doi:

DOI: 10.3724/SP.J.1249.2017.04372

Journal of Shenzhen University Science and Engineering (深圳大学学报理工版) 2017/34:4 PP.372-377

A-Weyl's theorem and hypercyclic property for bounded linear operators


Abstract:
Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. For TB(H), we call a-Weyl's theorem holds for T if σa(T)σea(T)=π00a(T), where σa(T) and σea(T) denote the approximate point spectrum and essential approximate point spectrum respectively, and π00a(T)={λ∈isoσa(t):0N(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.

Key words:linear operator theory,a-Weyl's theorem,approximate point spectrum,hypercyclic operators,operator function,Fredholm operator,spectrum set,Browder spectrum

ReleaseDate:2017-10-20 02:07:26



[1] Herrero D A. Limits of hypercyclic and supercyclic operators[J]. Journal of Functional Analysis, 1991, 99(1):179-190.

[2] Weyl H. Überbeschränkte 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.