Some Researches Associated With Inequalities Of Bounded Linear Operators On A Complex Hilbert Space

Operator theory is a very comprehensive research area currently, while operator inequality is an attractive research direction in operator theory, which is extremely active nationally and internationally. Recently, news results on operator inequalities are appeared rapidly. Meanwhile, there are many open problems in this area, such as the Bessis-Moussa-Villani trace conjecture, the CP-rank conjecture and the Grone-Merris conjecture on Laplacian spectra. Therefore, it is necessary to do further researches on operator inequalities.In this thesis, we focus on researching of operator version of Young and its reverse inequalities, Heinz inequalities and Kantorovich inequality under existing results. The main work contants the following topics.1. We obtained the improvements of the difference type of Young and its reverse inequalities for scalars.2. Utilizing the monotoncity of the operator functions and the the improvements of Young and its reverse inequalities for scalars, we deduced the improvements of the operator version 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 new lower and upper bounds for the difference between arithmetic mean operator and geometric mean operator.5. We obtained the improvements of Young and its reverse inequalities with Kantorovich constant for scalars. Meanwhile, we deduced the improvements of the operator version of Young and its reverse inequalities with Kantorovich constant and inequalities for Heinz means operator.6. Using the variant of the inequality of Hermite-Hadamand for continous convex functions, we got refinements of operator version of Heinz inequalities for unitarily invariant norms.7. Utilizing the improvements of the operator version of Heinz inequalities and the general C-P-R inequality, we obtained a refinement of Zhan’s inequality for operators on any dimensional Hilbert space.8. Using the improvement of Choi’s inequality for unital positive linear maps, we extended the operator version of Kantorovich inequality. In addition, we improved other famous inequalities.