Direct sums and the Szlenk index

Abstract

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.

The proof of Proposition~2.4 has changed, with some of the arguments transferred to the proof of an added-in lemma, Lemma~2.8. Changes have been made to the Applications section

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• Operator ideals associated with the Szlenk index