This is an announcement for the paper "Direct sums and the Szlenk index" by Philip A. H. Brooker.

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 $\varepsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.

Archive classification: math.FA

Submitted from: philip.brooker@anu.edu.au

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1003.5708

or

http://arXiv.org/abs/1003.5708