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The main results in this paper contribute to bringing 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-oscillation 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., F. Albiac and J.L. Ansorena acknowledge the support of the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional analysis methods in approximation theory and applications. M. Berasategui was supported by ANPCyT PICT-2018-04104. Open Access funding provided by Universidad Pública de Navarra.

Tsirelson's space 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 T , a subject that had not been previously explored
in the literature. More specifically, we show that Tsirelson's space
and its convexifications T (p) for 0 < p < 1 have uncountably
many non-equivalent greedy bases. We also investigate the conditional
basis structure of spaces T (p) in the range of 0 < p < 1 and
prove that they have uncountably many non-equivalent conditional
almost greedy bases., The research of both authors was supported by the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional Analysis Techniques in Approximation Theory and Applications.

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 (3) (1999), 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 appearance of the aforementioned foundational paper. 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., The research of F. Albiac and J.L. Ansorena was partially funded by the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional analysis methods in approximation theory and applications. The research of V.N. Temlyakov (Section 13) was supported by the Russian Science Foundation under Grant No. 23-71-30001, https://rscf.ru/project/23-71-30001/, and performed at Lomonosov Moscow State University.

Certain algebra norms on the algebra of functions on a two-point set are defined and used to construct a new family of parametric algebra norms on the space CC(K) of continuous complex-valued functions on a compact Hausdorff space. As a by-product of our work we transfer our construction to uniform algebras to obtain a new collection of norms with special properties., The research of F. Albiac and O. Blasco was partially funded by the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional analysis methods in approximation theory and applications. F. Albiac was also supported by the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. Open Access funding provided by Universidad Pública de Navarra.

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., The research of F. Albiac, J.L. Ansorena, and Ó. Blasco was funded by the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional analysis methods in approximation theory and applications. F. Albiac also acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. The University of Illinois partially supported the work of H. V. Chu and T. Oikhberg via Campus Research Board award 23026. Open Access funding provided by Universidad Pública de Navarra.

One 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., The authors acknowledge the support of the Spanish Ministry for Science and Innovation under Grant PID2022-138342NB-I00 for Functional Analysis Techniques in Approximation Theory and Applications. Open Access funding provided by Universidad Pública de Navarra.

[EN] Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman-Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances-contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples., This research was funded by the Agencia Estatal de Investigacion under grant number PID2022-138342NB-I00. The research of the first author was funded by the Universitat Politecnica de Valencia through the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-21). The research of the other authors was also funded by the European Union's Horizon Europe research and innovation program under the Grant Agreement No. 101059609 (Re-Livestock).

[EN] Consider L 0 , the F-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin's classical result on the factorization of sublinear continuous operators from a Banach space to L 0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces., The first author was supported by the National Science Centre, Poland, Project no. 2019/33/B/ST1/00165. This work was part of the project/grant R+D&I PID2022-138342NB-I00 funded by MCIN/AEI/10.13039/501100011033/ (Spain) , which financially supported the second author.

[EN] The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens-Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry., The first author was supported by a contract of the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-21), Universitat Politecnica de Valencia. This research was funded by the Agencia Estatal de Investigacion grant number PID2022-138342NB-I00. The research was funded by the European Union's Horizon Europe research and innovation program under the Grant Agreement No. 101059609 (Re-Livestock).

[EN] Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction., The first author was supported by a contract of the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-21), Universitat Politecnica de Valencia. This publication is part of the R & D & I project PID2020-112759GB-I00 funded by MCIN/AEI /10.13039/501100011033. This publication is part of the R & D & I project PID2022-138342NB-I00 funded by MCIN/AEI /10.13039/501100011033. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.