On a question of Furuta on chaotic order Original Research Article

The chaotic order Amuch greater-thanB among positive invertible operators A,B>0 on a Hilbert space is introduced by logAgreater-or-equal, slantedlogB. Uchiyamaʹs 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, Amuch greater-thanB if and only if image holds for all pgreater-or-equal, slanted1, rgreater-or-equal, slantedt, sgreater-or-equal, slanted1 and tset membership, variant[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 Agreater-or-equal, slantedB. Moreover, (Q) characterizes the spectral order by extending the bounds of t.