On a Question of Furuta on Chaotic Order, II

The chaotic order A ≫ B among positive invertible operators on a Hilbert space is introduced by log A ≥ logB. Related to the Furuta inequality for the chaotic order, Furuta posed the following question: For A;B > 0, A ≫ B if and only if holds for all p ≥ 1, r ≥ t, s ≥ 1 and t ∈ [0,1]? Recently he gave a counterexample to the "only if" part. In our preceding note, we pointed out that the condition (Q) characterizes the operator order A ≥ B. Moreover we showed that (Q) characterizes the chaotic order in some sense. The purpose of this note is to continue our preceding discussion on the operator inequality (Q) under the chaotic order. Among others, we prove that if A ≫ B for A, B > 0, then for p≥1, s≥1, r≥0 and t≤0, where A _s B = Aand particularly ♯_s=_s for s ∈(0,1).

1 2 (A−12 BA−1 2 )sA 1 2 and particularly ]s = \s for s 2 [0; 1].