In this paper, we consider an equivalence relation ∼ c on p (I), which is said to be " convex equivalent " for p ∈ [1, +∞) and a nonempty set I. We characterize the structure of all bounded linear operators T : p (I) → p (I) that strongly preserve the convex equivalence relation. We prove that the rows of the operator which preserve convex equivalent, belong to 1 (I). Also, we show that any bounded linear operators T : p (I) −→ p (I) which preserve convex equivalent, also preserve convex majorization.