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 and , let Tξ,η be the linear map which sends each xi to yi. We prove that if 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.