Abstract
The chaotic order A≫ B among positive invertible operators A, B>0 on a Hilbert space is introduced by log A⩾log B. 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) A r−t⩾ A r/2 A −t/2B pA −t/2 sA r/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.
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