# Conditional quasi-greedy bases in non-superreflexive Banach spaces

## TL;DR

This paper investigates the growth of conditionality constants for quasi-greedy bases in non-superreflexive Banach spaces, showing that optimal bounds are only guaranteed in superreflexive spaces and constructing examples with logarithmic growth.

## Contribution

It demonstrates that in non-superreflexive Banach spaces, quasi-greedy bases can have conditionality constants growing like log m, filling the gap between general and superreflexive cases.

## Key findings

- Quasi-greedy bases in non-superreflexive spaces can have conditionality constants ~ log m.
- In non-superreflexive spaces, there exist bases with conditionality constants ~ m.
- Spaces of type 2 and cotype q can have quasi-greedy bases with conditionality constants ~ log m.

## Abstract

For a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space the associated conditionality constants $k_{m}[\mathcal{B}]$ verify the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies $k_{m}[\mathcal{B}]=(\log m)^{1-\epsilon}$ for some $0<\epsilon<1$, and this is optimal. Our first goal in this paper will be to fill the gap in between the general case and the superreflexive case and investigate the growth of the conditionality constants in non-superreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space $\mathbb{X}$ is not superreflexive then there is a quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{Y}$ finitely representable in $\mathbb{X}$ with $k_{m}[\mathcal{B}] \approx \log m$. As a consequence we obtain that for every $2<q<\infty$ there is a Banach space $\mathbb{X}$ of type $2$ and cotype $q$ possessing a quasi-greedy basis $\mathcal{B}$ with $k_{m}[\mathcal{B}] \approx \log m$. We also tackle the corresponding problem for Schauder bases and show that if a space is non-superreflexive then it possesses a basic sequence $\mathcal{B}$ with $k_m[\mathcal{B}]\approx m$.