Searcharxiv⌕ Search

arXiv subjects

Félix Cabello Sánchez

Publications and source records attributed to Félix Cabello Sánchez.

At least 19 recordsLinked to original sources
arXiv2026

Interpolation of Directional Banach Spaces and Commutator Estimates

We generalize the complex interpolation formula $(X, X')_{\frac{1}{2}} = L^2$ from the context of Banach function spaces to that of directional Banach spaces. We also obtain a formula for the interpolation of matrix weighted spaces of vector valued functions via interpolation functors, and apply our formula to the particular case of interpolation of matrix weighted $L^p$ spaces by the real and complex methods. We present consequences regarding the matrix Muckenhoupt classes and commutator estimates of Calderón-Zygmund operators with matrix $BMO$ functions. We also characterize the logarithms of Muckenhoupt weights through higher order commutators.

math.FA↗
arXiv2026

The behaviour of quasi-linear maps on $C(K)$-spaces

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a $C(K)$ must be nontrivial on a subspace isomorphic to $c_0$. We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of $\ell_1$ and $c_0$.

math.FA↗
arXiv2025

Accessible operators on ultraproducts of Banach spaces

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces $X$ and $Y$, there is an "accessible" operator $X_U\to Y_U$ that is not the ultraproduct of a family of operators $X\to Y$ if and only if there is a short exact sequence of quasi-Banach spaces and operators $0\to Y\to Z\to X\to 0$ that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces $\ell_p$. The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.

math.FA↗
arXiv2024

Trimming the Johnson bonsai

We show that if $p>1$ every subspace of $\ell_p(Γ)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(Γ)$ for $0<p\leq 1$ that are not even isomorphic to any $\ell_p$-sum of separable spaces, notably the kernel of any quotient map $\ell_p(Γ)\to L_1(2^Γ)$ with $Γ$ uncountable. We involve the separable complementation property (SCP) and the separable extension property (SEP), showing that if $X$ is a Banach space of density character $\aleph_1$ with the SCP then the kernel of any quotient map $\ell_p(Γ)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP.

math.FA↗
arXiv2024

A forgotten theorem of Ortyński, not far off

We combine techniques of Ortyński \cite{orty} and Moreno and Plichko \cite{moreplic} to show that, for $0<p<\infty$, every subspace of $\ell_p(Γ)$ having density character $\mathfrak m$ has the form $\ell_p(\mathfrak m, H_m)$ for some family of subspaces $H_m\subset \ell_p$. The ideas of Ortyński are then applied to obtain unique positions for Lindenstrauss spaces.

math.FA↗
arXiv2022

Dynamics of the semigroup of contractive automorphisms of Banach spaces

Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.

math.FA↗
arXiv2021

The Twisted Hilbert Space Ideals

We study those operators on a Hilbert space that can be lifted / extended to any twisted Hilbert space. We prove that these form an ideal of operators which contains all the Schatten classes. We characterize those multiplication operators on $\ell_p$ that are liftable / extensible through centralizers.

math.FA↗
arXiv2021

An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $Δ$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(Δ, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(Δ, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(Δ, X)$ if $K$ is a metric compactum of finite covering dimension.

math.FA↗
arXiv2021

On Mazur rotations problem and its multidimensional versions

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last ten years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces Lp. The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.

math.FA↗
arXiv2021

Higher order derivatives of analytic families of Banach spaces

We show that the Rochberg spaces induced by complex interpolation form themselves complex interpolation scales, obtain the interpolated spaces and associated derivations. We present our results in the context of analytic families of Banach spaces and study the problem of determining the Rochberg spaces induced by these new families.

math.FA↗
arXiv2021

When Kalton and Peck met Fourier

The paper studies short exact sequences of Banach modules over the convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The main tool is the notion of a nonlinear $L_1$-centralizer, which in combination with the Fourier transform, is used to produce sequences of $L_1$-modules $0\rightarrow L_q \rightarrow Z \rightarrow L_p \rightarrow 0$ that are nontrivial as long as the general theory allows it, namely for $p\in (1,\infty], q\in[1,\infty)$. Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.

math.FA↗
arXiv2020

Homological dimensions of Banach spaces

The purpose of this paper is to lay the foundations for the study of the problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$, including the first example of a separable Banach space $\mathscr K$ so that $\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to compare $\Ext^2(\cdot, \cdot)$ in Banach and Quasi-Banach spaces.

math.FA↗
arXiv2020

Sailing over three problems of Koszmider

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $ω$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.

math.FA↗
arXiv2020

On the ${\Ext}^2$-problem for Hilbert spaces

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.

math.FA↗