|Table of Contents|

A-Weyl’s theorem and hypercyclic property for bounded linear operators(PDF)


Research Field:
Publishing date:


A-Weyl’s theorem and hypercyclic property for bounded linear operators
Yang Guozeng1 Kong Yingying2 and Cao Xiaohong2
1)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
O 177.2
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.


[1] Herrero D A. Limits of hypercyclic and supercyclic operators[J]. Journal of Functional Analysis, 1991, 99(1): 179-190.
[2] Weyl H. berbeschrnkte quadratische formen, deren differenz vollstetig ist[J]. Rendiconti del Circolo Matematico di Palermo, 1909, 27(1): 373-392.
[3] Coburn L A. Weyl’s theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3): 285-288.
[4] Berberian S K. An extension of Weyl’s theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3): 273-279.
[5] Cao Xiaohong, Guo Maozheng, Meng Bin. Drazin spectrum and Weyl’s, theorem for operator matrices[J]. Journal of Mathematical Research and Exposition, 2006, 26(3): 52-66.
[6] 青梅.无界算子的矩阵的谱和补问题[D].呼和浩特:内蒙古大学,2016.
Qing Mei. Spectra and completion problems of unbounded operator matrices[D]. Hohhot: Inner Mongolia University, 2016.(in Chinese)
[7] Kitson D, Harte R, Hernández C. Weyl’s theorem and tensor products: a counterexample[J]. Journal of Mathematical Analysis and Applications, 2011, 378(1): 128-132.
[8] 戴磊.Weyl型定理的判定及其稳定性[D].西安:陕西师范大学,2013.
Dai Lei. The judgment and the stability of Weyl type theorems[D]. Xi’an: Shaanxi Normal University.(in Chinese)
[9] 李娜娜.算子与其共轭的Weyl型定理的等价性判定[D].西安:陕西师范大学,2013.
Li Nana. The equivalence of Weyl’s theorem of operators and their conjugate[D]. Xi’an: Shaanxi Normal University.(in Chinese)
[10] Kato T. Perturbation theory for linear operators[M]. Berlin: Springer-Verlag Berlin Heidelberg, 1976: 11-12.
[11] Dunford N, Schwartz J T. Linear operators: part 1 general theory[M]. Berlin: Springer-Verlag Berlin Heidelberg, 1988: 5-6.
[12] H Radjavi, P Rosenthal. Invariant subspaces[M]. 2nd edit. New York, USA: Dover Publications, 2003: 63-64.
[13] Taylor A E. Theorems on ascent, descent, nullity and defect of linear operators[J]. Mathematische Annalen, 1966, 163(1): 18-49.
[14] Conway J B. A course in functional analysis[M]. 2nd edit. New York: Springer-Verlag, 2003: 181-182.
[15] Hassane Zguitti. A note on drazin invertibility for upper triangular block operators[J]. Mediterranean Journal of Mathematics, 2013, 10(3): 93-102.
[16] 戴磊,曹小红. 单值延拓性质与广义(ω)性质[J]. 陕西师范大学学报自然科学版,2011,39(2):32-41.
Dai Lei, Cao Xiaohong. The single valued extension property and generalized property (ω)[J]. Journal of Shaanxi Normal Unviersity Natural Science Edition, 2011, 39(2): 32-41.(in Chinese)
[17] 周婷婷.Weyl型定理及相关问题[D].长春:吉林大学,2014.
Zhou Tingting. Weyl’s theorem and related problems[D]. Changchun: Journal of Jilin University, 2014.(in Chinese)
[18] Sun Chenhui,Cao Xiaohong,Dai Lei. A Weyl-type theorem and perturbations[J]. Acta Mathtmatica Sinica, 2009, 52(1): 73-80.


Last Update: 2017-06-26