Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Some inequalities for unitarily invariant norms of matrices | Wang et al. Journal of Inequalities and Applications 2011 2011 10 http content 2011 1 10 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some inequalities for unitarily invariant norms of matrices Shaoheng Wang Limin Zou and Youyi Jiang Correspondence limin-zou@163. com School of Mathematics and Statistics Chongqing Three Gorges University Chongqing 404000 People s Republic of China SpringerOpen0 Abstract This article aims to discuss inequalities involving unitarily invariant norms. We obtain a refinement of the inequality shown by Zhan. Meanwhile we give an improvement of the inequality presented by Bhatia and Kittaneh for the Hilbert-Schmidt norm. Mathematical Subject Classification MSC 2010 15A60 47A30 47B15 Keywords Unitarily invariant norms Positive semidefinite matrices Convex function Inequality 1. Introduction Let Mm n be the space of m X n complex matrices and Mn Mn n. Let II - denote any unitarily invariant norm on Mn. So IIƯAVII A for all Ae Mn and for all unitary matrices U VeMn. For A ai7 eMn the Hilbert-Schmidt norm of A is defined by IIAII2 n n _ ị------------ E E hI21 T Ãh JE7 si A i 1 j 1 where tr is the usual trace functional and s1 A s2 A . sn-1 A sn A are the singular values of A that is the eigenvalues of the positive semidefinite matrix 1 A ẠA 2 arranged in decreasing order and repeated according to multiplicity. The Hilbert-Schmidt norm is in the class of Schatten norms. For 1 p the Schatten p-norm II-lip is defined as A p e s tr Al 11p. For k 1 . n the Ky Fan k-norm lldl fe is defined as Ek . j 1 Sj A . It is known that these norms are unitarily invariant and it is evident that each unitarily invariant norm is a symmetric guage function of singular values 1 p. 54-55 . Bhatia and Davis proved in 2 that if A B Xe Mn such that A and B are positive semidefinite and if 0 r 1 then 2011 Wang et al licensee Springer. This is an Open Access article distributed under
