Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A)...

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Bibliographic Details
Published in:Journal of approximation theory Vol. 297; p. 105970
Main Authors: Albiac, Fernando, Ansorena, José L., Blasco, Óscar, Chu, Hùng Việt, Oikhberg, Timur
Format: Journal Article
Language:English
Published: Elsevier Inc 01.01.2024
Subjects:
Greedy basis
Property (A)
Suppression unconditional basis
Symmetric basis
Thresholding greedy algorithm
46B15
Greedy basis
41A65
46B45
Symmetric basis
Thresholding greedy algorithm
Suppression unconditional basis
Property (A)
ISSN:0021-9045, 1096-0430
Online Access:Get full text
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Summary:We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2023.105970