The Characterization Of Hypercyclicity And Supercyclicity Of Bounded Operators

The spectral of operators is a generalization of the concept of eigenvalues of finite dimensional matrices. Under certain conditions, a large number of problems in mechanics, physics and engineering can be attributed to find roots of algebraic equations, differential equations, integral equations or differential integral equation-s. On the basis of the results of the study of solving these equations, the theory of spectral of general operator is formed gradually. Local spectral theory is an impor-tant promotion of normal operator theory, and the single valued extension property plays an important role in the study of local spectral theory. Perturbation theory of linear operator is closely linked with the disciplines of physics, and it has been an impressive key branch in operator theory.Based on the existing theories, the main content of the paper is the relationship between hypercyclicity and topological uniform descent of bounded linear operators. Also, we study the equivalent conditions of supercyclicty of bounded linear opera-tors.This paper contains three chapters:In Chapter1, given the historical background of this study, prepare knowledge and the definition of hypercyclic operators and supercyclic operators.In Chapter2, through the topological uniform descent of conjugate operators, we investigate the equivalence of hypercyclicity of bounded linear operators. At the same time, we use various spectral sets of operators to investigate the equivalent conditions of hypercyclicty of bounded linear operators.In Chapter3, the first, we introduce the concept of the supercyclic opera-tors. The second, using the topological uniform descent property and the single valued extension property, we investigate the equivalent conditions of supercyclicty of bounded linear operators.