Researches Of Some Inequalities For Bounded Linear Operators And Related Problems On Hilbert Space

The study of operator theory can be dating back to early in20th century. And operator theory is an important part of Functional analysis. Since operator inequality is a very important component of the operator theory, there are a lot of applications in differential equations, optimization theory, statistics and many other branches of mathematics.This paper contains four chapters:In the first part of the thesis, we mainly introduce the concept of "geometric mean",and discuss four different algorithms on two-by-two matrix, which include the entire process of calculating a living example at the end of Chapter1.In the second part of this thesis, we mainly introduce the origin and generalizations of the Ky-Fan inequality, and give a generalization of Ky-Fan Inequality for bounded linear operators. In addition, we try to establish an operator version of the Ky-fan inequality. Ultimately, we give an example to prove this definition to be incorrect.In the third part of this thesis, we find a new method to consider the best possibility of Furuta inequality, which is different with Tanahashi and then provide a new train of thought about the remainder problem of Furuta inequality.In the last part, we give a summary of this paper and list some fields worthy of studying about the Ky-Fan inequality and the remainder problem of Furuta inequanlity, and expect to work out more results through our joint efforts.