Research On The Hua's Inequalities And Moore-Penrose Inverses Of Operators In Hilbert C*-modules

Inequality is an indispensable basic tool in all fields of mathematics and science and technology.It not only holds a unique position in mathematics,but also provides us with a powerful tool to understand mathematics and an important part of modern mathematics.In addition,in the operator theory of Hilbert space,it is very important to study the closed property of range and the structure of the Moore-Penrose inverse of operators.Although the Hilbert C*-module is a generalization of the Hilbert space,but because of the complexity of C*-algebra is embedded in the Hilbert C*-module,leading to the above problems in Hilbert C*-module is more complex than Hilbert space,so it is of great significance to study the above topics in the Hilbert C*-module.The research contents of this paper can be divided into two parts.The first part is research on the Hua's inequality in Hilbert C*-modules,including norm Hua's type inequality and operator Hua's type inequality,etc.The second part is focused on the operators product Moore-Penrose inverse and reverse order law in Hilbert C*-module,including the operator product Moore-Penrose inverse equivalent representation and the reverse order law,derived from Moore-Penrose inverse reversibility of special operators and so on.The structure of this article is as follows.In chapter one,the research contents and significance of this paper,the basic con-cepts and properties of the Hilbert C*-module and its bounded adjointable operator are introduced.In chapter two,we first introduce the research background of Hua inequality,and then give some preparatory knowledge.The Hua's inequality is generalized in the Hilbert C*-module,which mainly includes the generalization of the norm Hua's type inequality and the operator Hua's type inequality.In chapter three,first,the research background of the operator product Moore-Penrose inverse is introduced,and some preparatory knowledge are given.By making full use of block-operator matrix representation,we study the product of two operators Moore-Penrose inverse,which mainly includes the following problems:?i??TS?^?equivalent representation;?ii?the equivalent conditions of S^?T^???TS?{1,2,3} and S^?T^??TS?{1,2,4};?iii?the necessary and sufficient conditions for Moore-Penrose operator product reverse order law.At the end of this chapter,the invertibility of two special operators is studied by Moore-Penrose inverse.As an application,a sufficient condition for the reverse order law is given.