Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

TL;DR

This paper constructs new examples of Banach spaces with quasi-greedy bases that have maximal or near-maximal conditionality constants, advancing understanding of the structure of such bases in various Banach spaces.

Contribution

The authors develop new techniques to produce examples of both non-superreflexive and superreflexive Banach spaces with quasi-greedy bases exhibiting large conditionality constants, including almost greedy bases.

Findings

Constructed non-superreflexive spaces with quasi-greedy bases having $k_m=O(\log m)$.

Constructed superreflexive spaces with quasi-greedy bases with $k_m=O((\log m)^{1-\epsilon})$ for any $\epsilon>0$.

Most constructed bases are almost greedy, showing the richness of such bases in classical Banach spaces.

Abstract

It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)^{1-\epsilon}$ for some $\epsilon>0$. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed in [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases $\mathcal{B}$ with $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and superreflexive classical Banach spaces having for every $\epsilon>0$ quasi-greedy bases $\mathcal{B}$ with $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)^{1-\epsilon}$. Moreover, in most cases those bases will be almost greedy.