A dichotomy for subsymmetric basic sequences with applications to Garling spaces

TL;DR

This paper establishes a dichotomy theorem for subsymmetric basic sequences in Banach and quasi-Banach spaces, and applies it to classify sequences in Garling spaces, revealing unique and continuum families of subsymmetric sequences.

Contribution

It introduces the notion of positioning and develops new tools leading to a general dichotomy theorem for spaces with subsymmetric bases.

Findings

Garling sequence spaces have a unique symmetric basic sequence.

Garling spaces have no symmetric basis.

Garling spaces possess a continuum of subsymmetric basic sequences.

Abstract

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.