Operator And Its Conjugate Wey1 Type Theorem Of Equivalence

In this paper, we study the property (ω) and the generalized property (ω), two variants of Weyl’s theorem. The judgement of equivalence for property (ω) and the judgement of equivalence for generalized property (ω) between operator and its conjugate on a Hilbert space are discussed by the relation between the two new spectrum sets σCI(T) and σCFI(T) defined in view of the consistency in invertibility and in the Fredholm index respectively. Also, the equivalence for the perturbation of property (ω) and the equivalence for the perturbation of generalized property (ω) are considered.This paper contains three chapters:In Chapter1, we define the property (ω), and establish for operator and its conjugate on a Hilbert space the sufficient and necessary conditions for which prop-erty (ω) hold by the relation between the two spectrum sets σCI(T) and σCFI(T). Meanwhile, the equivalence for the perturbation of property (ω) is considered.In chapter2, we study the judgement of equivalence for property (ω) between operator and its conjugate by means of the relationship between the set of topological uniform descent and the spectrum defined in Chapter1.In chapter3, we define the generalized property (ω), and the judgement of equivalence for generalized property (ω) between operator and its conjugate on a Hilbert space is discussed by the relation between the two spectrum sets σCI(T) and σCFI(T). In addition, the equivalence for the perturbation of generalized property (ω) is considered. Finally, we study the property generalized (ω) for hypcrcyclicity operators.