Direct sums and summability of the Szlenk index

We prove that the c0c0-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of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums.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 EE-direct sum of separable spaces provided that EE has a shrinking unconditional basis whose dual basis yields an asymptotic ℓpℓp structure in E⁎E⁎. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of c0c0 has power type 1 but non-summable Szlenk index.