Conditional quasi-greedy bases in non-superreflexive Banach spaces

Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw

Publication:
Conditional quasi-greedy bases in non-superreflexive Banach spaces

dc.contributor.author	Albiac Alesanco, Fernando José
dc.contributor.author	Ansorena, José L.
dc.contributor.author	Wojtaszczyk, Przemyslaw
dc.contributor.department	Estadística, Informática y Matemáticas	es_ES
dc.contributor.department	Estatistika, Informatika eta Matematika	eu
dc.date.accessioned	2019-08-26T11:13:29Z
dc.date.available	2019-08-26T11:13:29Z
dc.date.issued	2019
dc.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	en
dc.description.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 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.	en
dc.description.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.	en
dc.format.extent	21 p.
dc.format.mimetype	application/pdf	en
dc.identifier.doi	10.1007/s00365-017-9399-x
dc.identifier.issn	1432-0940
dc.identifier.uri	https://academica-e.unavarra.es/handle/2454/34670
dc.language.iso	eng	en
dc.publisher	Springer	en
dc.relation.ispartof	Constructive Approximation, 49 (1), 103-122	en
dc.relation.projectID	info:eu-repo/grantAgreement/MINECO//MTM2014-53009-P/ES/	en
dc.relation.projectID	info:eu-repo/grantAgreement/ES/1PE/MTM2016-76808-P	en
dc.relation.publisherversion	https://doi.org/10.1007/s00365-017-9399-x
dc.rights	© Springer Science+Business Media, LLC 2017	en
dc.rights.accessRights	info:eu-repo/semantics/openAccess
dc.subject	Thresholding greedy algorithm	en
dc.subject	Conditional basis	en
dc.subject	Conditionality constants	en
dc.subject	Quasi-greedy basis	en
dc.subject	Type	en
dc.subject	Cotype	en
dc.subject	Reflexivity	en
dc.subject	Superreflexivity	en
dc.subject	Super property	en
dc.subject	Finite representability	en
dc.subject	Banach spaces	en
dc.title	Conditional quasi-greedy bases in non-superreflexive Banach spaces	en
dc.type	info:eu-repo/semantics/article
dc.type.version	info:eu-repo/semantics/acceptedVersion	en
dc.type.version	Versión aceptada / Onetsi den bertsioa	es
dspace.entity.type	Publication
relation.isAuthorOfPublication	3a702006-6ba1-41ba-93bf-ea9fee1de239
relation.isAuthorOfPublication.latestForDiscovery	3a702006-6ba1-41ba-93bf-ea9fee1de239