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

 

Complete form of Furuta inequality


Authors: Jiangtao Yuan and Zongsheng Gao
Journal: Proc. Amer. Math. Soc. 136 (2008), 2859-2867
MSC (2000): Primary 47A63, 47B15, 47B20
Published electronically: April 7, 2008
MathSciNet review: 2399051
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Abstract: Let $ A$ and $ B$ be bounded linear operators on a Hilbert space satisfying $ A \ge B \ge 0$. The well-known Furuta inequality is given as follows: Let $ r\ge 0$ and $ p> 0$; then $ A^{\frac{r}{2}} A^{\min\{1,p\}} A^{\frac{r}{2}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{\min\{1,p\}+r}{p+r}}$. In order to give a self-contained proof of it, Furuta (1989) proved that if $ 1\geq r\ge 0$, $ p>p_{0}> 0$ and $ 2p_{0}+r\geq p>p_{0}$, then $ (A^{\frac{r}{2}} B^{p_{0}} A^{\frac{r}{2}})^{\frac{p+r}{p_{0}+r}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{p+r}{p+r}}$.

This paper aims to show a sharpening of Furuta (1989): Let $ r\ge 0$, $ p_{0}> 0$ and $ s=\min\{p, 2p_{0}+\min\{1,r\}\}$; then $ (A^{\frac{r}{2}} B^{p_{0}} A^{\frac{r}{2}})^{\frac{s+r}{p_{0}+r}} \ge(A^{\frac{r}{2}} B^p A^{\frac{r}{2}})^{\frac{s+r}{p+r}}$. We call it the complete form of Furuta inequality because the case $ p_{0}=1$ of it implies the essential part ($ p>1$) of Furuta inequality for $ \frac{1+r}{s+r}\in (0,1]$ by the famous Löwner-Heinz inequality. Afterwards, the optimality of the outer exponent of the complete form is considered. Lastly, we give some applications of the complete form to Aluthge transformation.


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Additional Information

Jiangtao Yuan
Affiliation: LMIB and Department of Mathematics, Beihang University, Beijing 100083, People’s Republic of China
Email: yuanjiangtao02@yahoo.com.cn

Zongsheng Gao
Affiliation: LMIB and Department of Mathematics, Beihang University, Beijing 100083, People’s Republic of China
Email: zshgao@buaa.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09446-X
PII: S 0002-9939(08)09446-X
Keywords: L{\"{o}}wner-Heinz inequality, Furuta inequality, positive operator, $q$-hyponormal operator, Aluthge transformation.
Received by editor(s): March 19, 2007
Published electronically: April 7, 2008
Additional Notes: This work is supported by the Innovation Foundation of Beihang University (BUAA) for PhD Graduates, the National Natural Science Fund of China (10771011), and National Key Basic Research Project of China Grant No. 2005CB321902.
Dedicated: Dedicated to the 20th anniversary of the birth of the Furuta inequality
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.