Continuity of Superposition Operators on the Double Sequence Spaces L-p

Abstract

In this paper, we define the superposition operator P-g where g : N-2 x R -> R by P-g ((x(ks))) = 1 (k, s, x(ks)) for all real double sequence (x(ks)). Chew & Lee [4] and Petranuarat & Kemprasit [7] have characterized P-g : l(p) -> l(1) and P-g : l(p) -> l(q) where 1 <= p, q < infinity, respectively. The main goal of this paper is to construct the necessary and sufficient conditions for the continuity of P-g : L-p -> L-1 and P-g : L-p -> L-q where 1 <= p, q < infinity.