Some inequalities for unitarily invariant norms of matrices

Transcription

1 Wang et al Journal of Inequalities and Applications 011, 011:10 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, , 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 (010) 15A60; 47A30; 47B15 Keywords: Unitarily invariant norms, Positive semidefinite matrices, Convex function, Inequality 1 Introduction Let M m,n be the space of m n complex matrices and M n =M n,n Let denote any unitarily invariant norm on M n So, UAV = A for all AÎM n and for all unitary matrices U,VÎM n For A =(a ij )ÎM n, the Hilbert-Schmidt norm of A is defined by n n n A = a ij = tr A = j=1 s j (A), i=1 j=1 where tr is the usual trace functional and s 1 (A) s (A) s n-1 (A) s n (A) are the singular values of A, that is, the eigenvalues of the positive semidefinite matrix 1 A = (AA, 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 p is defined as A p = ( n ) 1 /p j=1 sp j (A) = ( tr A p) 1/p For k = 1,,n, the Ky Fan k-norm (k) is defined as A (k) = k s j (A) j=1 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 [] that if A,B,XÎM n such that A and B are positive semidefinite and if 0 r 1, then 011 Wang et al; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

2 Wang et al Journal of Inequalities and Applications 011, 011:10 Page of 7 A 1/ XB 1 / A r XB 1 r + A 1 r XB r AX + XB (1:1) Let A,B,XÎM n such that A and B are positive semidefinite In [3], Zhan proved that A r XB r + A r XB r A X + taxb + XB, (1:) t + for any unitarily invariant norm and real numbers r,t satisfying 1 r 3,- <t The case r =1,t = 0 of this result is the well-known arithmetic-geometric mean inequality A 1 / XB 1 / AX + XB Meanwhile, for rî[0,1], Zhan pointed out that he can get another proof of the following well-known Heinz inequality A r XB 1 r + A 1 r XB r AX + XB by the same method used in the proof of (1) Let A,B,XÎM n such that A and B are positive semidefinite and suppose that ψ (v) = A 1+v XB 1 v + A 1 v XB 1+v (1:3) Then ψ is a convex function on [-1,1] and attains its minimum at v = 0 [4, p 65] In [5], for positive semidefinite n n matrices, the inequality AB 1 4 (A + B) (1:4) was shown to hold for every unitarily invariant norm Meanwhile, Bhatia and Kittaneh [5] asked the following Question Let A,BÎM n be positive semidefinite Is it true that s j (AB) 1 4 s j(a + B), j =1,,, n? The case n = is known to be true [5] (See also, [1, p 133], [6, p ], [7, p 198]) Obviously, if A,BÎM n are positive semidefinite and AB = BA, thenwehave j =1,,, n, j =1,,, n Some inequalities for unitarily invariant norms In this section, we first utilize the convexity of the function ψ (r) = A r XB r + A r XB r to obtain an inequality for unitarily invariant norms that leads to a refinement of the inequality (1) To do this, we need the following lemmas on convex functions Lemma 1 Let A,B,XÎM n such that A and B are positive semidefinite Then, for each unitarily invariant norm, the function

3 Wang et al Journal of Inequalities and Applications 011, 011:10 Page 3 of 7 ψ (r) = A r XB r + A r XB r is convex on [0,] and attains its minimum at r =1 Replace v+1 by r in (13) Lemma Let ψ be a real valued convex function on an interval [a,b] which contains (x 1,x ) Then for x 1 x x, we have ψ (x) ψ (x ) ψ (x 1 ) x x 1 x x 1ψ (x ) x ψ (x 1 ) x x 1 (:1) Since ψ is a convex function on [a,b], for a x 1 x x b, we have ψ (x 1 ) ψ (x) x 1 x ψ (x ) ψ (x) x x This is equivalent to the inequality (1) Theorem 1 Let A,B,XÎM n such that A and B are positive semidefinite If 1 r 3 and-<t, then A r XB r + A r XB r (r 0 1) AXB + 4 (1 r 0) A X + taxb + XB,(:) +t where r 0 = min{r,-r} If 1 r 1, then by Lemma 1 and Lemma, we have ( ) ( ) ψ (1) ψ ψ (1) ψ ψ (r) r That is ψ (r) (r 1) ψ (1) +(1 r) ψ It follows from (1) and (3) that ( ) 1 (:3) A r XB r + A r XB r 4 (1 r) (r 1) AXB + A X + taxb + XB +t

4 Wang et al Journal of Inequalities and Applications 011, 011:10 Page 4 of 7 If 1 r 3, then by Lemma 1 and Lemma, we have That is ( 3 ψ ψ (r) ) ψ (1) r 3 1 ( 3 ψ ψ (r) (3 r) ψ (1) +(r 1) ψ It follows from (1) and (4) that ) 3 ψ (1) 3 1 ( ) 3 (:4) A r XB r + A r XB r 4 (r 1) (3 r) AXB + A X + taxb + XB +t It is equivalent to the following inequality A r XB r + A r XB r (r 0 1) AXB + 4 (1 r 0) A X + taxb + XB +t This completes the proof Now, we give a simple comparison between the upper bound in (1) and the upper bound in () A X + taxb + XB (r 0 1) AXB 4 (1 r 0) A X + taxb + XB +t +t = (r 0 1) +t (r 0 1) +t A X + taxb + XB (r 0 1) AXB (+t) AXB (r 0 1) AXB =0 Therefore, Theorem 1 is a refinement of the inequality (1) Let A,B,XÎM n such that A and B are positive semidefinite Then, for each unitarily invariant norm, the function ϕ (v) = A v XB 1 v + A 1 v XB v is a continuous convex function on [0,1] and attains its minimum at v = 1 See [4, p 65] Then, by the same method above, we have the following result Theorem [8] Let A,B,XÎM n such that A and B are positive semidefinite If 0 v 1, then A v XB 1 v + A 1 v XB v A 4r 1/ 0 XB 1 / + (1 r 0 ) AX + XB, where r 0 = min{v,1-v} This is a refinement of the second inequality in (11) Next, we will obtain an improvement of the inequality (14) for the Hilbert-Schmidt norm To do this, we need the following lemma

