Determination Of The A-Weyl Theorem For Bounded Linear Operators

In functional analysis,operator theory,as an important part,is favored by many scholars.With the discovery of Weyl's theorem in spectral theory by H.Weyl in 1909,then scholars have extended and modified it,in which Harte and Lee defined Browder theorem,Rakoc?vic defined a-Browder theorem and a-Weyl's theorem.On the basis of the known theories,we give the definition of Kato type operators and define the new spectrum by the property of Kato type operators.Then by means of the new spectral set and the aggregation point of the new spectral set.We establish the sufficient and necessary conditions for bounded linear operators and operator functions to satisfy a-Browder theorem and a-Weyl's theorem are given.Finally,the theory is exemplified in the determination of a-Browder theorem and a-Weyl's theorem of hypercyclic operators and operator functions.This paper consists of four chapters.In Chapter 1,this paper describes the research background and related preparatory knowledge of this paper,introduces Weyl's theorem,a-Weyl's theorem,hpercyclic operator,Kato property and the definition of related spectral sets,and defines two new spectral sets based on Kato property of operator T;In Chapter 2,by using the newly defined spectra respectively,obtain the equivalent conditions of the bounded linear operator T satisfying a-Browder theorem,also study the conditions of the function calculus satisfying a-Browder theorem;In Chapter 3,by using one of the accumulation point of new spectra,obtain the necessary and sufficient conditions for T for which obey a-Weyl's theorem,and discuss the the conditions of the function calculus satisfying a-Weyl's theorem;In Chapter 4,finally,the theory in chapter2 and 3 is exemplified in the determination of a-Browder theorem and a-Weyl's theorem of hypercyclic operators and operator functions.