On the structure of the spreading models of a Banach space

Abstract

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 ${\ell }_1$ but none of them is isomorphic to ${\ell }_1$. 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 $\alpha <{\omega }_1$, a spreading model $({\tilde{x}}_i^{\alpha }{)}_i$ such that if $\alpha <\beta$ then $({\tilde{x}}_i^{\alpha }{)}_i$ is dominated by (and not equivalent to) $({\tilde{x}}_i^{\beta }{)}_i$. 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.

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