On uniqueness and plentitude of subsymmetric sequences

TL;DR

This paper investigates the diversity of subsymmetric basic sequences in Banach spaces, providing examples of spaces with unique or many subsymmetric sequences and criteria for their equivalence to classical bases.

Contribution

It introduces the first example of a space with a unique, non-symmetric subsymmetric basic sequence and establishes criteria for the existence of many such sequences.

Findings

The subsymmetrization of Tsirelson's space has a unique, non-symmetric subsymmetric basic sequence.

A criterion is provided for spaces to contain a continuum of nonequivalent subsymmetric basic sequences.

A criterion is given for subsymmetric sequences to be equivalent to classical basis of ll_p or c_0.

Abstract

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^*)$ of Tsirelson's original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a subsymmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to $Su(T^*)^*$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some $\ell_p$ or $c_0$.