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of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums.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 E-direct sum of separable spaces provided that E has a shrinking unconditional basis whose dual basis yields an asymptotic ℓp structure in E⁎. 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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