arXiv subjects

Fernando Albiac

Publications and source records attributed to Fernando Albiac.

At least 19 recordsLinked to original sources

arXiv2026

Structural consequences of the Schur $p$-property for Lipschitz-free $p$-spaces

Let $0<p<q\leq 1$. We show that the Schur $p$-property provides a powerful structural principle for Lipschitz-free $p$-spaces. Our main result asserts that $\mathcal{F}_p(M)$ has the Schur $p$-property for every $q$-metric space $M$; when $M$ is compact, it has the strong Schur $p$-property, with a constant depending only on $p$ and $q$. As consequences, $\mathcal{F}_p(M)$ is $\ell_p$-saturated, contains no isomorphic copy of an infinite-dimensional $r$-Banach space for $p<r\leq 1$, and every bounded operator from a $p$-Banach space into $\mathcal{F}_p(M)$ is either compact or fixes a copy of $\ell_p$. These results settle Questions 6.1, 6.2, 6.5, and 6.6 from our recent work [F. Albiac, J. L. Ansorena, J. B\'{\i}ma and M. C\'uth, Lipschitz free p-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction, J. Geom. Anal. 36 (2026), Paper No. 54] on the Schur $p$-property, together with several related problems. They also confirm a prediction of Kalton and the first-named author from 2009: no infinite-dimensional $q$-Banach space has the $p$-Lipschitz lifting property. Further applications reveal a sharp contrast between the linear and Lipschitz structures of nonlocally convex spaces.

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arXiv2026

Boundedness of the averaging projections in nonlocally convex Lorentz sequence spaces and applications to basis theory

We study the boundedness of averaging projections associated with symmetric Schauder bases in quasi-Banach spaces. Although this property is standard in the Banach setting, it is far from clear in the absence of local convexity and, indeed, fails for a broad class of quasi-Banach spaces with a symmetric basis, including $\ell_p$ for $0<p<1$. Our main result shows that, nevertheless, the canonical basis of an entire class of weighted Lorentz sequence spaces, including the spaces $\ell_{p,q}$ for $0<q<1<p<\infty$, has uniformly bounded averaging projections. Thus, bounded averaging projections do not characterize local convexity among quasi-Banach spaces with symmetric bases. As applications, we obtain new consequences for the structure of special bases. In particular, as a byproduct of our approach, we derive new examples of conditional and almost greedy bases in nonlocally convex spaces.

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arXiv2026

Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$

We show that inclusions of $p$-metric spaces always produce genuine linear embeddings at the level of Lipschitz-free $p$-spaces. More precisely, for every $0<p<1$ and every inclusion $ \mathit{N}\subset \mathit{M}$ of $p$-metric spaces, the canonical map from $ \mathit{F}_p(\mathit{N})$ into $ \mathit{F}_p( \mathit{M})$ is always an isomorphic embedding, as it plainly happens for $p=1$. Our proof relies on a versatile extension procedure for $p$-Banach-valued Lipschitz maps, allowing us to control the geometry of canonical molecules and uncover a rigidity principle governing the structure of Lipschitz free $p$-spaces. As an application, we prove that, given $0<p<q\le 1$, the natural envelope map from the Lipschitz-free $p$-space $ \mathit{F}_p( \mathit{M})$ to its $q$-Banach envelope $ \mathit{F}_q( \mathit{M})$ is one-to-one. These results give positive answers to two foundational questions that were originally raised by Kalton in [Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335], and provide tools for furthering the understanding of subspace structures, hereditary properties, and geometric invariants in Lipschitz-free $p$-spaces.

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arXiv2026

Isometric renormings for greedy bases in Banach spaces, with applications to the Haar System in $L_p[0,1]$, $1<p<\infty$

We investigate the problem of improving the greedy-type constant of a basis by means of an equivalent renorming of the ambient Banach space. Our main result shows that if a Banach space admits an unconditional and bidemocratic basis whose fundamental function satisfies certain regularity properties, then the space can be renormed so that the basis becomes isometrically greedy. The renorming simultaneously ensures lattice $1$-unconditionality, isometric bidemocracy, and allows prescribing the fundamental function up to a suitable regularization. As a principal application, we resolve a long-standing problem posed by Albiac--Wojtaszczyk in 2006 by proving that for each $1<p<\infty$ the $L_p$-normalized Haar system can be made $1$-greedy under an equivalent norm of $L_p$. Further applications include isometric greedy renormings for bases of Besov spaces, mixed-norm direct sums, and for a wide class of subsymmetric and conditional bases, including spreading models and the canonical basis of Schlumprecht space. These results show that isometric greedy renormings arise in far greater generality than previously known.

