Open Access
Embeddability of ℓp and bases in Lipschitz free p-spaces for 0 < p ≤ 1
(Elsevier, 2020) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our goal in this paper is to continue the study initiated by the authors in of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0 < p ≤ 1, denoted Fp(M). Here we develop new techniques to show that, by analogy with the case p = 1, the space p embeds isomorphically in Fp(M) for 0 < p < 1. Going further we see that despite the fact that, unlike the case p = 1, this embedding need not be complemented in general, complementability of p in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in Fp([0, 1]), we obtain examples of p-Banach spaces for p < 1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis.
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 Matematika
We 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.
Open Access
Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases
(Instytut Matematyczny, 2020) Albiac Alesanco, Fernando José; Ansorena, José L.; Dilworth, S. J.; Kutzarova, Denka; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non-superreflexive mixed-norm sequence space. As a by-product, we give applications to the study of conditional Schauder bases and conditional almost greedy bases in this new class of Banach spaces.
Open Access
Lipschitz free spaces isomorphic to their infinite sums and geometric applications
(American Mathematical Society, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct _1-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over Zd is isomorphic to its _1-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to l1. Moreover, following new ideas of Bruè et al. from [J. Funct. Anal. 280 (2021), pp. 108868, 21] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of p-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases p < 1 and p = 1.
Open Access
On certain subspaces of p for 0 < p ≤ 1 and their applications to conditional quasi-greedy bases in p-Banach spaces
(Springer, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We construct for each 0
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 Matemáticas
Our 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).
Open Access
Lipschitz free p-spaces for 0 < p < 1
(Springer, 2020) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
This paper initiates the study of the structure of a new class of p-Banach spaces, 0
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 Matemáticas
We 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].
Open Access
Building highly conditional almost greedy and quasi-greedy bases in Banach spaces
(Elsevier, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Dilworth, S. J.; Kutzarova, Denka; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k(m)[B](m=1)(infinity) of its conditionality constants verifies the estimate k(m)[B] = O(log m) and that if the reverse inequality log m =O(k(m)[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k(m)[B] =O(log m)(1-epsilon) for some epsilon > 0. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k(m)[B] = O(log m) and superreflexiye classical Banach spaces having for every epsilon > 0 quasi-greedy bases B with k(m)[B] = O(log m)(1-epsilon). Moreover, in most cases those bases will be almost greedy.
Open Access
New parameters and Lebesgue-type estimates in greedy approximation
(Cambridge University Press, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Berná, Pablo M.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The purpose of this paper is to quantify the size of the Lebesgue constants (𝑳𝑚)∞ 𝑚=1 associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters (𝒌𝑚)∞ 𝑚=1 determines the growth of (𝑳𝑚)∞ 𝑚=1. Multiple theoretical applications and computational examples complement our study.