Albiac Alesanco, Fernando José
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Albiac Alesanco
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Fernando José
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Estadística, Informática y Matemáticas
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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas
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Publication Open Access The uniqueness of unconditional basis of the 2-convexified Tsirelson space, revisited(Springer, 2024-10-13) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaOne of the hallmarks in the study of the classification of Banach spaces with a unique (normalized) unconditional basis was the unexpected result by Bourgain, Casazza, Lindenstrauss, and Tzafriri from their 1985 Memoir that the 2-convexified Tsirelson space T(2) had that property (up to equivalence and permutation). Indeed, on one hand, finding a “pathological” space (i.e., not built out as a direct sum of the only three classical sequence spaces with a unique unconditional basis) shattered the hopeful optimism of attaining a satisfactory description of all Banach spaces which enjoy that important structural feature. On the other hand it encouraged furthering a research topic that had received relatively little attention until then. After forty years, the advances on the subject have shed light onto the underlying patterns shared by those spaces with a unique unconditional bases belonging to the same class, which has led to reproving the original theorems with fewer technicalities. Our motivation in this note is to revisit the aforementioned result on the uniqueness of unconditional basis of T(2) from the current state-of-art of the subject and to fill in some details that we missed from the original proof.Publication Open Access Uniqueness of unconditional basis of Hp(T) ⊕ 2 and Hp(T) ⊕ T (2) for 0 < p < 1(Elsevier, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasOur goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair (X, Y) formed by a quasi-Banach space X and a Banach space Y which guarantee that every unconditional basis of their direct sum X ⊕ Y splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces Hp(Td) ⊕ T (2) and Hp(Td) ⊕ 2, for p ∈ (0, 1) and d ∈ N, have a unique unconditional basis (up to equivalence and permutation).Publication Open Access On the permutative equivalence of squares of unconditional bases(Elsevier, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasWe prove that if the squares of two unconditional bases are equivalent up to a permutation, then the bases themselves are permutatively equivalent. This settles a twenty-five year-old question raised by Casazza and Kalton in [13]. Solving this problem provides a new paradigm to study the uniqueness of unconditional basis in the general framework of quasi-Banach spaces. Multiple examples are given to illustrate how to put in practice this theoretical scheme. Among the main applications of this principle we obtain the uniqueness of unconditional basis up to permutation of finite sums of spaces with this property, as well as the first addition to the scant list of the known Banach spaces with a unique unconditional bases up to permutation since [14].Publication Open Access Addendum to "uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces"(Springer, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaAfter [Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces, Positivity 26 (2022), Paper no. 35] was published, we realized that Theorem 4.2 therein, when combined with work of Casazza and Kalton (Israel J. Math. 103:141-175, 1998) , solves the long-standing problem whether there exists a quasi-Banach space with a unique unconditional basis whose Banach envelope does not have a unique unconditional basis. Here we give examples to prove that the answer is positive. We also use auxiliary results in the aforementioned paper to give a negative answer to the question of Bourgain et al. (Mem Am Math Soc 54:iv+111, 1985)*Problem 1.11 whether the infinite direct sum l(1)(X) of a Banach space X has a unique unconditional basis whenever X does.