The Judgment And The Stability Of Weyl Type Theorems

Spectral theory of linear operators has been an important re-search topic and an active branch in operator theory. It has an important theoretical and practical value in quantum mechanics, modern science and technology, modern physics and other applied mathematics. Weyl type theorems become an active research direction in linear operator spectrum theory in the last two decades, and single-valued extension property plays an important role in the study of Weyl type theorems. Using the method of spectral theory, this dissertation involves the judgment and stability of Weyl type theorems. The ranges of the judgment of generalized property (ω), the relationship between Weyl type theorems and cyclic operators, op-erator matrix of generalized property (ω), and the stability of single-valued extension property and Weyl type theorems are mainly involved in this dis-sertation.In Chapter1, we introduce the historical background and current situ-ation, give some of the symbols and concepts used later, and list the main conclusions of this paper.In Chapter2, using the spectrum introduced by the property of Saphar and Kato, we establish for a bounded linear operator defined on a Banach space a sufficient and necessary condition for which generalized property (ω) and generalized a-Weyl’s theorem hold, then using single-valued exten-sion property and consistent in Fredholm and index property, respectively, we study the generalized property (ω).In Chapter3, using spectral characteristics of algebraically*-para- normal operators, Weyl type theorems and hypercyclicity for algebraically*-paranormal were studied, then we consider the relationship between gen-eralized property (ω) and hypercyclicity(supercyclicity).In Chapter4, according to the features of topological uniform descent of diagonal operators A and B, we give the characterization of generalized property (to) for operator matrix.In Chapter5, we study the stability of single-valued extension property and Weyl type theorems. Using the spectrum introduced by consistent in Fredholm and index property, we study the stability of single-valued ex-tension property under compact perturbation; then using characteristics of the generalized Kato resolvent set, we investigate the stability of Weyl type theorems under compact perturbation.The results from the dissertation consist of the following seven state-ments.(1) We discuss the relationship between generalized property (ω) and generalized a-Weyl’s theorem, and give a sufficient and necessary condi-tion for which generalized property (ω) and generalized a-Weyl’s theorem hold.(2) Using single-valued extension property and consistent in Fred-holm and index property, respectively, we study the generalized property (ω), and give the characteristics of generalized property (ω).(3) According to the discussion of algebraically*-paranormal oper-ators, we study Weyl type theorems and cyclicity of algebraically*-para-normal operators.(4) We study generalized property (ω1) and generalized property (ω) by means of a new spectrum, and discuss the hypercyclic(supercyclic) of operators for which generalized property (ω1) holds. (5) According to the features of topological uniform descent of diag-onal operators A and B, we give the characterization of generalized property (ω) for operator matrix.(6) Using the spectrum introduced by consistent in Fredholm and in-dex property, we study the stability of single-valued extension property under compact perturbation.(7) Using the characteristic of generalized Kato resolvent set, we in-vestigate the stability of Wey1type theorems under compact perturbation.