Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

TL;DR

This paper proves that Garling sequence spaces are not superreflexive by embedding a non-superreflexive subspace, and explores implications for the existence of special conditional bases in these spaces.

Contribution

It demonstrates the non-superreflexivity of Garling sequence spaces and applies this to the study of conditional bases within these spaces.

Findings

Garling sequence spaces contain a complemented non-superreflexive subspace

Results imply the existence of certain conditional Schauder bases

Applications to the structure of bases in Banach spaces

Abstract

In this paper we settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non superreflexive mixed-norm sequence space. As a by-product of our work, we give applications of this result to the study of conditional Schauder bases and conditional almost greedy bases in this new class of Banach spaces.