The Perturbations For The Single Valued Extension Property And The A-weyl’s Theorem

The research of spectrum theory has been highly valued by the mathematicians and physicists and the local spectrum theory has become one of focuses, the single valued extension property plays an important role in the study of the local spectrum theory, therefore it is very important to study the single valued extension property.In this paper, first, we give the definition of generalized Kato decomposition and define a new spectrum by the property of generalized Kato decomposition. Then using this new spectrum, we investigate the stability of the single valued extension property under compact perturbations and we characterize2×2upper triangular operator matrices for which the single valued extension property is stable under compact perturbations. Finally, similar to the study method of the compact perturbations for the single valued extension property, we investigate the stability of the a-Browder’s theorem and the a-Weyl’s theorem under compact perturbations and we characterize2×2upper triangular operator matrices for which the a-Browder’s theorem or the a-Weyl’s theorem is stable under compact perturbations by means of this new spectrum again.This paper contains three chapters:In Chapter1, we introduce its historical background and some basic definitions which are always needed in this paper.In Chapter2, first, we give the definition of generalized Kato decomposition and define a new spectrum by the property of generalized Kato decomposition. Then using this new spectrum, we investigate the stability of the single valued extension property under compact perturbations and we characterize those operators for which the single valued extension property is stable under some compact perturbation or under all small compact perturbations or under all compact perturbations. In addition, we also characterize2×2upper triangular operator matrices for which the single valued extension property is stable under compact perturbations by means of the conclusions we have reached. In Chapter3, similar to the study method of the compact perturbations for the single valued extension property, we investigate the stability of the a-Browder’s theorem and the a-Weyl’s theorem under compact perturbations and we character-ize those operators for which the a-Browder’s theorem or the a-Weyl’s theorem is stable under all small compact perturbations or under all compact perturbations. Finally, we also characterize2×2upper triangular operator matrices for which the a-Browder’s theorem or the a-Weyl’s theorem is stable under compact perturbations by means of the conclusions we have reached.