On L(?)wner Partial Order Of Operators And Matrix Inequalities

Matrix(Operator) inequality is an important research hotspot in matrix theory. In recent decades, Matrix(Operator) inequality has a wide range of applications in bio-mathematics, economics, control theory, computer image processing, probabilistic statistics, etc. Actually, there exist many unsolved problems in terms of matrix(operator) inequality, for instance Zhan’s conjecture, Bhatia-Kittaneh’s conjecture, Brualdi-Li’s conjecture, Lin’s conjecture, S-matrix’s conjecture and so on. Thus, the study on matrix(operator) inequality has been of interest to a great many of researchers.In this paper, we study some results on operator partial ordering for positive linear maps, determinantal inequalities for special matrices, Heinz inequalities for unitarily invariant norms and coneigenvalues. We discuss the main problems as follows:1. We present p( p ?2) power of operator Kantorovich and Wielandt inequalities and reverse operator AM-GM inequalities by Ando-Zhan inequality [Math. Ann. 1999] and matrix(operator) arithmetic-geometric mean inequalities [Linear Algebra Appl. 2000]. These operator inequalities generalize the results due to Lin [J. Math. Anal. Appl. 2013; Studia Math. 2013], Bhatia and Davis [Comm. Math. Phys. 2000].2. By Schur complement of positive definite block matrix and determinantal inequality zoom techniques, we discuss Fischer-type determinantal inequalities for Accretive-dissipative operator matrices which are refinements of Lin’s results [Linear Algebra Appl. 2013].3. We study some determinantal inequalities for Hadamard and Fan products of special matrices. Firstly, we give determinantal inequalities for Hadamard product of an M-matrix and finite number of inverse M-matrices, which are the generalizations of Chen’s inequalities [Linear Algebra Appl. 2007]. Secondly, we prove a determinantal inequality for Hadamard product of arbitrarily finite number of positive definite matrices, which is also a generalization of Chen’s result [Linear Algebra Appl. 2003]. Thirdly, we present determinantal inequalities for Hadamard product of finite number of H-matrices, which are the generalizations of the results for Hadamard product of two H-matrices due to Lynn [Proc. Cambridge Philos. 1964]. Finally, we obtain the determinantal inequalities for Fan product of finite number of M-matrices which generalize Ando’s result [Linear Multilinear Algebra. 1980].4. We discuss the matrix arithmetic-geometric mean inequalities for unitarily invariant norms which are the refinements of a result due to Bhatia and Kittaneh [Linear Algebra Appl. 2000]. In the same time, we also investigate matrix Heinz inequalities. The inequalities are refinements of Zhan’s results [SIAM J. Matrix Anal. Appl. 1998].5. We obtain some inequalities for coneigenvalues and singular values. They extend some inequality relations for classical eigenvalues and singular values.