On reversing of the modified Young inequality

Abstract

In the present paper, by Haagerup theorem, we show that if $A \in \mathbb{M}_{n}$ is a non scalar strictly positive matrix and $\nu \in (0,1)$ be a real number such that $ \nu \neq \frac{1}{2},$ then there exists $X \in \mathbb{M}_{n}$ such that $$\|A^{\nu}XA^{1-\nu}\| > \| \nu AX + (1- \nu)XA\|.$$