(omega) 'nature And Properties Of Generalized (omega')

In this paper, we study the property (ω’) and the generalized property (ω’), two variants of weyl’s theorem introduced by V.Rakocevic. By means of the new spectrum defined in view of the property of consistence in Fredholm and index, we establish for a bounded linear operator T defined on a Hilbert space the sufficient and necessary conditions for which the generalized property (ω’) and property (ω’) hold. The relationships between weyl type theorem and the single valued extension property are discussed. Similarly, the relationships between the generalized weyl type theorem and the single valued extension property (abbreviated by SVEP) are also discussed. Finally, the property (ω’) and the generalized property (ω’) for power finite rank operator and the dominant operator are considered.This paper contains four chapters:In Chapter1, we establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which the generalized property (ω’) holds by means of the variant of the essential approximate point spectrum and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the generalized property (ω’) for power finite rank operator is discussed.In Chapter2, similar with chapter1we establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which property (ω’) holds, meanwhile, property (ω’) for power finite rank operator is also discussed.In Chapter3, the relationship among property (ω’), Weyl theorem and a-Weyl theorem is discussed by SVEP and conjugate operator, and the property (ω’) for the dominant operator is considered.In Chapter4, the relationship among the generalized property (ω’), the gener-alized Weyl theorem and the generalized a-Weyl theorem is also discussed by SVEP and conjugate operator, then the generalized property (ω’) for the dominant oper-ator is also considered.