Direct sums and summability of the Szlenk index

摘要

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of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums.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 e-direct sum of separable spaces provided that e has a shrinking unconditional basis whose dual basis yields an asymptotic l(p) structure in e*. As a corollary, we show that the Thirelson direct sum of infinitely many copies of c(0) has power type 1 but non-summable Szlenk index. (C) 2016 Elsevier Inc. All rights reserved.