On the complexity of some classes of Banach spaces and non-universality

TL;DR

This paper investigates the complexity of various classes of separable Banach spaces, computing their descriptive set-theoretic complexity and demonstrating non-Borelness for several properties, with applications to non-universality results.

Contribution

It provides the first detailed complexity analysis of multiple Banach space classes and establishes their non-Borel nature, advancing understanding of their structural properties.

Findings

Complexity of Banach-Saks and related properties computed.

Certain classes are shown to be non-Borel in the space of Banach spaces.

Applications to non-universality results are provided.

Abstract

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks opera- tors is complemented in its bounded operators, are all non Borel in SB. At last, we give several applications of those results to non-universality results.