Inequalities for the powers of nonnegative Hermitian operators

Abstract:

In the set of bounded Hermitian operators from a Hilbert space $H$ into itself, we define three types of ordering by means of the cones of nonnegative, positive definite and positive invertible operators respectively. Our main theorem shows that for all three types of ordering, if $A$ is “greater” than $B$, then ${A^r}$ is “greater” than ${B^r}$ for all real numbers $r \leq 1$. This generalizes the results of Heinz [3] and Kato [4].