Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Furuta inequality with negative powers

Author: Kôtarô Tanahashi
Journal: Proc. Amer. Math. Soc. 127 (1999), 1683-1692
MSC (1991): Primary 47B15
Published electronically: February 11, 1999
MathSciNet review: 1476395
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Abstract: Let $ A, B \in B(H)$ be bounded linear operators on a Hilbert space $H$ 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 $p , q, r $ satisfy $ p+2r\leq (1+2r)q$ and $ 1\leq q$. In this paper, we show this inequality is valid if negative real numbers $ p, q, r $ satisfy a certain condition. Also, we investigate the optimality of that condition.

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Additional Information

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

Keywords: L\"{o}wner-Heinz inequality, the Furuta inequality, positive operator
Received by editor(s): September 29, 1995
Received by editor(s) in revised form: June 12, 1996, August 6, 1996, October 23, 1996, April 3, 1997, and September 4, 1997
Published electronically: February 11, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society