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# Some inequalities for unitarily invariant norms of matrices

- Shaoheng Wang
^{1}, - Limin Zou
^{1}Email author and - Youyi Jiang
^{1}

**2011**:10

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

© Wang et al; licensee Springer. 2011

**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 norms
- Positive semidefinite matrices
- Convex function
- Inequality

## 1. Introduction

*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

*A*∈

*M*

_{ n }and for all unitary matrices

*U*,

*V*∈

*M*

_{ n }. For

*A*= (

*a*

_{ ij })∈

*M*

_{ n }, the Hilbert-Schmidt norm of

*A*is defined by

*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

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].

*A*,

*B*,

*X*∈

*M*

_{ n }such that

*A*and

*B*are positive semidefinite and if 0 ≤

*r*≤ 1, then

*r*,

*t*satisfying 1 ≤ 2

*r*≤ 3,-2 <

*t*≤ 2. The case

*r*= 1,

*t*= 0 of this result is the well-known arithmetic-geometric mean inequality

*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).

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

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

### Question

Let *A*,*B*∈*M*_{
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*,*B*∈*M*_{
n
} are positive semidefinite and *AB* = *BA*, then we have
,
.

## 2. Some inequalities for unitarily invariant norms

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

*A*,

*B*,

*X*∈

*M*

_{ 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

### Proof

This is equivalent to the inequality (2.1).□

### Theorem 2.1

*A*,

*B*,

*X*∈

*M*

_{ n }such that

*A*and

*B*are positive semidefinite. If 1 ≤ 2

*r*≤3 and -2 <

*t*≤ 2, then

where *r*_{0} = min{*r*,2-*r*}.

### Proof

This completes the proof.□

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

*A*,

*B*,

*X*∈

*M*

_{ 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]

where *r*_{0} = 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]

### Theorem 2.3

### Proof

This completes the proof.□

*A*,

*B*,

*X*∈

*M*

_{ 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]

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

### Theorem 2.5

### Proof

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

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## Copyright

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.