Some Inequalities For Non-commutative Banach Function Spaces

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 op-erator. Firstly, we give some notations and introduce the definitions of projection,τ-measurable operator, unitarily invariant norm and non-commutative Banach functionspace norm, etc. Then we mainly introduce the singular value ofτ-measurable operatorand non-commutative Banach function space.In chapter 2,using the properties of singular values ofτ-measurable operators,wegeneralize the results on matrices expA,exp(ReA),exp(A + B) in [1] to the case oftheτ-measurable operators. It contains three propositions as bellow: , here A,B be self-adjoint. (c)If A, B be self-adjoint, then converges to as t converges to 0 decreasingly. In these propositions, A, B be elements of a vonNeumann algebra M and be a non-commutative Banach function space norm.In chapter 3,using the properties of singular values ofτ-measurable operators,westudy Schwarz inequalities forτ-measurable operators A*XB in the set-up of the non-commutative Banach function spaces. Firstly, we will prove the inequality(d) , hereA, B areτ-measurable operators andAB is a self-adjoint operator. At thebase of this inequality, we'll try to get two inequalities bellow...