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arXiv · 1511.07632

Direct sums and summability of the Szlenk index

Szymon Draga·Tomasz Kochanek
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Abstract

We prove that the $c_0$-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the $\mathfrak{E}$-direct sum of separable spaces provided that $\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\ell_p$ structure in $\mathfrak{E}^\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.

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Szymon Draga, Tomasz Kochanek. 2015-11-24. Direct sums and summability of the Szlenk index. https://arxiv.org/abs/1511.07632

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