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.