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 γ and ψ such that A≤B implies f(B)≤f(A)+γ1ℋ andf(A)≤f(B)+ψ1ℋ, for the self-adjoint operators A, B on a Hilbert space ℋ with identity operator 1ℋ 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.