Concerning summable Szlenk index

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from [Draga and Kochanek 2016] regarding the behavior of summability of Szlenk index under $c_0$ direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from [Draga and Kochanek 2017]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic $c_0$ finite dimensional decomposition, which generalizes a result from [Odell et al 2008]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For $1\leqslant p\leqslant \infty$, we prove precise results regarding the summability of the Szlenk index of an $\ell_p$ direct sum of a collection of operators.