Research On Norm Equalities Of Elementary Operators And Operator Partial Orders

Operator theory is an extremely important branch of functional analysis and it has a wide range of applications in the fields of science and engineering technology.The operators on the Hilbert space is the fundation of research on operator theory and operator algebras,and elementary operator is a kind of important operators.O ne of hot issues that elementary operator involved is the study of norm equalities of elementary operators.Research on the problem of operator partial orders is the generalization of that of matrices partial orders.In recent years,many scholars have studied and obtained various results on elementary operators and opertar partial orders.In this paper,in order to further generalize and improve the problems of norm equalities of elementary operators and operator partial orders,we use the knowledge of operator theory and operator algebras and study two types of problems.First,norm equalities of elementary operators on the Hilbert space are studied.Relations between norm equalities of elementary operators and numerical range are given.Second,using operator block matrices thechnique,some characterizations and properties of operator Sharp partial order on B(H)which is the set of all bounded linear operators on a complex seperable Hilbert space H,and relations between the operator Sharp partial order and other operator partial orders are given.The main content of this paper is divided into three chapters.In chapter 1,firstly,the background,development and research status at home and abroad of operator theory and operator algebras are introduced.Secondly,purpose and significance of this thesis are given.Finally,elementary knowledge and notations which are related with this thesis are presented.In chapter 2,norm equalities of elementary operators on B(H)are studied.In this chapter,firstly,some related concepts and properties are presented.Secondly,using the ideal of analysis and properties of norm of operators,relations between norm equalities of elementary operators and numerical range are given.Finally,some conditions and properties that elementary operators attain norm equalities are obtained.In chapter 3,the operator Sharp partial order on B(H)is studied.Firstly,using operator block matrice thechnique,some characterizations of operator Sharp partial order on B(H)are obtained.Secondly,some properties of operator Sharp partial order on B(H)are studied.Finally,relations between operator Sharp partial order and others are given.