  For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion
of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the
usual notion of summable Szlenk index. We define for an arbitrary
weak$^*$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the
martingale type norm of an operator. We prove that this quantity is determined
by norming sets and determines $\xi$-Szlenk power type and
$\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the
behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the
strongest way to injective tensor products of Banach spaces. Furthermore, we
combine this fact with a result of Schlumprecht to prove that a separable
Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm
can be embedded into a Banach space with a shrinking basis and the same
behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a
Banach space with a shrinking basis and the same $\xi$-Szlenk power type.
Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms
under $\ell_r$ direct sums. This allows us to give an alternative proof of a
result of Brooker regarding Szlenk indices of $\ell_p$ and $c_0$ direct sums of
operators.

Functional Analysis