Research On Operator Inequality For Young Type And Sector Matrices

Operator theory,which was born in the early 20th century,is an important part of functional analysis.Nowadays,it not only has a wide range of practical applications in many theoretical research fields such as matrix theory,calculus equation,operational research and control theory,statistics,but also has been deeply involved in many fields such as quantum mechanics,differential dynamics system and so on.So it's a promising research direction.As an important branch of operator theory,the research of operator inequality,especially some classical operator inequalities such as Young type and some kinds of mean inequalities,is very important.In recent years,the research on this topic has made great progress,and the articles in this field are also emerging.On this basis,we have made in-depth research and given some improvements and innovation,and also obtained some further results.In this work,we mainly give a generalization of refinements and reverses for Young type inequality,a new Young type involving Heinz mean and a completed refinement of some operator inequalities for positive linear map.Besides,some results about Heron mean inequality for sector matrices are presented too.According to the above contents,the paper is mainly divided into the following parts:1.Chapter 1 introduces the research background and related concepts and basic properties.2.Chapter 2 gives some improved methods of Young type and its inverse inequality in the sense of operator and matrix,and the related results are generalized as well.3.Chapter 3 explores a new kind of Young type involving Heinz operator mean inequality.Besides,we optimizes them with Kantorovich constant,Specht's rate and operator iteration,respectively.And then,we give some applications on Hilbert Schmidt(Frobenius)norm and trace norm.4.Chapter 4 utilizes operator norm to optimize and generalize some operator inequalities with positive linear map.5.Chapter 5 extends the Heron mean inequality for positive definite matrices to sector matrices.Meanwhile,its applications in determinant,unitary invariant norm and singular value are given as well.Besides,the related new properties involving positive linear map and the relationship between the Heron mean of reverse of sector matrices and the reverse of Heron mean for sector matrices are also obtained.