数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 539-551.doi: 10.1016/S0252-9602(12)60036-8

• 论文 • 上一篇    下一篇

WEYL’S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

M.H. M. Rashid   

  1. Department of Mathematics and Statistics, Faculty of Science, P.O. Box(7), Mu’tah University, Mu’tah-Jordan
  • 收稿日期:2009-05-14 修回日期:2010-12-27 出版日期:2012-03-20 发布日期:2012-03-20

WEYL’S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

M.H. M. Rashid   

  1. Department of Mathematics and Statistics, Faculty of Science, P.O. Box(7), Mu’tah University, Mu’tah-Jordan
  • Received:2009-05-14 Revised:2010-12-27 Online:2012-03-20 Published:2012-03-20

摘要:

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T*∈ SC, then generalized a-Browder’s theorem holds for f(T) for every ∈Hol(σ(T)). (ii) If T or T* ∈ HC has topological uniform descent at all λ∈ iso(σ(T)), then generalized Weyl’s theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈ HC has topological uniform descent at all
λ∈ E(T), then T satisfies generalized Weyl’s theorem. (iv) Let T ∈HC. If T satisfies the growth condition Gd(d ≥ 1), then generalized Weyl’s theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈ SC, then, fSBF±(T)) = σSBF±(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈ HC. If T − λI has finite ascent at every λ ∈ Ea(T) and if F is of finite rank on H such that TF = FT, then T +F obeys generalized a-Weyl’s theorem.

关键词: Weyl’s theorem, hypercyclic operators, supercyclic operators

Abstract:

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T*∈ SC, then generalized a-Browder’s theorem holds for f(T) for every ∈Hol(σ(T)). (ii) If T or T* ∈ HC has topological uniform descent at all λ∈ iso(σ(T)), then generalized Weyl’s theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈ HC has topological uniform descent at all
λ∈ E(T), then T satisfies generalized Weyl’s theorem. (iv) Let T ∈HC. If T satisfies the growth condition Gd(d ≥ 1), then generalized Weyl’s theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈ SC, then, fSBF±(T)) = σSBF±(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈ HC. If T − λI has finite ascent at every λ ∈ Ea(T) and if F is of finite rank on H such that TF = FT, then T +F obeys generalized a-Weyl’s theorem.

Key words: Weyl’s theorem, hypercyclic operators, supercyclic operators

中图分类号: 

  • 47A53