Advances in Mathematics ( IF 1.494 ) Pub Date : 2020-01-08 , DOI: 10.1016/j.aim.2019.106962
Spiros A. Argyros; Pavlos Motakis

Let ${\left({e}_{i}\right)}_{i}$ denote the unit vector basis of ${\ell }_{p}$, $1\le p<\infty$, or ${c}_{0}$. We construct a reflexive Banach space with an unconditional basis that admits ${\left({e}_{i}\right)}_{i}$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-${\ell }_{p}$ or ${c}_{0}$ subspace. This solves a problem of E. Odell. We also construct a space with a unique ${\ell }_{1}$ spreading model and no subspace with a uniformly unique ${\ell }_{1}$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

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