Property (ω') For Bounded Linear Operator

In this paper, we study the property (ω'), a variant of Weyl's theorem. We establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which property (ω') holds. Meanwhile, We discuss the perturbation and application of property (ω'). In addition, the equivalence of property (ω') and Weyl's theorem are considered.This paper contains four chapters:In Chapter 1, we define the property (ω'), and establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which property (ω') holds by means of the new spectrum. In addition, the perturbation of property (ω') is discussed. Finally, we study the property (ω') for hypercyclicity (or supercyclicity) operators.In chapter 2, we study the equivalence of property (ω') and Weyl's theorem.In chapter 3, the property (ω') is considered by means of the single valued exten-sion property, and establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which property (ω') holds. Meanwhile, the application of property (ω') is considered by the main results of this chapter.In chapter 4, we study the equivalence and judgement for Weyl type theorem of operators on a Hilbert space by means of the relationship between the set of topological uniform descent and the new spectrum defined in Chapter 1.