Concerning the Szlenk index

Ryan M. Causey¹

¹ Department of Mathematics Miami University Oxford, OH 45056, U.S.A.

Studia Mathematica, Tome 236 (2017) no. 3, pp. 201-244

Résumé

We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three-space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove the optimality of a previous universality result, and compute the Szlenk index of the injective tensor product of separable Banach spaces.

Détail

DOI : 10.4064/sm8197-10-2016

Keywords: discuss pruning coloring lemmas regular families discuss several applications these lemmas computing szlenk index certain * compact subsets dual separable banach space applications include estimates szlenk index minkowski sums infinite direct sums separable banach spaces constant reduction three space properties consider using regular families construct banach spaces prescribed szlenk index consequence characterization which countable ordinals occur szlenk index banach space prove optimality previous universality result compute szlenk index injective tensor product separable banach spaces