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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| Published in: | Journal of approximation theory Vol. 297; p. 105970 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 105970 |
| Author | Blasco, Óscar Oikhberg, Timur Albiac, Fernando Chu, Hùng Việt Ansorena, José L. |
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| Keywords | 46B15 Greedy basis 41A65 46B45 Symmetric basis Thresholding greedy algorithm Suppression unconditional basis Property (A) |
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| References | Albiac, Ansorena, Berná, Wojtaszczyk (b4) 2021; 560 Dilworth, Odell, Schlumprecht, Zsák (b8) 2010; 148 Wojtaszczyk (b14) 2003 Albiac, Wojtaszczyk (b6) 2006; 138 Albiac, Ansorena (b1) 2016; 201 Livshits (b12) 2010; 201 Albiac, Kalton (b5) 2016; vol. 233 Konyagin, Temlyakov (b10) 1999; 5 Dilworth, Odell, Schlumprecht, Zsák (b9) 2011; 163 Wojtaszczyk (b13) 2000; 107 Albiac, Ansorena, Berasategui, Berná, Lassalle (b3) 2023; 57 Albiac, Ansorena (b2) 2017; 30 Dilworth, Kalton, Kutzarova, Temlyakov (b7) 2003; 19 Lindenstrauss, Tzafriri (b11) 1977 Albiac (10.1016/j.jat.2023.105970_b6) 2006; 138 Albiac (10.1016/j.jat.2023.105970_b4) 2021; 560 Dilworth (10.1016/j.jat.2023.105970_b7) 2003; 19 Albiac (10.1016/j.jat.2023.105970_b1) 2016; 201 Wojtaszczyk (10.1016/j.jat.2023.105970_b14) 2003 Albiac (10.1016/j.jat.2023.105970_b5) 2016; vol. 233 Wojtaszczyk (10.1016/j.jat.2023.105970_b13) 2000; 107 Albiac (10.1016/j.jat.2023.105970_b3) 2023; 57 Albiac (10.1016/j.jat.2023.105970_b2) 2017; 30 Konyagin (10.1016/j.jat.2023.105970_b10) 1999; 5 Dilworth (10.1016/j.jat.2023.105970_b9) 2011; 163 Lindenstrauss (10.1016/j.jat.2023.105970_b11) 1977 Dilworth (10.1016/j.jat.2023.105970_b8) 2010; 148 Livshits (10.1016/j.jat.2023.105970_b12) 2010; 201 |
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| SubjectTerms | Greedy basis Property (A) Suppression unconditional basis Symmetric basis Thresholding greedy algorithm |
| Title | Counterexamples in isometric theory of symmetric and greedy bases |
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