A Characterization of Chaotic Order

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.