The structure of greedy-type bases in Tsirelson's space and its convexifications

TL;DR

This paper explores the structure of greedy and conditional bases in Tsirelson's space and its convexifications, revealing uncountably many non-equivalent bases and deepening understanding of its complex Banach space properties.

Contribution

It is the first to analyze the greedy-type basis structure of Tsirelson's space and its convexifications, showing the existence of uncountably many non-equivalent bases.

Findings

Tsirelson's space has uncountably many non-equivalent greedy bases.

Convexifications of Tsirelson's space also have uncountably many non-equivalent bases.

The spaces possess uncountably many non-equivalent conditional almost greedy bases.

Abstract

Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in 1974 soon to become one of the most significant counterexamples in the theory. Its structure broke the ideal pattern that analysts had conceived for a generic Banach space, thus giving rise to the era of pathological examples. Since then, many authors have contributed to the study of different aspects of this special space with an eye on better understanding its idiosyncrasies. In this paper we are concerned with the greedy-type basis structure of $\mathcal{T}$, a subject that had not been previously explored in the literature. More specifically, we show that Tsirelson's space and its convexifications $\mathcal{T}^{(p)}$ for $0<p<\infty$ have uncountably many non-equivalent greedy bases. We also investigate the conditional basis structure of spaces $\mathcal{T}^{(p)}$ in the range of $0<p<\infty$ and prove that they have uncountably many non-equivalent conditional almost greedy bases.