Characterization of Two-sided Order Preserving of Convex Majorization on ℓp(I)

Abstract

In this paper, we consider an equivalence relation c on ℓp(I); which is said to be “convex
equivalent” for p \in [1;+\infinity) and a nonempty set I: We characterize the structure of all bounded linear
operators T : ℓp(I)\to ℓp(I) that strongly preserve the convex equivalence relation. We prove that the
rows of the operator which preserve convex equivalent, belong to ℓ1(I): Also, we show that any operators
T : ℓp(I)\to ℓp(I) which preserve convex equivalent, also preserve convex majorization.