A characterization of chaotic order. (8th June 2006)

• Record Type: Journal Article
• Authors: Yang, Changsen; Gao, Fugen
• Is Part Of: Journal of inequalities and applications. Volume 2006(2006)
• Publication Date: 2006-06-08
• Subjects: Inequalities (Mathematics) -- Periodicals; 512.97
• DOI: 10.1155/JIA/2006/79123
• Languages: English
• ISSNs: 1029-242X

Abstract

Abstract : The chaotic orderA ≫ B among positive invertible operatorsA, B > 0 on a Hilbert space is introduced bylog ⁡ A ≥ log ⁡ B . Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out thatA ≫ B if and only if‖ B p A − p / 2 B − p / 2 ‖ A p ≥ B p holds for any0 < p < p 0, wherep 0 is any fixed positive number. On the other hand, for any fixedp 0 > 0, we also show that there exist positive invertible operatorsA, B such that‖ B p A − p / 2 B − p / 2 ‖ A p ≥ B p holds for anyp ≥ p 0, butA ≫ B is not valid.