an: Zbl 1186.47014
au: Seo, Yuki
ti: On a reverse of Ando-Hiai inequality.
la: EN
so: Banach J. Math. Anal. 4, No. 1, 87-91, electronic only (2010).
py: 2010
dt: J
cc: *47A63 47A30 47A64
ut: Ando-Hiai inequality; positive operator; geometric mean
ab: 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.