Determination Of (?) Properties Of Bounded Linear Operators

Spectrum theory is an important branch of operator theory and operator algebra.It is closely related to other disciplines,and widely used in physics,quantum mechanics and other disciplines.The problem of Weyl type theorem in spectrum theory has attracted much attention in recent years.The research on this problem and related issues has become a hot topic in spectrum theory.In this dissertation,using the relationship between new defined spectrum and other spectrums,we mainly study the variant of Weyl's theorem,namely property(?).Firstly,using the property of consistency in invertibility,we define a new spectrum to give the judgments of property(?)which an operator and its functional calculus hold.Then,using the property of consistency in Fredholm and index,we define another new spectrum to consider property(?)of an operator and its adjoint operator.Meanwhile,we discuss the finite rank perturbation of property(?).The thesis is divided into three chapters as follows:In Chapter 1,we introduce the research background and relevant basic theoretical knowledge.We also illustrate the concepts of different kinds of spectrums,property(?)and property(?1),and define two new spectrums.In Chapter 2,using the new spectrum defined by the property of consistency in invertibility,we give the judgments of an operator and its functional calculus which the property(?)hold.In Chapter 3,using the new spectrum defined by the property of consistency in Fredholm and index,we discuss the decision conditions for an operator and its adjoint operator which the property(?1)hold.Then,we study the decision conditions for an operator and its adjoint operator which the property(?)hold separately.Last,we investigate the finite rank perturbation of property(?).