Open Access

A characterization of chaotic order

Journal of Inequalities and Applications20062006:79123

https://doi.org/10.1155/JIA/2006/79123

Received: 15 November 2005

Accepted: 4 January 2006

Published: 8 June 2006

Abstract

The chaotic order among positive invertible operators on a Hilbert space is introduced by . Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that if and only if holds for any , where is any fixed positive number. On the other hand, for any fixed , we also show that there exist positive invertible operators , such that holds for any , but is not valid.

[123456]

Authors’ Affiliations

(1)
Department of Mathematics, Henan Normal University

References

  1. Ando T: On some operator inequalities. Mathematische Annalen 1987,279(1):157–159. 10.1007/BF01456197MathSciNetView ArticleMATHGoogle Scholar
  2. Fujii M, Furuta T, Kamei E: Furuta's inequality and its application to Ando's theorem. Linear Algebra and Its Applications 1993, 179: 161–169.MathSciNetView ArticleMATHGoogle Scholar
  3. Yamazaki T: Characterizations ofand normaloid operators via Heinz inequality. Integral Equations and Operator Theory 2002,43(2):237–247. 10.1007/BF01200255MathSciNetView ArticleMATHGoogle Scholar
  4. Furuta T: Results undercan be derived from ones underby Uchiyama's method—associated with Furuta and Kantorovich type operator inequalities. Mathematical Inequalities & Applications 2000,3(3):423–436.MathSciNetView ArticleMATHGoogle Scholar
  5. Uchiyama M: Some exponential operator inequalities. Mathematical Inequalities & Applications 1999,2(3):469–471.MathSciNetView ArticleMATHGoogle Scholar
  6. Tanahashi K: Best possibility of the Furuta inequality. Proceedings of the American Mathematical Society 1996,124(1):141–146. 10.1090/S0002-9939-96-03055-9MathSciNetView ArticleMATHGoogle Scholar

Copyright

© C. Yang and F. Gao 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.