Asymptotic structure and the existence of noncompact operators between Banach spaces

Abstract

We investigate the problem of the existence of a noncompact operator T : X 0 ⊆ X → Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for ξ = 〈 x i 〉 1 n ∈ { X } n and η = 〈 y i 〉 1 n ∈ { Y } n , let T ξ , η be the linear map which sends each x i to y i . We prove that if inf { ‖ T ξ , η ‖ : ξ ∈ { X } n , η ∈ { Y } n } > 1 for some n ∈ N then every T : X 0 ⊆ X → Y is compact. If for n = 2 all such maps have norm 1 we show the existence of a noncompact T : X 0 ⊆ X → Y .