Conditional Quasi-Greedy Bases in Non-superreflexive Banach Spaces

For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants k m [ B ] verify the estimate k m [ B ] = O ( log m ) . Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-gre...

Full description

Saved in:
Bibliographic Details
Published in:Constructive approximation Vol. 49; no. 1; pp. 103 - 122
Main Authors: Albiac, Fernando, Ansorena, José L., Wojtaszczyk, Przemysław
Format: Journal Article
Language:English
Published: New York Springer US 15.02.2019
Springer Nature B.V
Subjects:
Analysis
Banach spaces
Mathematics
Mathematics and Statistics
Numerical Analysis
Cotype
Reflexivity
Finite representability
Thresholding greedy algorithm
Banach spaces
Superreflexivity
Conditional basis
Type
46B15
Super property
41A65
Quasi-greedy basis
Conditionality constants
ISSN:0176-4276, 1432-0940
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants k m [ B ] verify the estimate k m [ B ] = O ( log m ) . 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 k m [ B ] = O ( ( log m ) 1 - ϵ ) for some 0 < ϵ < 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 k m [ B ] ≈ log m . 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 k m [ B ] ≈ log m . 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 k m [ B ] ≈ m .
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-017-9399-x