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A note on norm inequalities for positive operators
© Huang et al.; licensee Springer. 2014
Received: 20 March 2014
Accepted: 23 April 2014
Published: 8 May 2014
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).
Let be the space of complex matrices. Let denote any unitarily invariant norm on . We shall always denote the singular values of A by , that is, the eigenvalues of the positive semidefinite matrix , arranged in decreasing order and repeated according to multiplicity. Let be Hermitian; the order relation means, as usual, that is positive semidefinite.
where is the Frobenius norm of X. For more information on the Schatten p-norm and its applications, the reader is referred to .
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).
This completes the proof. □
This completes the proof. □
Remark 2.4 If A and B are positive semidefinite, then by inequality (2.6), we get (1.1).
we find that inequality (2.7) is a strengthening of inequality (2.8).
Remark 2.6 Recently, Zou [, Theorem 2.2] and Zou and He [, Theorem 2.2] gave some generalizations of inequality (1.1) for normal matrices. Our result is different from the ones obtained by these authors.
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  09) and the National Natural Science Foundation of China (No. 61263034).
- Bhatia R, Kittaneh F: Norm inequalities for positive operators. Lett. Math. Phys. 1998, 43: 225–231. 10.1023/A:1007432816893MathSciNetView ArticleMATHGoogle Scholar
- Kittaneh F: 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: Schatten p -norm inequalities related to a characterization of inner product spaces. Math. Inequal. Appl. 2010, 13: 235–241.MathSciNetMATHGoogle Scholar
- Bhatia R: Positive Definite Matrices. Princeton University Press, Princeton; 2007.MATHGoogle Scholar
- Bhatia R: Matrix Analysis. Springer, New York; 1997.View ArticleMATHGoogle Scholar
- Albadawi H: Matrix inequalities related to Hölder inequality. Banach J. Math. Anal. 2013, 7: 162–171.MathSciNetView ArticleMATHGoogle Scholar
- Zou L: Inequalities for unitarily invariant norms and singular values. Int. J. Appl. Math. Stat. 2013, 46: 308–313.MathSciNetGoogle Scholar
- Zou L, He C: Weak majorization inequalities for singular values. Oper. Matrices 2013, 7: 733–737.MathSciNetView ArticleMATHGoogle Scholar
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