  For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and
sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk
index not exceeding $\omega^\alpha$. It follows from our results that the
Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural
way by the behaviour of the $\epsilon$-Szlenk indices of its summands. Our
methods give similar results for $c_0$-direct sums.
