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

Abstract

It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (km[B])m=1∞ of its conditionality constants verifies the estimate km[B]=O(log⁡m) and that if the reverse inequality log⁡m=O(km[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate km[B]=O(log⁡m)1−ϵ for some ϵ>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 by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with km[B]=O(log⁡m) and superreflexive classical Banach spaces having for every ϵ>0 quasi-greedy bases B with km[B]=O(log⁡m)1−ϵ. Moreover, in most cases those bases will be almost greedy.