Continuity Of Superposition Operators On The Double Sequence Spaces ℒ_{p}

Abstract

In this paper, we define the superposition operator P_{g} where
g:ℕ²×ℝ→ℝ by P_{g}((x_{ks}))=g(k,s,x_{ks}) for all real double sequence (x_{ks}). Chew & Lee [3] and Petranuarat & Kemprasit [6] have characterized P_{g}:l_{p}→l₁ and P_{g}:l_{p}→l_{q} where 1≤p,q<∞, respectively. The main goal of this paper is to construct the necessary and sufficient conditions for the continuity of P_{g}:ℒ_{p}→ℒ₁ and P_{g}:ℒ_{p}→ℒ_{q} where 1≤p,q<∞.