On The Deformation Of Weyl's Theorem Generalized (¦Ø ') Nature

The present research of spectrum theory has been highly valued by the math-ematicians and physicists, especially the Weyl type theorems related to the distri-bution of eigenvalue. In recent years, M.Berkani and J.J.Kolila enriched the forms of Wcyl's theorem by defining the generalized Weyl type theorems. Based on the research mentioned above, we define and study the generalized property (ω'), a new variation of Weyl's theorem. Although this variation is related to Weyl's theorem, there is a great deal of difference between it and Weyl's theorem. Therefore the study of variation is not simple generalization of Weyl's theorem.Jacobson theorem suggested a new idea for the property of consistency in Fred-holm and index. We find that it is closely linked with generalized Weyl type theorems and generalized property (ω') when the equivalent characterizations are considered. This paper's studies, for the first time, combine the property of consistency in Fred-holm and index and the generalized property (ω'). By means of the new spectrum defined in view of the property of consistency in Fredholm and index, we discuss the generalized property (ω'), and deeply research on its determining theorem, stability, and the generalized property (ω') for adjoint operators and operational calculus, and so on.This paper contains three chapters:The first chapter deals with its historical background, some notations and def-initions in this paper.The second chapter provides the definition of the property of consistency in Fredholm and index as well as the corresponding spectrum, then it is studied that the determining theorem and the equivalent characterizations of generalized Weyl theorem and generalized a-Weyl theorem.The third chapter discusses the generalized property (ω') using the new spec-trum defined in chapter 2. And chapter 3 composes of five parts:3.1 We establish for a bounded linear operator defined on a Hilbert space the sufficient and necessary conditions for which the generalized property (ω') holds.3.2 The sufficient and necessary conditions for which the generalized property (ω') of operational calculus holds are given. 3.3 The equivalent characterizations of generalized property (ω') between oper-ators and their adjoint operators are studied.3.4 We consider the preservation of generalized property (ω') under a finite rank perturbation. In addition, a nilpotent perturbation is studied.3.5 The theory is exemplified in the case of some special classes of operators.