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Direct sums and summability of the Szlenk index

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of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums.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.