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The best possibility of the grand Furuta inequality

Authors:	Kô tarô Tanahashi

Institution:	Department of Mathematics, Tohoku College of Pharmacy, Komatsushima, Aoba-ku, Sendai 981-8558, Japan

Abstract:	Let $A, B \in B(H)$ be invertible bounded linear operators on a Hilbert space satisfying $O\leq B \leq A$ , and let 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 and the Ando-Hiai inequality ( ). 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 with $O\leq B \leq A$ which do not satisfy $\left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha }$ .

Keywords:	The L\"owner-Heinz inequality the Furuta inequality the grand Furuta inequality

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