Some Inequalities For Commutators Of τ-measurable Positive Operators

This paper contains three chapters, the outline of the paper is arranged as follows:Chapter 1 mainly introduces some notations, definitions and some properties of operator.Firstly, we give some-notations and introduce the definitions of projection,τ-meas urable operator, unitarily invariant norm and non-commutative Banach function space norm, etc. Then we mainly introduce the singular value ofτ-measurable operator and non-commutative Banach function space.In chapter 2, considering the imitate relation of singular value and the non-commutative Banach function space norm, we begin the chapter with researching the property of singularvalue ofτ-measurable operator. Thus some non-commutative Banach function space norm inequalities for commutators ofτ-measurable positive operators are given. In this chapter, we introduce the operator matrix. Using the properties of singular values ofτ-measurable operators, we generalize the F. Kittaneh's results in [1] to the case of the non-commutative Banach function space.In chapter 3, we generalize the inequality |||A^pX - XB^p|||≥||||AX - XB|^p||| forcommutators of inequality:|||A^p-B^p|||≥||||A-B|^p|||,which p≥1,|X|=(X^*X)^1/2byfurther using the properties of singular values ofτ-measurable operators to the case of the non-commutative Banach function space.