Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 79123, 6 pages
doi:10.1155/JIA/2006/79123
A characterization of chaotic order
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.
Abstract
The chaotic order A≫B among positive invertible operators A,B>0 on a Hilbert space is introduced by logA≥logB. Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that A≫B if and only if ‖BpA−p/2B−p/2‖Ap≥Bp 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 ‖BpA−p/2B−p/2‖Ap≥Bp holds for any p≥p0, but A≫B is not valid.