Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 79123, 6 pages
doi:10.1155/JIA/2006/79123

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

Abstract

The chaotic order A≫B among positive invertible operators A,B>0 on a Hilbert space is introduced by log⁡A≥log⁡B. 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.