Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces

Resumen

For a conditional quasi-greedy basis (Formula presented.) in a Banach space, the associated conditionality constants (Formula presented.) verify the estimate (Formula presented.). 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 (Formula presented.) for some (Formula presented.), and this is optimal. Our first goal in this paper will be to fill the gap between the general case and the superreflexive case and investigate the growth of the conditionality constants in nonsuperreflexive 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 (Formula presented.) is not superreflexive, then there is a quasi-greedy basis (Formula presented.) in a Banach space (Formula presented.) finitely representable in (Formula presented.) with (Formula presented.). As a consequence, we obtain that for every (Formula presented.), there is a Banach space (Formula presented.) of type 2 and cotype q possessing a quasi-greedy basis (Formula presented.) with (Formula presented.). We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence (Formula presented.) with (Formula presented.). © 2017 Springer Science+Business Media, LLC