Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 79123, 6 pages

A characterization of chaotic order

Changsen Yang and Fugen Gao

Department of Mathematics, Henan Normal University, Xinxiang 453007, Henan, China

Received 15 November 2005; Accepted 4 January 2006

Copyright © 2006 Changsen Yang and Fugen Gao. 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.


The chaotic order AB among positive invertible operators A,B>0 on a Hilbert space is introduced by logAlogB. Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that AB if and only if BpAp/2Bp/2ApBp holds for any 0<p<p0, where p0 is any fixed positive number. On the other hand, for any fixed p0>0, we also show that there exist positive invertible operators A, B such that BpAp/2Bp/2ApBp holds for any pp0, but AB is not valid.