This is an announcement for the paper “Building highly conditional almost greedy and quasi-greedy bases in Banach spaces” by Fernando Albiachttps://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1, Jose L. Ansorenahttps://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1, Stephen Dilworthhttps://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1, Denka Kutzarovahttps://arxiv.org/find/math/1/au:+Kutzarova_D/0/1/0/all/0/1.
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 $k_m[\mathcal{B}=O(\log m)$ and that if the reverse inequality $\logm=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 $k_m[\mathcal{B}=O(\log m)^{1-\epsilon}$ 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 $k_m[\mathcal{B}=O(\log m)$ and superreflexive classical Banach spaces having for every $\epsilon>0$ quasi-greedy bases $\mathcal{B}$ with $k_m[\mathcal{B}=O(\log m)^{1-\epsilon}$. Moreover, in most cases those bases will be almost greedy.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.08351