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arXiv2026

Unconditional structure of Banach spaces with few operators

This article was initially motivated by our goal to show that the Banach space $\mathbb{G}$ constructed by Gowers in [W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc. 26 (1994), no. 6, 523-530] to settle Banach's hyperplane problem has a unique unconditional basis. This uniqueness result served as a springboard to ask whether further structural insights could be derived by rigging Gowers' original construction. As it turned out, the $p$-convexification of $\mathbb{G}$ for $1< p<\infty$, $p\not=2$, provides a family of Banach spaces, each of them with a unique unconditional basis containing block bases whose spreading models are not equivalent to the unit vector basis of $\ell_1$, $\ell_2$, or $c_0$. This solves in the negative a forty-year-old open problem raised by Bourgain et al. in their 1985 \textit{Memoir}, [J. Bourgain, P. G. Casazza, J. Lindenstrauss, and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Mem. Amer. Math. Soc. 54 (1985), no. 322, iv+111] where they studied the uniqueness of unconditional structure in infinite direct sums of those three spaces with the aim to classify all Banach spaces with a unique unconditional basis. As a by-product of our work, we also disprove the conjecture in structure theory that a space having a unique unconditional basis must be isomorphic to its square, and evince that when a Banach space $\mathbb{X}$ with an unconditional basis has few operators, then the space itself and all its complemented subspaces have a unique unconditional structure.

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arXiv2025

When Greedy Approximation Breaks: Counterexamples in Quasi-Banach Spaces

We construct two counterexamples that resolve long-standing open problems on greedy approximation theory with respect to bases, posed in [F. Albiac et al., Dissertationes Math. 560 (2021)] and restated in [F. Albiac, J. L. Ansorena, V. Temlyakov, J. Approx. Theory 307 (2025)]. Our first result exhibits a quasi-Banach space $\mathbb{X}$ with an almost greedy basis which, when transported to the Banach envelope of $\mathbb{X}$, ceases to be quasi-greedy. This shows that the passage to the Banach envelope, although it preserves linear and lattice structure, may radically disrupt the performance of the thresholding greedy algorithm, to the extent that in some respects it could perform better in a quasi-Banach space than in its Banach envelope. Our second result constructs an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $\mathbb{Y}$ which fails to be a Schauder basis under any reordering. Together, these examples highlight that local convexity and the Banach envelope construction play an unexpectedly active role in shaping greedy approximation phenomena, revealing structural differences between Banach and quasi-Banach spaces that go beyond the classical theory of bases.

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arXiv2025

On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$

With the aim to better understand the intricate geometry of the class of Lipschitz free $p$-spaces $\mathcal{F}_p(\mathcal{M})$ when $0<p<1$, in this note we study their Banach envelopes and prove that if $0<p<1$ and $ \mathcal{M}$ is a metric space then the Banach envelope map of $\mathcal{F}_p(\mathcal{M})$ is one-to-one, thus solving in the positive a problem raised by Kalton in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of $ \mathcal{F}_p(\mathcal{M})$ separates the points of $\mathcal{F}_p(\mathcal{M})$.

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arXiv2025

Lipschitz free $p$-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction

The geometric analysis of non-locally convex quasi-Banach spaces presents rich and nuanced challenges. In this paper, we introduce the Schur $p$-property and the strong Schur $p$-property for $0 < p \leq 1$, providing new tools to deepen the understanding of these spaces, and the Lipschitz free $p$-spaces in particular. Moreover, by developing an adapted version of the compact reduction principle, we prove that Lipschitz free $p$-spaces over discrete metric spaces possess the approximation property, thereby answering positively a question raised by Albiac et al. in arXiv:2005.06555v2.

