The best possibility of the grand Furuta inequality

Tanahashi, Kôtarô

doi:10.1090/S0002-9939-99-05261-2

The best possibility of the grand Furuta inequality
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by Kôtarô Tanahashi

Proc. Amer. Math. Soc. 128 (2000), 511-519

DOI: https://doi.org/10.1090/S0002-9939-99-05261-2

Published electronically: July 6, 1999

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Abstract:

Let $A, B \in B(H)$ be invertible bounded linear operators on a Hilbert space $H$ satisfying $O\leq B \leq A$ , and let $p, r, s , t$ be real numbers satisfying $1 < s, 0 < t < 1 , t \leq r , 1 \leq p .$ Furuta showed that if $0 < \alpha \leq \dfrac { 1-t+r}{ (p-t)s + r}$, then $\left \{ A^{\frac {r}{2}} \left ( A^{ -\frac {t}{2}} B^{p} A^{ -\frac {t}{2}} \right )^{s} A^{\frac {r}{2}} \right \}^{\alpha } \leq A^{ \left \{ (p-t)s + r \right \} \alpha }$. This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality $(t=0)$ and the Ando-Hiai inequality ( $t=1, r = s$ ). In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if $\dfrac { 1-t+r}{ (p-t)s + r} < \alpha$, then there exist invertible matrices $A,B$ with $O\leq B \leq A$ which do not satisfy $\left \{ A^{\frac {r}{2}} \left ( A^{ -\frac {t}{2}} B^{p} A^{ -\frac {t}{2}} \right )^{s} A^{\frac {r}{2}} \right \}^{\alpha } \leq A^{ \left \{ (p-t)s + r \right \} \alpha }$.

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