Asymptotic structure and the existence of noncompact operators between Banach spaces ✩

We investigate the problem of the existence of a noncompact operator T : X0 ⊆ X →Y in terms of the asymptotic structure of separable Banach spaces X and Y . More precisely, for ξ = xi n1 ∈ {X}n and η = yi n1 ∈ {Y }n, let Tξ,η be the linear map which sends each xi to yi . We prove that if inf{ Tξ,η : ξ ∈ {X}n, η ∈ {Y }n} > 1 for some n ∈ N then every T : X0 ⊆ X→Y is compact. If for n = 2 all such maps have norm 1 we show the existence of a noncompact T : X0 ⊆ X→Y . © 2007 Elsevier Inc. All rights reserved.