5 Wang et al Journal of Inequalities and Applications 011, 011:10 Page 5 of 7 Lemma 3[9] Let A,B,XÎM n such that A and B are positive semidefinite If 0 v 1, then A v XB 1 v AX v XB 1 v Theorem 3 Let A,B,XÎM n such that A and B are positive semidefinite If 0 v 1, then A v XB 1 v + ( AX v XB 1 v) AX 4v + XB 4(1 v) + A v XB 1 v Let S = AX 4v + XB 4(1 v) + A v XB 1 v ( A v XB 1 v + ( AX v XB 1 v) ) So, S = AX 4v + XB 4(1 v) + A v XB 1 v 4 A v XB 1 v ( AX v XB 1 v) 4 4 A v XB 1 v ( AX v XB 1 v) = AX 4v + XB 4(1 v) A v XB 1 v ( AX v XB 1 v) 4 4 A v XB 1 v ( AX v XB 1 v) By Lemma 3, we have That is, S AX 4v + XB 4(1 v) AX v XB (1 v) ( AX v XB 1 v) 4 4 A v XB 1 v ( AX v XB 1 v) S ( AX v XB 1 v) ( ( AX v + XB 1 v) ( AX v XB 1 v) 4 A v XB 1 v ) =4 ( AX v XB 1 v) ( AX v XB 1 v A v XB 1 v ) 0 Hence, AX 4v + XB 4(1 v) + A v XB 1 v ( A v XB 1 v + ( AX v XB 1 v) ) This completes the proof Let A,B,XÎM n such that A and B are positive semidefinite, for Hilbert-Schmidt norm, the following equality holds: AX + XB A = AX + XB + 1/ XB 1 / Taking v = 1 in Theorem 3, and then we have the following result

6 Wang et al Journal of Inequalities and Applications 011, 011:10 Page 6 of 7 Theorem 4[10] Let A,B,XÎM n such that A and B are positive semidefinite Then A 1 / XB 1 / ( + AX ) XB AX + XB Bhatia and Kittaneh proved in [5] that if A,BÎM n are positive semidefinite, then A 3 / B 1 / + A 1 / B 3 / 1 (A + B) (:5) Now, we give an improvement of the inequality (14) for the Hilbert-Schmidt norm Theorem 5 Let A,BÎM n be positive semidefinite Then AB + ( 1 A 3 / B 1 / A 1 / B ) 3 / 1 (A + B) 4 Let X = A 1 / B 1 / Then, by Theorem 4, we have ( AB + A 3 / B 1 / A 1 / B ) 3 / A 3 / B 1 / + A 1 / B 3 / (:6) It follows form (5) and (6) that ( AB + A 3 / B 1 / A 1 / B ) 3 / 1 (A + B) That is, AB + ( 1 A 3 / B 1 / A 1 / B ) 3 / 1 (A + B) 4 This completes the proof Acknowledgements The authors wish to express their heartfelt thanks to the referees and Professor Vijay Gupta for their detailed and helpful suggestions for revising the manuscript At the same time, we are grateful for the suggestions of Yang Peng This research was supported by Natural Science Foundation Project of Chongqing Science and Technology Commission (No CSTC, 010BB0314), Natural Science Foundation of Chongqing Municipal Education Commission (No KJ101108), and Scientific Research Project of Chongqing Three Gorges University (No 10ZD-16) Authors contributions SW and LZ designed and performed all the steps of proof in this research and also wrote the paper YJ participated in the design of the study and suggest many good ideas that made this paper possible and helped to draft the first manuscript All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 11 January 011 Accepted: 0 June 011 Published: 0 June 011

7 Wang et al Journal of Inequalities and Applications 011, 011:10 Page 7 of 7 References 1 Zhan, X: Matrix Theory Higher Education Press, Beijing (008) (in Chinese) Bhatia, R, Davis, C: More matrix forms of the arithmetic-geometric mean inequality SIAM J Matrix Anal Appl 14, (1993) doi:101137/ Zhan, X: Inequalities for unitarily invariant norms SIAM J Matrix Anal Appl 0, (1998) doi:101137/ S Bhatia, R: Matrix Analysis Springer-Verlag, New York (1997) 5 Bhatia, R, Kittaneh, F: Notes on matrix arithmetic geometric mean inequalities Linear Algebra Appl 308, (000) doi:101016/s (00) Bhatia, R, Kittaneh, F: The matrix arithmetic geometric mean inequality revisited Linear Algebra Appl 48, (008) 7 Bhatia, R: Positive Definite Matrices Princeton University Press, Princeton (007) 8 Kittaneh, F: On the convexity of the Heinz means Integr Equ Oper Theory 68, (010) doi:101007/s Kittaneh, F: Norm inequalities for fractional powers of positive operators Lett Math Phys 7, (1993) doi:101007/bf Kittaneh, F, Manasrah, Y: Improved Young and Heinz inequalities for matrices J Math Anal Appl 361, 6 69 (010) doi:101016/jjmaa doi:101186/109-4x Cite this article as: Wang et al: Some inequalities for unitarily invariant norms of matrices Journal of Inequalities and Applications :10 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropencom