On the Structure of the Spreading Models of a Banach Space

Androulakis, G. ; Odell, E. ; Schlumprecht, Th. ; Tomczak-Jaegermann, N.

Canadian journal of mathematics, Tome 57 (2005) no. 4, pp. 673-707

Résumé

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 ${{\left( \tilde{x}_{i}^{\left( \alpha\right)} \right)}_{i}}$ such that if $\alpha \,<\,\beta $ then ${{\left( \tilde{x}_{i}^{\left( \alpha\right)} \right)}_{i}}$ is dominated by (and not equivalent to) ${{\left( \tilde{x}_{i}^{\left( \beta\right)} \right)}_{i}}$ . 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.

Détail

DOI : 10.4153/CJM-2005-027-9

Mots-clés : 46B03

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