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Best possibility of the Furuta inequality

Authors:	Kô tarô Tanahashi

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

Abstract:	Let $0\le p,q,r\in\mathbb R, p+2r\le(1+2r)q$ , and $1\le q$ . Furuta (1987) proved that if bounded linear operators $A,B\in B(H)$ on a Hilbert space $(\dim(H)\ge 2)$ satisfy $0\le B\le A$ , then $(A^r B^p A^r)^{1/q} \le A^{(p+2r)/q}$ . In this paper, we prove that the range $p+2r\le (1+2r)q$ and $1\le q$ is best possible with respect to the Furuta inequality, that is, if or , then there exist $A,B\in B(\mathbb R^2)$ which satisfy $0\le B\le A$ but $(A^r B^p A^r)^{1/q}\nleq A^{(p+2r)/q}$ .

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

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