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Journal of Operator Theory

Volume 68, Issue 2, Fall 2012  pp. 405-442.

Asplund operators and the Szlenk index

Authors:  Philip A.H. Brooker
Author institution: Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia

Summary:  For $\alpha$ an ordinal, we investigate the class $\szlenkop{\alpha}$ consisting of all operators whose Szlenk index is an ordinal not exceeding $\omega^\alpha$. We show that each class $\szlenkop{\alpha}$ is a closed operator ideal and study various operator ideal properties for these classes. The relationship between the classes $\szlenkop{\alpha}$ 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.

Keywords:  Szlenk index, operator ideal, Asplund operator, factorization property, space ideal

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