More about operator order preserving

Abstract

It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing functions do. In this article, we employ a convex approach to discuss operator order preserving or conversing functions. As an easy consequence of more general results, we find nonnegative constants $\gamma$ and $\psi$ such that $A\le B$ implies

$$f\left(B\right)\le f\left(A\right)+\gamma {\mathbf{1}}_{\mathcal{\mathscr{H}}}\text{ and}f\left(A\right)\le f\left(B\right)+\psi {\mathbf{1}}_{\mathcal{\mathscr{H}}},$$

for the self-adjoint operators $A$, $B$ on a Hilbert space $\mathcal{\mathscr{H}}$ with identity operator ${\mathbf{1}}_{\mathcal{\mathscr{H}}}$ and for the convex function $f$ whose domain contains the spectra of both $A$ and $B$.

The connection of these results to the existing literature will be discussed and the significance will be emphasized by some examples.