Direct sums and the Szlenk index

Philip A. H. Brooker

doi:10.1016/j.jfa.2010.12.016

Direct sums and the Szlenk index

Functional Analysis 2011-02-11 v2

Authors: Philip A. H. Brooker

View on arXiv ↗ PDF ↗ DOI ↗

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.

Cite

@article{arxiv.1003.5708,
  title  = {Direct sums and the Szlenk index},
  author = {Philip A. H. Brooker},
  journal= {arXiv preprint arXiv:1003.5708},
  year   = {2011}
}

Comments

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

Direct sums and the Szlenk index

Abstract

Cite

Comments

Related papers