2023/04/12 by Fernando Albiac, José L. Ansorena, Albiac, Fernando +7 · 3 citations
Mathematics · #41A65 (Primary) 46B15 #46B45 (Secondary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2304.05888
openalex publication_date 2023/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of 1-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] 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 [F. Albiac and J.L. Ansorena, Characterization of 1-almost greedy bases, Rev. Mat. Complut. 30 (2017), no. 1, 13-24]. 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.