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# A note on norm inequalities for positive operators

- Jiewu Huang
^{1}, - Yang Peng
^{2}and - Limin Zou
^{2}Email author

**2014**:171

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

© Huang et al.; licensee Springer. 2014

**Received: **20 March 2014

**Accepted: **23 April 2014

**Published: **8 May 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.

## Keywords

- unitarily invariant norms
- singular values
- weak majorization

## 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.

*m*,

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

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

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

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.

## Declarations

### 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).

## Authors’ Affiliations

## References

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**Inequalities for the Schatten**p**-norm. III.***Commun. Math. Phys.*1986,**104:**307–310. 10.1007/BF01211597MathSciNetView ArticleMATHGoogle Scholar - Hirzallah O, Kittaneh F, Moslehian MS:
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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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.