Some Matrix Inequalities And An Application In Quantum Uncertainty Theory

We study some problems about positive semidefinite(block)matrices,contrac-tive matrices and sector matrices.And then,Some unitarily invariant norm,matrix mean and determinantal inequalities are obtained.Finally,we also investigate quan-tum uncertainty principle.Our main results are as follows:1.We showe Hayajneh and Kittaneh's conjecture on unitarily invariant norm by using Schur complement,PPT matrix and majorization.This indirectly solves a problem posed by Bourin.2.Lin and Zhou[72]investigate Hilbert-Schmidt and unitarily invariant norm inequalities for accretive-dissipative matrix.One of their results is comparing the size of unitarily invariant norm of accretive-dissipative matrix and its diagonal block.And then,we extend their's result to sector matrix.3.Choi[25]prove a reversed Fischer's type determinant inequalities for positive definite matrices.Firstly,we give a alternative proof for Choi's result,then,an analogue of Choi's result is presented involving Hadamard product.4.We obtain a generalized Holder type eigenvalue inequalities by extending Hua determinant inequalities[64],This inequality is also a extension of Marcus's result[94].5.The Brunn-Minkowski inequality is one of the most important geometric inequalities.There is a vast amount of work on its generalizations and on its con-nections with other areas.Ky Fan[34]gives a generalization of this inequality,Yuan and Leng[77]further give a generalization of the inequality.The original proof of Yuan and Leng seems to be lengthy.We firstly give a simple proof for Yuan and Leng's main result,then,we show some generalizations of the matrix form of the Brunn-Minkowski inequality to the case of matrices whose numerical ranges are contained in a sector.6.We establish geometric mean-harmonic mean inequalities for sector matrices.Some singular value and unitarily invariant norm inequalities for sector matrices are obtained by using the above matrix mean inequality.7.Mond and Pecaric[92]showed the following mixed AM-GM and GM-HM inequalities for two positive definite matrices.Two analogous inequalities of Mond and Pecaric's results for sector matrices are presented.8.Quantum uncertainty principle is one of the basic principle in quantum mechanics,it describes that any observational value can not be observed two non-commutative observable measurements at the same time.We generalize Ko and Yoo's results[60]and establish a new uncertainty relation.Our result extends some of the existing results.