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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 best possibility
of the grand Furuta inequality


Author: Kôtarô Tanahashi
Journal: Proc. Amer. Math. Soc. 128 (2000), 511-519
MSC (1991): Primary 47B15
Published electronically: July 6, 1999
MathSciNet review: 1654088
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A, B \in B(H)$ be invertible bounded linear operators on a Hilbert space $H$ satisfying $O\leq B \leq A$ , and let $ p, r, s , t $ be real numbers satisfying $ 1 < s, 0 < t < 1 , t \leq r , 1 \leq p . $ Furuta showed that if $ 0 < \alpha \leq \dfrac{ 1-t+r}{ (p-t)s + r} $, then $ \left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha } $. This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality $(t=0)$
and the Ando-Hiai inequality ( $ t=1, r = s $ ).

In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if $ \dfrac{ 1-t+r}{ (p-t)s + r} < \alpha $, then there exist invertible matrices $A,B$ with $O\leq B \leq A$ which do not satisfy $ \left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha } $.


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Additional Information

Kôtarô Tanahashi
Email: tanahasi@tohoku-pharm.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05261-2
PII: S 0002-9939(99)05261-2
Keywords: The L\"owner-Heinz inequality, the Furuta inequality, the grand Furuta inequality
Received by editor(s): September 27, 1997
Received by editor(s) in revised form: March 31, 1998
Published electronically: July 6, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society