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 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 Uniqueness of unconditional basis of ℓ2⊕T(2)(American Mathematical Society, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaWe provide a new extension of Pitt’s theorem for compact operators between quasi-Banach lattices which permits to describe unconditional bases of finite direct sums of Banach spaces X1 · · · Xn as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each Xi has a unique unconditional basis (up to equivalence and permutation), then X1 · · · Xn has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space ℓ2⊕T(2) has a unique unconditional basis.Publication Open Access Democracy of quasi-greedy bases in p-Banach spaces with applications to the efficiency of the thresholding greedy algorithm in the hardy spaces Hp(Dd)(Cambridge University Press, 2023) Albiac Alesanco, Fernando José; Ansorena, José L.; Bello, Glenier; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a -Banach space for 0 < p < p are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy Hp(D) space for 0 < p < 1 are democratic while, in contrast, no quasi-greedy basis of Hp(Dd) for d > 2 is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.Publication Open Access The Tsirelson space tau (p) has a unique unconditional basis up to permutation for 0 < p < 1(Hindawi Publishing Corporation, 2009) Albiac Alesanco, Fernando José; Leránoz Istúriz, María Camino; Matemáticas; MatematikaWe show that the p-convexified Tsirelson space tau((p)) for 0 < p < 1 and all its complemented subspaces with unconditional basis have unique unconditional basis up to permutation. The techniques involved in the proof are different from the methods that have been used in all the other uniqueness results in the nonlocally convex setting. Copyright (C) 2009 F. Albiac and C. Leranoz.