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 (k(m)[B](m=1)(infinity) of its conditionality constants verifies the estimate k(m)[B] = O(log m) and that if the reverse inequality log m =O(k(m)[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 k(m)[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 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 k(m)[B] = O(log m) and superreflexiye classical Banach spaces having for every epsilon > 0 quasi-greedy bases B with k(m)[B] = O(log m)(1-epsilon). Moreover, in most cases those bases will be almost greedy.