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

DOI:
https://doi.org/10.1090/S0002-9939-08-09446-X

Published electronically:
April 7, 2008

MathSciNet review:
2399051

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and be bounded linear operators on a Hilbert space satisfying . The well-known Furuta inequality is given as follows: Let and ; then . In order to give a self-contained proof of it, Furuta (1989) proved that if , and , then .

This paper aims to show a sharpening of Furuta (1989): Let , and ; then . We call it the complete form of Furuta inequality because the case of it implies the essential part () of Furuta inequality for 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:
https://doi.org/10.1090/S0002-9939-08-09446-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.