Weyl-type Theorems And Cyclicity Studies For Bounded Linear Operators

Spectral theory is an important branch of operator algebra in operator theory.It is closely related to other disciplines.It is widely used in physics,quantum mechanics and other disciplines.In recent years,Weyl type theorem in spectral theory has received much attention.Research on this and related issues has become a hot issue in spectral theory.In this dissertation,based on the theories which have existed,we define two new spectra with the help of the property of consistency in invertibility.Based on these two spectra,we give the necessary and sufficient conditions for T which Weyl type theorem holds,and discuss the relationship between the condition"T ? HC(H)" and the condition "T is a-isoloid and satisfies a-Weyl's theorem".This dissertation consists of four chapters.In Chapter 1,the academic background,research status and preliminary knowledge of this paper are introduced.Moreover,the concepts of Weyl type theorem,hypercyclic operator and operator which is consistency of invertibility(short for CI)are introduced.Based on the property of consistent of invertibility,we define two new spectra ?1(T)and ?2(T);In Chapter 2,by using the newly defined spectra respectively,we obtain the equivalent conditions of the bounded linear operator T satisfying Weyl's theorem,also study the conditions of the function calculus satisfying Weyl's theorem;In Chapter 3,by using the spectrum ?2(T),we obtain the necessary and sufficient conditions for T for which obeys a-Browder's theorem and a-Weyl's theorem respectively,and discuss the perturbation of a-Weyl's theorem for T;In Chapter 4,by using the spectrum ?2(T),we obtain a way of deciding the function calculus f(T)is in HC(H).Further,we study the relationship between a-Weyl's theorem of bounded linear operators and the function calculus and their hypercyclicity.