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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Furuta inequality
in Banach $*$-algebras


Authors: Kôtarô Tanahashi and Atsushi Uchiyama
Journal: Proc. Amer. Math. Soc. 128 (2000), 1691-1695
MSC (1991): Primary 47A05, 47B15
Published electronically: 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 $H $ 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.


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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: http://dx.doi.org/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
Published electronically: 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
Article copyright: © Copyright 2000 American Mathematical Society