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On the Structure of the Spreading Models of a Banach Space

Open Access article
 Printed: Aug 2005
  • G. Androulakis
  • E. Odell
  • Th. Schlumprecht
  • N. Tomczak-Jaegermann
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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.
MSC Classifications: 46B03 show english descriptions Isomorphic theory (including renorming) of Banach spaces 46B03 - Isomorphic theory (including renorming) of Banach spaces

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