2 results
On the complete separation of unique
$\ell _{1}$ spreading models and the Lebesgue property of Banach spaces
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 13 December 2024, pp. 1-23
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We construct a reflexive Banach space
$X_{\mathcal {D}}$ with an unconditional basis such that all spreading models admitted by normalized block sequences in 
$X_{\mathcal {D}}$ are uniformly equivalent to the unit vector basis of 
$\ell _1$, yet every infinite-dimensional closed subspace of 
$X_{\mathcal {D}}$ fails the Lebesgue property. This is a new result in a program initiated by Odell in 2002 concerning the strong separation of asymptotic properties in Banach spaces.
A QUANTITATIVE EXTENSION OF SZLENK’S THEOREM
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 20 February 2019, pp. 338-343
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- October 2019
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We show that for a bounded subset
$A$ of the 
$L_{1}(\unicode[STIX]{x1D707})$ space with finite measure 
$\unicode[STIX]{x1D707}$, the measure of weak noncompactness of 
$A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space 
$L_{1}$ has a subsequence whose arithmetic means are norm convergent.
