The Stability Judgment Of Weyl's Theorem For Bounded Linear Operators

As one of the most basic theories of modern mathematics,the spectrum theory of linear operators has important applications in mathematics,physics,engineering and so on.It is also an important branch of functional analysis in modern times.In recent decades,there has been a series of works on the spectum theory of linear operators.In 1909,H.Weyl discovered the characteristics of spectrum set of Her-mitian operators(this feature was later called Weyl’s theorem),many scholars at home and abroad,such as R.Harte,H.Radjvi,and so on,have transformed and generalized the Weyl’s theorem.In addition,Weyl’s theorem and the stability of Weyl’s theorem for operator functions have also been studied by scholars.In particular,with the exploration of some powerful new tools,local spactrum theory has greatly enriched the operator spactrum structure.The local spectrum theory is an important generalization in the theory of liner operators,and the single valued extension property is an impor-tant tool for studying local spectrum theory.Therefore,it has important research value.On the basis of the existing theories,this paper defines a new spectrum set,the operators satisfying the stability of Weyl’s theorem is deeply studied,and the relationship between the stability of Weyl’s theorem for operator functions of T and that of T is found.The full text is divided into three chapters,the specific contents are as follows:In the first chapter,It introduces the background and basic knowledge of this paper,and gives the definitions of Weyl’s theorem,stability of Weyl’s theorem and related spectrum set.In Chapter 2,we discuss the method to determine the stability of Weyl’s theo-rem for operator T by the new spectrum set.In Chapter 3,The necessary and sufficient conditions for operator T to satisfy the stability of Weyl’s theorem are discussed by the new spectrum set.At the same time,It is also illustrated that the conditions given in the theorem are indispensable.