有界线性算子的a-Weyl定理及亚循环性

1)郑州师范学院数学与统计学院,河南郑州 450044; 2)陕西师范大学数学与信息科学学院,陕西西安710062

线性算子理论; a-Weyl定理; 逼近点谱; 亚循环算子; 算子函数; Fredholm算子; 谱集; Browder谱

A-Weyl's theorem and hypercyclic property for bounded linear operators
Yang Guozeng1, Kong Yingying2, and Cao Xiaohong2

Yang Guozeng1, Kong Yingying2, and Cao Xiaohong21)School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, Henan Province, P.R.China; 2)Shaanxi Normal University, Institute of Mathematics and Information Science, Xi'an 710062, Shaanxi Province, P.R.China

linear operator theory; a-Weyl's theorem; approximate point spectrum; hypercyclic operators; operator function; Fredholm operator; spectrum set; Browder spectrum

DOI: 10.3724/SP.J.1249.2017.04372

备注

设H为无限维复可分的Hilbert空间, B(H)为H上的有界线性算子的全体.称T∈B(H)满足a-Weyl定理,若σa(T)\σea(T)=πa00(T), 其中, σa(T)和σea(T)分别表示算子T∈B(H)的逼近点谱和本质逼近点谱, πa00(T)={λ∈isoσa(T):0<dim N(T-λI)<∞}. 通过定义新的谱集,给出了算子函数满足a-Weyl定理的判定方法,研究了当T为亚循环算子时, 算子函数满足a-Weyl定理的充要条件.

Let H be an infinite dimensional separable complex Hilbert space and B(H)be the algebra of all bounded linear operators on H. For T∈B(H), we call a-Weyl's theorem holds for T if σa(T)\σea(T)=πa00(T), where σa(T)and σea(T)denote the approximate point spectrum and essential approximate point spectrum respectively, and πa00(T)={λ∈isoσa(t):0<dim N(T-λI)<∞}. Using the new defined spectrum, we investigate a-Weyl's theorem for operator function. Meanwhile, we characterize the sufficient and necessary conditions for operator function satisfying a-Weyl's theorem if T is a hypercyclic operator.

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