Albiac Alesanco, Fernando JoséAnsorena, José L.Blasco, ÓscarChu, Hùng ViệtOikhberg, Timur2024-05-072024-05-072024Albiac, F., Ansorena, J. L., Blasco, Ó., Chu, H. V., Oikhberg, T. (2024) Counterexamples in isometric theory of symmetric and greedy bases. Journal of Approximation Theory, 297, 1-20. https://doi.org/10.1016/j.jat.2023.105970.0021-904510.1016/j.jat.2023.105970https://academica-e.unavarra.es/handle/2454/48077We 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.application/pdfeng© 2023 The Author(s). This is an open access article under the CC BY license.Greedy basisProperty (A)Suppression unconditional basisSymmetric basisThresholding greedy algorithminfo:eu-repo/semantics/article2024-05-07Acceso abierto / Sarbide irekiainfo:eu-repo/semantics/openAccess