Locally boundedness and continuity of superposition operators on double sequence spaces Cr0

Let R be set of all real numbers, N be the set of all natural numbers and N2 = N×N. In this paper, we define the superposition operator Pg where g: N2×R → R by Pg ((xks)) = g(k,s,xks) for all real double sequence (xks). Chew & Lee [4] and Petranuarat & Kemprasit [11] have characterized Pg: c0 → l1 and Pg: c0 → lq where 1 ≤ q < ∞, respectively. The main aim of this paper is to construct the necessary and sufficient conditions for the boundedness and continuity of Pg: Cr0 → L1 and Pg: Cr0 → Lp where 1≤ p< ∞.