Research On Two Kinds Of Trace Norm Equality

As the basic concept of operator theory and quantum mechanics,the trace of matrix has important mathematical properties and applications. In recent years, many interesting results on matrix trace inequalities in finite Hilbert space has been obt ained.As further extension of these conclusions,Some important trace inequalities had been extended to infinite dimensional Hilbert space.This thesis is the continuation of the research in this aspect,and we will mainly study two types of norm equalities on trace class operators.First,we will use spectral decomposition of operator and some relevant knowledge of trace to give the sufficient and necessary proof of tr（|A|_n）=tr(|A|_n-1A).Second,we will use a method that’s different to reference[8] to prove that the trace-norm triangle equality ||A+B||₁=||A||₁+||B||₁ is equivalent to the triangle equality |A+B|=|A|+|B|. Last,by using the technic of block operator matrix,we will study the structuree of A when the equality ||A||₁+||A*||₁=||A+A*|| holds.These will be three bright spots of this article.This thesis is divided into three chapters,the concrete structure is as follows:Chapter 1 First,we will introduce background and results of this thesis.Sec-ond,we will give relevant concepts,properties and theorems of trace-class operators.Chapter 2 For a single trace-class operator A,we will give equivalent charac-terizations for equalities that are similar to the equality tr（|A|n）=tr(|A|_n-1A).Chapter 3 First,we will prove that the equality ||A+B||=||A||₁+||B||₁ is equivalent to the triangle equality of operators |A+B|=|A|+|B| that is researched in literature[8].Second,as further research,we will st udy equivalent conditions of the equality ||A+B||₁=||A-B||₁=||A||₁+||B||₁,which is revelent to the famous Clarkson’s inequality in literature[11].Finally,by using the technic of bloek operator matrix,we will give the structure of A that satisfies the equality ||A||₁+||A*||₁= ||A+A*|| 1,then,we will give the proof of the equality’s necessary and sufficient conditions.