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Some inequalities for unitarily invariant norms of matrices

Journal of Inequalities and Applications20112011:10

https://doi.org/10.1186/1029-242X-2011-10

Received: 11 January 2011

Accepted: 20 June 2011

Published: 20 June 2011

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 normsPositive semidefinite matricesConvex functionInequality

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, for all AM n and for all unitary matrices U,VM n . For A = (a ij )M n , the Hilbert-Schmidt norm of A is defined by
where tr is the usual trace functional and s 1 (A) ≥ s 2 (A) ≥ ... ≥ s n-1 (A) ≥ s n (A) are the singular values of A, that is, the eigenvalues of the positive semidefinite matrix , 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 is defined as
For k = 1,...,n, the Ky Fan k-norm is defined as

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,XM n such that A and B are positive semidefinite and if 0 ≤ r ≤ 1, then
(1.1)
Let A,B,XM n such that A and B are positive semidefinite. In [3], Zhan proved that
(1.2)
for any unitarily invariant norm and real numbers r,t satisfying 1 ≤ 2r ≤ 3,-2 < t ≤ 2. The case r = 1,t = 0 of this result is the well-known arithmetic-geometric mean inequality
Meanwhile, for r[0,1], Zhan pointed out that he can get another proof of the following well-known Heinz inequality

by the same method used in the proof of (1.2).

Let A,B,XM n such that A and B are positive semidefinite and suppose that
(1.3)

Then ψ is a convex function on [-1,1] and attains its minimum at v = 0 [4, p. 265].

In [5], for positive semidefinite n × n matrices, the inequality
(1.4)

was shown to hold for every unitarily invariant norm. Meanwhile, Bhatia and Kittaneh [5] asked the following.

Question

Let A,BM n be positive semidefinite. Is it true that

, ?

The case n = 2 is known to be true [5]. (See also, [1, p. 133], [6, p. 2189-2190], [7, p. 198].)

Obviously, if A,BM n are positive semidefinite and AB = BA, then we have , .

2. Some inequalities for unitarily invariant norms

In this section, we first utilize the convexity of the function

to obtain an inequality for unitarily invariant norms that leads to a refinement of the inequality (1.2). To do this, we need the following lemmas on convex functions.

Lemma 2.1

Let A,B,XM n such that A and B are positive semidefinite. Then, for each unitarily invariant norm, the function

is convex on [0,2] and attains its minimum at r = 1.

Proof

Replace v+1 by r in (1.3).□

Lemma 2.2

Let ψ be a real valued convex function on an interval [a,b] which contains (x1,x2). Then for x1xx2, we have
(2.1)

Proof

Since ψ is a convex function on [a,b], for ax1xx2b, we have

This is equivalent to the inequality (2.1).□

Theorem 2.1

Let A,B,XM n such that A and B are positive semidefinite. If 1 ≤ 2r ≤3 and -2 <t ≤ 2, then
(2.2)

where r0 = min{r,2-r}.

Proof

If , then by Lemma 2.1 and Lemma 2.2, we have
That is
(2.3)
It follows from (1.2) and (2.3) that
If , then by Lemma 2.1 and Lemma 2.2, we have
That is
(2.4)
It follows from (1.2) and (2.4) that
It is equivalent to the following inequality

This completes the proof.□

Now, we give a simple comparison between the upper bound in (1.2) and the upper bound in (2.2).

Therefore, Theorem 2.1 is a refinement of the inequality (1.2).

Let A,B,XM n such that A and B are positive semidefinite. Then, for each unitarily invariant norm, the function

is a continuous convex function on [0,1] and attains its minimum at . See [4, p. 265]. Then, by the same method above, we have the following result.

Theorem 2.2.[8]

Let A,B,XM n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

where r0 = min{v,1-v}. This is a refinement of the second inequality in (1.1).

Next, we will obtain an improvement of the inequality (1.4) for the Hilbert-Schmidt norm. To do this, we need the following lemma.

Lemma 2.3.[9]

Let A,B,XM n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

Theorem 2.3

Let A,B,XM n such that A and B are positive semidefinite. If 0 ≤ v ≤ 1, then

Proof

Let
So,
By Lemma 2.3, we have
That is,
Hence,

This completes the proof.□

Let A,B,XM n such that A and B are positive semidefinite, for Hilbert-Schmidt norm, the following equality holds:

Taking in Theorem 2.3, and then we have the following result.

Theorem 2.4.[10]

Let A,B,XM n such that A and B are positive semidefinite. Then
Bhatia and Kittaneh proved in [5] that if A,BM n are positive semidefinite, then
(2.5)

Now, we give an improvement of the inequality (1.4) for the Hilbert-Schmidt norm.

Theorem 2.5

Let A,BM n be positive semidefinite. Then

Proof

Let
Then, by Theorem 2.4, we have
(2.6)
It follows form (2.5) and (2.6) that
That is,

This completes the proof.□

Declarations

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, 2010BB0314), Natural Science Foundation of Chongqing Municipal Education Commission (No. KJ101108), and Scientific Research Project of Chongqing Three Gorges University (No. 10ZD-16).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, People's Republic of China

References

  1. Zhan X: Matrix Theory. Higher Education Press, Beijing; 2008.Google Scholar
  2. Bhatia R, Davis C: More matrix forms of the arithmetic-geometric mean inequality. SIAM J Matrix Anal Appl 1993, 14: 132–136. 10.1137/0614012MathSciNetView ArticleGoogle Scholar
  3. Zhan X: Inequalities for unitarily invariant norms. SIAM J Matrix Anal Appl 1998, 20: 466–470. 10.1137/S0895479898323823View ArticleGoogle Scholar
  4. Bhatia R: Matrix Analysis. Springer-Verlag, New York; 1997.View ArticleGoogle Scholar
  5. Bhatia R, Kittaneh F: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl 2000, 308: 203–211. 10.1016/S0024-3795(00)00048-3MathSciNetView ArticleGoogle Scholar
  6. Bhatia R, Kittaneh F: The matrix arithmetic-geometric mean inequality revisited. Linear Algebra Appl 2008, 428: 2177–2191.MathSciNetView ArticleGoogle Scholar
  7. Bhatia R: Positive Definite Matrices. Princeton University Press, Princeton; 2007.Google Scholar
  8. Kittaneh F: On the convexity of the Heinz means. Integr Equ Oper Theory 2010, 68: 519–527. 10.1007/s00020-010-1807-6MathSciNetView ArticleGoogle Scholar
  9. Kittaneh F: Norm inequalities for fractional powers of positive operators. Lett Math Phys 1993, 27: 279–285. 10.1007/BF00777375MathSciNetView ArticleGoogle Scholar
  10. Kittaneh F, Manasrah Y: Improved Young and Heinz inequalities for matrices. J Math Anal Appl 2010, 361: 262–269. 10.1016/j.jmaa.2009.08.059MathSciNetView ArticleGoogle Scholar

Copyright

© Wang et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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.

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