On the Structure of the Spreading Models of a Banach Space

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space X. In particular we give an example of a reflexive X so that all spreading models of X contain l1 but none of them is isomorphic to l1. We also prove that for any countable set C of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of C. In certain cases this ensures that X admits, for each α < ω1, a spreading model (x̃ (α) i )i such that if α < β then (x̃ (α) i )i is dominated by (and not equivalent to) (x̃ (β) i )i. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.