TAILIEUCHUNG - báo cáo hóa học: " Some inequalities for unitarily invariant norms of matrices"

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

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