New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality

Abstract

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by Lemos and Soares in 2018. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by Bhatia, Lim and Yamazaki in 2016, and recently proved by Dinh, Dumitru and Franco in 2017 for the Schatten p-norm with $1\le p\le \mathrm{\infty }$.

Keywords:

• Positive semi-definite matrix
• Hermitian matrix
• log-majorization
• eigenvalues
• singular values
• norm inequalities

Mathematics Subject Classifications:

• 15A42
• 47A63
• 47A64
• 47A30

Acknowledgements

The authors sincerely thank the handling Editor and the Editor-in-Chief for their valuable suggestions and useful comments. Also, many thanks go to reviewers for their constructive reports. The first three authors acknowledge financial support from the Lebanese University research grant programmes.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Université Libanaise [small grant].