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ISSN 1088-6826(e) ISSN 0002-9939(p)

The Furuta inequality in Banach -algebras

Author(s): Kôtarô Tanahashi; Atsushi Uchiyama
Journal: Proc. Amer. Math. Soc. 128 (2000), 1691-1695.
MSC (1991): Primary 47A05, 47B15
Posted: September 30, 1999
MathSciNet review: 1654084
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Abstract: Let $0 < p, q, r \in \mathbb{R}$ be real numbers with $p+2r\leq (1+2r)q$ and $1\leq q.$ Furuta (1987) proved that if bounded linear operators $A, B \in B(H)$ on a Hilbert space satisfy $O\leq B \leq A$ , then $B^{\frac{p+2r}{q}} \leq (B^{r}A^{p}B^{r})^{\frac{1}{q}}$ . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

References:

1.: T. Furuta, $A \geq B \geq O$ assures $(B^{r}A^{p}B^{r})^{\frac{1}{q}} \geq B^{\frac{p+2r}{q}}$ for $r\geq 0, p \geq 0, q \geq 1$ with $(1+2r)q\geq (p+2r)$ , Proc. Amer. Math. Soc., 101 (1987), 85-88.MR 89b:47028
2.: T. Furuta, An elementary proof of an order preserving inequality, Proc. Japan Acad., 65 (1989), 126.MR 90g:47029
3.: T. Furuta, Two operator functions with monotone property, Proc. Japan Acad., 111 (1991), 511-516.MR 91f:47023
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6.: T. Okayasu, Heinz's inequality in Banach -algebras, ( preprint ).
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Additional Information:

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

Atsushi Uchiyama
Affiliation: Mathematical Institute, Tohoku University, Aoba-ku, Sendai 980-8578, Japan

DOI: 10.1090/S0002-9939-99-05262-4
PII: S 0002-9939(99)05262-4
Keywords: The L\"owner-Heinz inequality, the Furuta inequality
Received by editor(s): February 12, 1998
Received by editor(s) in revised form: July 13, 1998
Posted: September 30, 1999
Additional Notes: This research is partially supported by Grant-in-Aid Scientific Research (K. Tanahashi, No. 10640185).
Communicated by: David R. Larson
Copyright of article: Copyright 2000, American Mathematical Society

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