- Research
- Open access
- Published:

# A note on norm inequalities for positive operators

*Journal of Inequalities and Applications*
**volume 2014**, Article number: 171 (2014)

## Abstract

In this short note, we present a generalization of a norm inequality due to Bhatia and Kittaneh (Lett. Math. Phys. 43:225-231, 1998), which is also a refinement and a generalization of a result obtained by Kittaneh (Commun. Math. Phys. 104:307-310, 1986).

**MSC:**15A42, 47A63.

## 1 Introduction

Let {M}_{n} be the space of n\times n complex matrices. Let \parallel \cdot \parallel denote any unitarily invariant norm on {M}_{n}. We shall always denote the singular values of *A* by {s}_{1}(A)\ge \cdots \ge {s}_{n}(A)\ge 0, that is, the eigenvalues of the positive semidefinite matrix |A|={(A{A}^{\ast})}^{1/2}, arranged in decreasing order and repeated according to multiplicity. Let A,B\in {M}_{n} be Hermitian; the order relation A\ge B means, as usual, that A-B is positive semidefinite.

Let A,B\in {M}_{n} be positive semidefinite. Bhatia and Kittaneh [[1], Theorem 2.2] proved that for any positive integer *m*,

Let A,B\in {M}_{n}. Kittaneh [[2], Theorem 2.2] proved that

where {\parallel X\parallel}_{F} is the Frobenius norm of *X*. For more information on the Schatten *p*-norm and its applications, the reader is referred to [3].

In this note, we present a generalization of inequality (1.1), which is also a refinement and a generalization of (1.2).

## 2 Main results

Now, we show the generalization of inequality (1.1).

**Theorem 2.1** *Let* A,B\in {M}_{n} *and suppose that* *p*, *q* *be real numbers with* p>1 *and* \frac{1}{p}+\frac{1}{q}=1. *Then for any positive integer* *m*,

*Proof* Let A,B\in {M}_{n} with polar decompositions A=U|A| and B=V|B|. It is known [[4], p.15] that

It follows that

So, by Proposition 1.3.2 of [[4], p.13], we have

for some contraction *K*. For k=1,\dots ,n, by using Horn’s inequality [[5], p.72], we know that

Let

and

Then inequality (2.2) is equivalent to

Since weak log-majorization implies weak majorization, we get

Thanks to the Fan dominance principle [[5], p.93], we know that inequality (2.3) is equivalent to

By Hölder’s inequality for unitarily invariant norms [[5], p.95] (see also [[6], Theorem 2.4]), we obtain

It follows from (2.4) and (2.5) that

This completes the proof. □

**Remark 2.2** Inequality (2.1) is a norm version of the following scalar triangle inequality:

**Corollary 2.3** *Let* A,B\in {M}_{n}. *Then for any positive integer* *m*,

*Proof* Putting p=q=2 in inequality (2.1), we have

It follows from inequality (1.1) and this last inequality that

This completes the proof. □

**Remark 2.4** If *A* and *B* are positive semidefinite, then by inequality (2.6), we get (1.1).

**Remark 2.5** For m=1, by inequality (2.1), we get

In particular,

which is a generalization and a refinement of inequality (1.2). For the usual operator norm, it is known that

which is sharp. Since

we find that inequality (2.7) is a strengthening of inequality (2.8).

**Remark 2.6** Recently, Zou [[7], Theorem 2.2] and Zou and He [[8], Theorem 2.2] gave some generalizations of inequality (1.1) for normal matrices. Our result is different from the ones obtained by these authors.

## References

Bhatia R, Kittaneh F:

**Norm inequalities for positive operators.***Lett. Math. Phys.*1998,**43:**225–231. 10.1023/A:1007432816893Kittaneh F:

**Inequalities for the Schatten***p***-norm. III.***Commun. Math. Phys.*1986,**104:**307–310. 10.1007/BF01211597Hirzallah O, Kittaneh F, Moslehian MS:

**Schatten***p***-norm inequalities related to a characterization of inner product spaces.***Math. Inequal. Appl.*2010,**13:**235–241.Bhatia R:

*Positive Definite Matrices*. Princeton University Press, Princeton; 2007.Bhatia R:

*Matrix Analysis*. Springer, New York; 1997.Albadawi H:

**Matrix inequalities related to Hölder inequality.***Banach J. Math. Anal.*2013,**7:**162–171.Zou L:

**Inequalities for unitarily invariant norms and singular values.***Int. J. Appl. Math. Stat.*2013,**46:**308–313.Zou L, He C:

**Weak majorization inequalities for singular values.***Oper. Matrices*2013,**7:**733–737.

## Acknowledgements

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript. This research was supported by the program of Guizhou Science and technology department and Guizhou Minzu University (No. Qian Science co-J word LKM [2011] 09) and the National Natural Science Foundation of China (No. 61263034).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The main idea of this paper was proposed by JH, YP, and LZ. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Huang, J., Peng, Y. & Zou, L. A note on norm inequalities for positive operators.
*J Inequal Appl* **2014**, 171 (2014). https://doi.org/10.1186/1029-242X-2014-171

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1029-242X-2014-171