Some Researches Associated With Several Classical Operator Inequalities And Their Applications

Operator theory is an important part of the theory of functional analysis, arising in early twentieth century. It not only deeps into the matrix theory, operational research and control theory, statistics and so on, but also is a very broad field of study and has a widen practical application in many areas such as quantum mechanics, differential dynamic systems and so on. As an important branch in theory operator, the research of operator inequality is particularly important, especially some classic operator inequalities. Recently, more and more new inequality versions are revealed, the methods and techniques also are diversified and these operators inequalities are further expand the applications of cross-disciplinary application. Therefore, it is necessary to do further researches on operator inequalities.In this thesis, we focus on researching of several classical operator inequalities and their applications by means of the convexity and concavity of continuous function, the spectral decomposition of matrix, the unitary invariance of Hilbert-Schmidt norm, function calculations and other tools. The main work contains the following topics:1. We generalized the range of arithmetic-geometric mean operator inequalities and presented the corresponding series of quasi-arithmetic-geometric mean operator inequalities.2. Utilizing the monotoncity of the operator functions and the improvements of Young and its reverse inequalities for scalars, we gave the corresponding improvements of the operator versions of Young and its reverse inequalities.3. Using the unitary invariance of the Hilbert-Schmidt norm and the improvements of Young and its reverse inequalities for scalars, we got the new improvements of the matrix version of the improvements of Young and its reverse inequalities.4. We obtained the improvements of Young and its reverse inequalities with Kantorovich constant for scalars by introducing the Kantorovich constant. Meanwhile, we deduced the improvements of the multi-parameters operator version of Young and its reverse inequalities with Kantorovich constant and inequalities for Heinz means operator.5. Based on the property of continuous convex function, we gave some unitary invariant norm inequalities between Heinz means operator and Heron means operator.6. We presented a property of the continuous s-convex function with multi-parameters on the co-ordinates, and obtained the corresponding multi-parameters Hermite-Hadamard integral operator inequalities.7. Using the definition of absolute distance in statistical, we generalized the Samuelson inequality with parameters and obtained the locations of eigenvalues of complex matrix and the characteristic roots of complex coefficients polynomial by applied the improved Samuelson inequality.