A Special Kind Of Operator Matrix Convexity And The Characterization Of Operator Variance

In quantum information and operator theory,operator matrices and operator variances have important research value.In 1955,Professor Hua Luogeng proposed a special operator matrix called Hua-type operator matrices and got several deter-minant inequalities of Hua-type operator matrices.Since then,many scholars have not only carried out a more in-depth study of the Hua-type operator matrices but also obtained many valuable conclusions.In addition,some interesting conclusions have been obtained due to the study of the variances of operator by many scholars.These results play a great role in further study of the variances of operator.In this note,we define a class of new operator matrices ???_n^?m?and ???_n^?m?by remove inverse of the Hua-type operator matrices.The norm bounds and convex properties of special operator matrices ???_n^?m?and ???_n^?m?are investigated.A operator A ?B?H?,A = B + iC is the Cartesian decomposition of A,for the real part B and the imaginary part C,we study some equivalence relations about the variances and the absolute variances.Finally,we get that a necessary and sufficient condition of the variance of |A| is equivalent to the absolute variance of A when Hilbert space H is finite dimension.The main contents are as follows:In Chapter 1,we mainly introduce some common symbols,definitions and the-orems,which are the theoretical basis of this article.In Chapter 2,we mainly study the norm bounds and convex properties of special operator matrices ???_n^?m?and ???_n^?m?.When the Hilbert K infinite dimensional,we firstly show that ???_n^?m?= Hn+1?m?and ???_n^?m?= Sn+1?m?,for m,n=1,2,….Then we get that ???_n^?m?is a convex and compact set in ?*topology.Moreover,some norm bounds for ???_n^?m?and ???_n^?m?are given.In Chapter 3,as we all know,a operator A ? B?H?,A = B + iC is the Cartesian decomposition of A,for the real part B and the imaginary part C,we discuss some equivalence relations about the variances and the absolute variances.Finally,we get that a necessary and sufficient condition of the variance of |A| is equivalent to the absolute variance of A when Hilbert space H is finite dimension.