华东师范大学学报（自然科学版） ›› 2025, Vol. 2025 ›› Issue (1): 13-27.doi: 10.3969/j.issn.1000-5641.2025.01.002

• 数学 • 上一篇    下一篇

有界线性算子的Fa-Weyl定理与a-Weyl定理

李偲萌, 张邺*(), 曹小红   

  1. 陕西师范大学 数学与统计学院, 西安　710119
  • 收稿日期:2023-11-28 出版日期:2025-01-25 发布日期:2025-01-20
  • 通讯作者: 张邺 E-mail:zhangye@snnu.edu.cn
  • 基金资助:
    国家自然科学基金(11971283)

Fa-Weyl’s theorem and a-Weyl’s theorem for bounded linear operators

Simeng LI, Ye ZHANG*(), Xiaohong CAO   

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an　710119, China
  • Received:2023-11-28 Online:2025-01-25 Published:2025-01-20
  • Contact: Ye ZHANG E-mail:zhangye@snnu.edu.cn

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摘要：

Fa-Weyl 定理和 a-Weyl 定理是 Weyl 定理的两种变形, Weyl 型定理的研究对于谱理论研究十分重要. 通过定义新的谱集, 给出了在 Hilbert 空间上的有界线性算子$T $同时满足 Fa-Weyl 定理和 a-Weyl 定理的等价条件. 此外, 还讨论了算子$T $在有限秩摄动下的 Fa-Weyl 定理和 a-Weyl 定理.

关键词: Fa-Weyl 定理, a-Weyl 定理, 摄动, 谱

Abstract:

Both Fa-Weyl’s theorem and a-Weyl’s theorem are the variants of Weyl’s theorem. The study of Weyl’s type theorems is very important for spectral theory. By defining a new spectral set in this paper, sufficient and necessary conditions for a bounded linear operator $T $ definded on a Hilbert space to satisfy the Fa-Weyl’s theorem and the a-Weyl’s theorem are established. In addition, we discuss the Fa-Weyl’s theorem and the a-Weyl’s theorem of bounded linear operator $T $ under a finite rank perturbation.

Key words: Fa-Weyl’s theorem, a-Weyl’s theorem, perturbation, spectrum

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