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.

In this work we consider all bounded linear operators T : c 0 → c 0 that preserve convex equivalent relation ∼ c on c 0 and we denote by P ce (c 0) the set of such operators. If T strongly preserves convex equivalent, we denote them by P sce (c 0). Some interesting properties of P ce (c 0) are given. For T ∈ P ce (c 0), we show that all rows of T belong to 1 and for any j ∈ N, we have 0 ∈ Im(Te j), also there are a, b ∈ Im(Te j) such that co(Te j) = [a, b]. It is shown that all row sums of T belong to [a, b]. We characterize the elements of P ce (c 0), and some interesting results of all T ∈ P sce (c 0) are given, for example we prove that a = 0 < b or a < 0 = b. Also the elements of P sce (c 0) are characterized. We obtain the matrix representation of T ∈ P sce (c 0) does not contain any zero row. Some relevant examples are given.