- Open Access
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).
- unitarily invariant norms
- singular values
- weak majorization
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).
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).
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