Research Of The Operator Inequalities And Related Problems On Hilbert Space

Operator theory is an important part of Functional analysis,it stems from the early 20th century by the integral equation continue studies.It discuss the linear Hilbert space,bounded operator algebra,geometry,analysis,and spectrum properties mainly.Operator inequality is an important branch of the operator theory,there are a lot of applications in differential equations,optimization theory,statistics and many other branches of mathematics.In 1934,L(o|¨)wner made a famous operator inequality which called L(o|¨)wner-Heinz inequality after,On this basis,in 1987 the Japanese mathematician Furuta made famous Furuta inequality,which has attracted wide attention.In 1979,D.W.Nicholson gives necessary and sufficient conditions of the matrix AB+BA to be positive matrix.Firstly this paper will carry out strictly to its extension, and then discuss necessary and sufficient conditions in terms of these operators AB+BA to be positive operators from the operator angle for,as well as its characterization of some of the applications are.At the same time,we also cited several examples described them in objective.This article also discusses the singular value inequality operator,today,the matrix singular value inequality has been an important research topic in inequality,the paper extend matrix singular value inequality to the operator field and focuses on the arithmetic geometric mean inequality.Finally.this paper discuss the operator power mean equivalent of furuta inequality form,it analyse the scope of the establishment of the Furuta inequality from the operator point of view of power mean,and provides a new approach for Furuta inequality index of the scope of the establishment of the study.