Research On Weyl's Theorem Of Operator T~n

Weyl's theorem is one of the most important branches of operator spectrum theory,and draws much attention from scholars.In recent years,many authors have used the single-valued extension property as a basic condition to study Weyl's theorem,which provides new thoughts.We continue this work.In this paper,we give new judgments of Weyl's theorem for bounded linear operators T and T^*;in addition,we discuss the relationship between T and T²?or T and Tn?about Weyl's theorem and its perturbations.The paper contains four chapters:In chapter 1,we introduce the research's background and current situation,the relevant fundamental knowledge and the definitions of Weyl's theorem,the stabil-ity of Weyl's theorem and the single-valued extension property?SVEP for short?.Moreover,We define a new spectrum _?1?T?based on SVEP.In chapter 2,using the new spectrum defined before,we give new judgments of Weyl's theorem for bounded linear operator T and its adjoint T^*,respectively.After that,we find the equivalent conditions in which Weyl's theorem holds for both T and T^*.In chapter 3,first,we provide the equivalent conditions for Weyl's theorem holding for both T and T².Then we illustrate that each condition is essential by examples.After that we discuss the judgments of the stability of Weyl's theorem.We give the conditions of the stability of Weyl's theorem holding for T when T² satisfies it.In chapter 4,we generalize the conclusions about T and T² to T and Tⁿ.