On a question of Furuta on chaotic order

AbstractThe chaotic order A≫B among positive invertible operators A,B>0 on a Hilbert space is introduced by logA⩾logB. Uchiyama's method brings us the Furuta inequality for the chaotic order from the Furuta inequality. Related to this, Furuta posed the following question: For A,B>0, A≫B if and only if(Q)Ar−t⩾Ar/2A−t/2BpA−t/2sAr/2(r−t)/((p−t)s+r)holds for all p⩾1, r⩾t, s⩾1 and t∈[0,1]? Recently he gave a counterexample to the “only if” part. In this note, we point out that condition (Q) characterizes the operator order A⩾B. Moreover, (Q) characterizes the spectral order by extending the bounds of t