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On the Structure of the Spreading Models of a Banach Space

Published online by Cambridge University Press:  20 November 2018

G. Androulakis ,
E. Odell ,
Th. Schlumprecht  and
N. Tomczak-Jaegermann
G. Androulakis
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A., e-mail: giorgis@math.sc.edu
E. Odell
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, U.S.A., e-mail: odell@math.utexas.edu
Th. Schlumprecht
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A., e-mail: schlump@math.tamu.edu
N. Tomczak-Jaegermann
Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 e-mail: nicole@ellpspace.math.ualberta.ca
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Abstract

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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 ${{\left( \tilde{x}_{i}^{\left( \alpha \right)} \right)}_{i}}$ 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.

Keywords

46B03
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 4 , 01 August 2005 , pp. 673 - 707
Copyright
Copyright © Canadian Mathematical Society 2005

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