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arXiv2024

Isomorphisms between vector-valued $H_p$-spaces for $0<p\le 1$ and uniqueness of unconditional structure

The aim of this paper is twofold. On the one hand, we manage to identify Banach-valued Hardy spaces of analytic functions over the disc $\mathbb{D}$ with other classes of Hardy spaces, thus complementing the existing literature on the subject. On the other hand, we develop new techniques that allow us to prove that certain Hilbert-valued atomic lattices have a unique unconditional basis, up to normalization, equivalence and permutation. Combining both lines of action we show that that $H_p(\mathbb{D},\ell_2)$ for $0<p<1$ has a unique atomic lattice structure. The proof of this result relies on the validity of some new lattice estimates for non-locally convex spaces which hold an independent interest.

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arXiv2024

Conditional bases with Property~(A)

Property~(A) is a week symmetry condition that plays a fundamental role in the characterization of greedy-type bases in the isometric case, i.e., when the constants involved in the study of the efficiency of the thresholding greedy algorithm in Banach spaces are sharp. In this note we build examples of Banach spaces with Schauder bases that have Property~(A) but fail to be unconditional, thus settling a long standing problem in the area. As a by-product of our work we hone our construction to produce counterexamples that solve other open questions in the isometric theory of greedy bases.

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arXiv2024

Twenty-five years of greedy bases

Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.~V.~Konyagin and V.~N.~Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (1999), no. 3, 365--379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the foundational paper mentioned above appeared. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.

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arXiv2023

Linear versus nonlinear forms of partial unconditionality of bases

The main results in this paper contribute to bring to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory.

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arXiv2023

Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] 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 [F. Albiac and J.L. Ansorena, Characterization of $1$-almost greedy bases, Rev. Mat. Complut. 30 (2017), no. 1, 13-24]. 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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arXiv2022

The structure of greedy-type bases in Tsirelson's space and its convexifications

Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in 1974 soon to become one of the most significant counterexamples in the theory. Its structure broke the ideal pattern that analysts had conceived for a generic Banach space, thus giving rise to the era of pathological examples. Since then, many authors have contributed to the study of different aspects of this special space with an eye on better understanding its idiosyncrasies. In this paper we are concerned with the greedy-type basis structure of $\mathcal{T}$, a subject that had not been previously explored in the literature. More specifically, we show that Tsirelson's space and its convexifications $\mathcal{T}^{(p)}$ for $0<p<\infty$ have uncountably many non-equivalent greedy bases. We also investigate the conditional basis structure of spaces $\mathcal{T}^{(p)}$ in the range of $0<p<\infty$ and prove that they have uncountably many non-equivalent conditional almost greedy bases.

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arXiv2022

Elton's near unconditionality of bases as a threshold-free form of greediness

Elton's near unconditionality and quasi-greediness for largest coefficients are two properties of bases that made their appearance in functional analysis from very different areas of research. One of our aims in this note is to show that, oddly enough, they are connected to the extent that they are equivalent notions. We take advantage of this new description of the former property to further the study of the threshold function associated with near unconditionality. Finally, we made a contribution to the isometric theory of greedy bases by characterizing those bases that are $1$-quasi-greedy for largest coefficients.

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arXiv2022

Existence of almost greedy bases in mixed-norm sequence and matrix spaces, including Besov spaces

We prove that the sequence spaces $\ell_p\oplus\ell_q$ and the spaces of infinite matrices $\ell_p(\ell_q)$, $\ell_q(\ell_p)$ and $(\bigoplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which are isomorphic to certain Besov spaces, have an almost greedy basis whenever $0<p<1<q<\infty$. More precisely, we custom-build almost greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth-Kalton-Kutzarova method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101], which was originally designed for constructing almost greedy bases in Banach spaces, to make it valid for direct sums of mixed-normed spaces with nonlocally convex components. Additionally, we prove that the fundamental functions of all almost greedy bases of these spaces grow as $(m^{1/q})_{m=1}^\infty$.

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arXiv2022

Democracy of quasi-greedy bases in $p$-Banach spaces with applications to the efficiency of the TGA in the Hardy spaces $H_p(\mathbb{D}^d)$

We use new methods, specific of non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0<p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p(\mathbb{D})$ for $0<p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p(\mathbb{D}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [F. Albiac, J. L. Ansorena, and P. Wojtaszczyk, \textit{Quasi-greedy bases in $\ell_p$ ($0<p<1$) are democratic}, J. Funct. Anal. \textbf{280} (2021), no. 7, 108871, 21]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

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arXiv2022

Sparse approximation using new greedy-like bases in superreflexive spaces

This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.

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