Research On Singular Value Inequalities Of Compact Operators And Norm Inequalities Of Measurable Operators

Operator theory plays an important role in mathematics and other sciences,and has a wide range of applications.The theory of bounded linear operators on Hilbert space and Banach space is the basis of operator theory and operator algebra.Compact operators and measurable operators are two kinds of operators which are important for operator theory.The singular values of compact operators is one of the research hotspots of operator theory.Non-commutative Lp space is an important part of functional analysis,and its properties are the research hotspot of current functional analysis.The research on the measurable operator and its norm inequality is an important research field of operator theory.In recent years,many scholars have carried out extensive research on the related problems such as the singular value and norm of compact operators and measurable operators,and have obtained a lot of results.In this paper,using the relevant knowledge and techniques of operator theory and operator algebra,we study furtherly the singular value inequalities of compact operators and the norm inequalities of measurable operators.Three kinds of problems are studied in this paper.Firstly,using the technique of the operator block matrix,the singular value inequality of compact operators on a Hilbert space is studied.Secondly,the properties of non-commutative Lp spaces are studied by using properties of von Neumann algebra.Thirdly,using the properties of measurable operators on a von Neumann algebra M,the norm inequalities of measurable operators on M are studied.The main content of this paper is divided into four parts.In the first part,the origin and development of functional analysis,operator theory and operator algebra are introduced.Then we introduce the research status of compact operator and measurable operator at home and abroad.Finally we introduce the content,purpose and related preliminary knowledge of this dissertation.In the second part,the singular value inequalities of compact operators on a Hilbert space are studied.Some concepts and properties are introduced.Then,using the properties of compact operators and technique of operator block matrices,some singular value inequalities of compact operators are obtained.In the third part,using the properties of the von Neumann algebra,some properties of the non-commutative space are obtained.In the fourth part,the norm and the properties of the measurable operator are introduced.Then a series of norm inequalities are obtained.Finally the equivalence of several singular value inequalities of measurable operators is proved.