Conditional quasi-greedy bases in non-superreflexive Banach spaces
Date
2019Version
Acceso abierto / Sarbide irekia
Type
Artículo / Artikulua
Version
Versión aceptada / Onetsi den bertsioa
Impact
|
10.1007/s00365-017-9399-x
Abstract
For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). 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 sp ...
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For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). 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 km[B]=O((logm)1-E) for some 0<E<1, 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 X is not superreflexive, then there is a quasi-greedy basis B in a Banach space Y finitely representable in X with km[B]approximate to logm. As a consequence, we obtain that for every 2<q<, there is a Banach space X of type 2 and cotype q possessing a quasi-greedy basis B with km[B]approximate to logm. We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence B with km[B]approximate to m. [--]
Subject
Thresholding greedy algorithm,
Conditional basis,
Conditionality constants,
Quasi-greedy basis,
Type,
Cotype,
Reflexivity,
Superreflexivity,
Super property,
Finite representability,
Banach spaces
Publisher
Springer
Published in
Constructive Approximation, 49 (1), 103-122
Description
This is a post-peer-review, pre-copyedit version of an article published in Constr Approx (2019) 49:103–122. The final authenticated version is available online at: https://doi.org/10.1007/s00365-017-9399-x
Departament
Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas /
Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila
Publisher version
Sponsorship
F. Albiac and J. L. Ansorena were partially supported by the Spanish Research Grant Analisis Vectorial, Multilineal y Aplicaciones, Reference Number MTM2014-53009-P. F. Albiac also acknowledges the support of Spanish Research Grant Operators, lattices, and structure of Banach spaces, with reference MTM2016-76808-P. P. Wojtaszczyk was partially supported by National Science Centre, Poland Grant UMO-2016/21/B/ST1/00241.