Characterizations of log A>=log B and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:”If A, B>=0and X(element of)B(H), then ||AXB||^ r ||X||^1–r >=||A^ r XB^ r ||holds for r(element of) [0, 1].” In this paper, we shall show the following two applications of this inequality: Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logA>=logB) by a norm inequality. Secondly, we shall study the condition under which ||T||=||T~||, where T~=|T|^1/2 U|T|^1/2 is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=||T||) via Aluthge transformation.