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A characterization of chaotic order

The chaotic order among positive invertible operators on a Hilbert space is introduced by. Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that if and only if holds for any, where is any fixed positive number. On the other hand, for any fixed, we also show that there exist positive invertible operators, such that holds for any, but is not valid.