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On a reverse of Ando-Hiai inequality. (English) Zbl 1186.47014
Summary: We show a complement of the Ando-Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in[0,1]$. If $A\sharp_\alpha B\le I$, then $$A^r\sharp_\alpha B^r\le \|(A\sharp_\alpha B)^{-1}\|^{1-r}I \quad\text{for all }0<r\le 1,$$ where $I$ is the identity operator and $\|\cdot\|$ stands for the operator norm.

47A63Operator inequalities
47A30Operator norms and inequalities
47A64Operator means, shorted operators, etc.
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