Asplund operators and the Szlenk index

For a an ordinal, we investigate the classSZα consisting of all operators whose Szlenk index is an ordinal not exceeding ωα. We show that each class SZα is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes SZα and several well-known closed operator ideals is investigated and quantitative factorization results in terms of the Szlenk index are obtained for the class of Asplund operators.