in the normal case.

发表机构

• University of Gafsa（加夫萨大学）
• Faculty of Sciences of Gafsa, University of Gafsa（加夫萨大学理学院）
• University of Sfax（萨法克斯大学）
• University of Sousse（苏塞大学）
• Higher Institute of Applied Sciences and Technology of Sousse（苏塞应用科学与技术高等研究院）

机构由 AI 辅助整理，请以论文原文为准。