On ultra-products of some families of composition operators between certain finite dimensional ℓp spaces

Abstract

Let 0 and lt; σ and lt; 1 and 1 and lt; p, r and lt; ∞ be such that 1/r + (1- σ)/p' = 1. We show that for every continuous linear map T between Banach spaces E, F such that its restriction to every finite dimensional subspace N of E factorizes through a chain of type ℓ∞ (ΩN,μN) → ℓr ∞ (ΩN,μN) → ℓ 1 (ΩN,μN)+ ℓp((ΩN,μN) where (ΩN,μN) is a discrete measure space with a finite number of atoms, there is a σ- finite measure space (Ω, μ) such that T ∈ L(E, F") factorizes through the chain of "continuous spaces ℓ∞ (ΩN,μN) → ℓr ∞ (ΩN,μN) → ℓ 1 (ΩN,μN)+ ℓp((ΩN,μN)