Weyl-type Theorem And Cyclicity Of Bounded Linear Operators

The operator spectral theory,as an important branch of operator theory,has attracted the attention of many scholars at home and abroad.In recent years,many scholars have used Kato property as a tool to study Weyl type theorem and provided many new ideas.In this paper,based on the theories which have existed,we define a new spectra with the help of the property of Kato.By defining a new spectral set,we study the Weyl type theorem of bounded liner operators,and get the relationship between condition "T ?(?)" and the Weyl type theorem,as well as the spectral mapping theorem of the spectral set of the corresponding spectral set of operators.This paper 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 and hypercyclic operator.Based on the property of Kato,we define two new spectra.In Chapter 2,by using the newly defined spectra respectively,we obtain a new judgement for the bounded linear operator T satisfying Weyl's theorem,also study the judgement for the operator function satisfying Weyl's theorem.In Chapter 3,by using the characteristic of the accumulation point of the newly defined spectra set,we discuss a judgement for the bounded linear operators and their functions satisfying Weyl's theorem again.In Chapter 4,by using one of the new spectra,we study the relationship between Weyl's theorem of bounded linear operators and its functional and their hypercyclicity.