Continuity of Superposition Operators on the Double Sequence Spaces of Maddox $${\mathcal{L}}\left( p \right)$$ L p

Abstract

Petranuarat and Kemprasit (Southeast Asian Bull Math 21:139–147, 1997) characterized continuity of the superposition operator acting from the sequence space l p into l q where 1 ≤ p, q < ∞. Sagir and Gungor defined the superposition operator P g by P g (x) = (g(k, s, x ks )) ,=1 ∞ for all real double sequences (x ks ) where $$g:{\mathbb{N}}^{2} \times {\mathbb{R}} \to {\mathbb{R}}$$ and gave continuity of the superposition operator acting from the double sequence spaces $${\mathcal{L}}_{p}$$ into $${\mathcal{L}}_{q}$$ for 1 ≤ p, q < ∞. In this paper, we characterize the continuity of the superposition operator acting from Maddox double sequence spaces $${\mathcal{L}}\left( p \right)$$ into $${\mathcal{L}}\left( q \right)$$ where p = (p ks ) and q = (q ks ) are bounded double sequences of positive numbers.