Some inequalities for unitarily invariant norms of matrices

Wang Shaoheng, Zou Limin, Jiang, Jiang Youyi

Description

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 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

Sujets

• Convex function
• Inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 11
• Langue: English

Extrait

Wanget al.Journal of Inequalities and Applications2011,2011:10 http://www.journalofinequalitiesandapplications.com/content/2011/1/10

R E S E A R C HOpen Access Some inequalities for unitarily invariant norms of matrices * Shaoheng Wang, Limin Zouand Youyi Jiang

* Correspondence: liminzou@163. com School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404000, People’s Republic of China

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 LetM m,n be the space ofm×ncomplex matrices andM n = M n,n . Let ∙ denote any unitarily invariant norm onM n . So,  UAV  =  A  for allAÎM n and for all unitary matricesU,VÎM n . ForA= (a ij )ÎM n , the HilbertSchmidt norm ofAis defined by   n n   n  2 2 2      A  = a= tr | A | =s ( A ) 2 ij j j=1 i=1j=1

wheretris the usual trace functional ands 1 (A)≥s 2 (A)≥...≥s n1 (A)≥s n (A) are the singular values ofA, that is, the eigenvalues of the positive semidefinite matrix 1 , arranged in decreasing order and repeated according to multiplicity. The ∗ 2 | A | = ( AA ) HilbertSchmidt norm is in the class of Schatten norms. For 1≤p<∝, theSchatten p norm ∙ p is defined as 1p /  n   p 1p p /  A  =s ( A ) =tr | A | p j j=1

Fork= 1,...,n, theKy Fan knorm ∙ ( k) is defined as    A ( k) =s j ( A ) =1

It is known that these norms are unitarily invariant, and it is evident that each unita rily invariant norm is a symmetric guage function of singular values [1, p. 5455]. Bhatia and Davis proved in [2] that ifA,B,XÎM n such thatAandBare positive semidefinite and if 0≤r≤1, then

© 2011 Wang et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.