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The Furuta inequality with negative powers

Authors:	Kô tarô Tanahashi

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

Abstract:	Let $A, B \in B(H)$ be bounded linear operators on a Hilbert space satisfying $O\leq B\leq A$ . Furuta showed the operator inequality $(A^{r}B^{p}A^{r})^{\frac{1}{q}}\leq$ $A^{\frac{p+2r}{q}}$ as long as positive real numbers satisfy $p+2r\leq (1+2r)q$ and $1\leq q$ . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

Keywords:	L\"{o}wner-Heinz inequality the Furuta inequality positive operator

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