### 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} ^{∞} of its conditionality constants verifies the estimate k_{m}[B]=O(logm) and that if the reverse inequality logm=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(logm)^{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 k_{m}[B]=O(logm) and superreflexive classical Banach spaces having for every ϵ>0 quasi-greedy bases B with k_{m}[B]=O(logm)^{1−ϵ}. Moreover, in most cases those bases will be almost greedy.

Original language | English (US) |
---|---|

Pages (from-to) | 1893-1924 |

Number of pages | 32 |

Journal | Journal of Functional Analysis |

Volume | 276 |

Issue number | 6 |

DOIs | |

State | Published - Mar 15 2019 |

### Keywords

- Almost greedy basis
- Conditionality constants
- Quasi-greedy basis
- Subsymmetric basis

### ASJC Scopus subject areas

- Analysis

## Fingerprint Dive into the research topics of 'Building highly conditional almost greedy and quasi-greedy bases in Banach spaces'. Together they form a unique fingerprint.

## Cite this

*Journal of Functional Analysis*,

*276*(6), 1893-1924. https://doi.org/10.1016/j.jfa.2018.08.015