A characterization of chaotic order. (8th June 2006) - Yang, Changsen; Gao, Fugen
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.