Operator-norm inequalities and interpolated trace inequalities

Abstract

In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.

Keywords:

• unitarily invariant norm
• operator inequality
• trace norm
• Hilbert–Schmidt norm
• positive operator

AMS Subject Classifications:

• 15A60
• 47A30
• 47A63
• 47B15

Acknowledgements

The authors are thankful to Prof. Fuad Kittaneh for many interesting discussions, in particular for introducing us to Bourin’s questions. Also, we would like to thank the referee for his valuable suggestions and comments. This work has been done while the second author is in a leave from Princess Sumaya University For Technology to Sharjah University. The author is debited to both institutions for their support.