Direct sums and summability of the Szlenk index

2015/11/24 by Szymon Draga, Draga, Szymon, Tomasz Kochanek +1
Mathematics · #46B03 #46B20 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B03 #msc:46B20

paper · pdf · doi:10.48550/arxiv.1511.07632

26 pp

arxiv created 2015/11/24 · arxiv updated 2015/11/25

Abstract

We prove that the c₀-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of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums.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 \mathfrakE-direct sum of separable spaces provided that \mathfrakE has a shrinking unconditional basis whose dual basis yields an asymptotic ℓ_p structure in \mathfrakE^∗. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of c₀ has power type 1 but non-summable Szlenk index.

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