ON THE ANDO–HIAI–OKUBO TRACE INEQUALITY

• Published in: Journal of operator theory 2017, Vol.77 (1), p.77-86
• Main Authors: HAYAJNEH, MOSTAFA, HAYAJNEH, SAJA, KITTANEH, FUAD
• Format: Article
• Language: English
• ISSN: 0379-4024; 1841-7744
• DOI: 10.7900/jot.2015dec23.2096

Let A and B be positive semidefinite matrices. It is shown that |Tr(AwBzA 1-w B 1-z)| ≤ Tr(AB) for all complex numbers w, z for which |Re w – ½| + |Re z – ½| ≤ ½. This is a generalization of a trace inequality due to T. Ando, F. Hiai, and K. Okubo for the special case when w, z are real numbers, and a recent trace inequality proved by T. Bottazzi, R. Elencwajg, G. Larotonda, and A. Varela when w = z with ¼ ≤ Re z ≤ ¾. As a consequence of our new trace inequality, we prove that ∥AwBz + B 1-z̄ A 1-w̄∥₂ ≤ ∥AwBz + A 1-w̄ B 1-z̄∥₂ for all complex numbers w, z for which |Re w – ½| + |Re z – ½| ≤ ½. This is a generalization of a recent norm inequality proved by M. Hayajneh, S. Hayajneh, and F. Kittaneh when w, z are real numbers.