arXiv subjects

José Rodríguez

Publications and source records attributed to José Rodríguez.

At least 19 recordsLinked to original sources

arXiv2026

On weak convergence in Köthe-Bochner function spaces

Let $E$ be an order continuous Köthe function space over a non purely atomic probability measure $μ$ and let $X$ be a Banach space, with topological duals $E^*$ and $X^*$, respectively. Let $E(X)$ and $E^*(X^*)$ be the corresponding Köthe-Bochner function spaces and consider $E^*(X^*)$ as a subspace of $E(X)^*$. We prove that if $X^*$ fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence $(f_n)$ in $E(X)$ such that $\langle φ,f_n\rangle \to 0$ for every $φ\in E^*(X^*)$; in particular, the closed unit ball of $E^*(X^*)$ is not a James boundary for $E(X)$. This extends a result by B. Cascales and A.J. Pallarés [Collect. Math. 45 (1994), 263--270] on the case $E=L_1(μ)$ and allows us to answer a question posed recently by S. Dwivedi [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM 120 (2026), 71].

math.FA↗

arXiv2026

Seat Arrangement Problems under B-utility and W-utility

In the Seat Arrangement problem the goal is to allocate agents to vertices in a graph such that the resulting arrangement is optimal or fair in some way. Examples include an arrangement that maximises utility or one where no agent envies another. We introduce two new ways of calculating the utility that each agent derives from a given arrangement, one in which agents care only about their most preferred neighbour under a given arrangement, and another in which they only care about their least preferred neighbour. We also present a new restriction on agent's preferences, namely 1-dimensional preferences. We give algorithms, hardness results, and impossibility results for these new types of utilities and agents' preferences. Additionally, we refine previous complexity results, by showing that they hold in more restricted settings.

cs.DS↗

arXiv2026

Course Allocation with Credits via Stable Matching

In the {\sc Course Allocation} problem, there are a set of students and a set of courses at a given university. University courses may have different numbers of credits, typically related to different numbers of learning hours, and there may be other constraints such as courses running concurrently. Our goal is to allocate the students to the courses such that the resulting matching is stable, which means that no student and course(s) have an incentive to break away from the matching and become assigned to one another. We study several definitions of stability and for each we give a mixture of polynomial-time algorithms and hardness results for problems involving verifying the stability of a matching, finding a stable matching or determining that none exists, and finding a maximum size stable matching. We also study variants of the problem with master lists of students, and lower quotas on the number of students allocated to a course, establishing additional complexity results in these settings.

cs.DS↗

arXiv2025

A Diestel-Faires type result for multimeasures

Let $X$ be a real Banach space and let $Y \subseteq X^*$ be a linear subspace having the Orlicz-Thomas property, that is, for each $σ$-algebra $Σ$ and for each map $ν:Σ\to X$, the countable additivity of the composition $x^*\circ ν$ for all $x^*\in Y$ implies the countable additivity of $ν$. We show that the Orlicz-Thomas property allows to test countable additivity of set-valued maps. Namely, if $M$ is a map defined on a $σ$-algebra $Σ$ whose values are convex, $σ(X,Y)$-compact, bounded non-empty subsets of $X$, then the following statements are equivalent: (i) $M$ is a strong multimeasure, that is, for every disjoint sequence $(A_n)_{n}$ in $Σ$ the series of sets $\sum_n M(A_n)$ is unconditionally convergent and the equality $M(\bigcup_n A_n)=\sum_n M(A_n)$ holds. (ii) $M$ is a multimeasure, that is, for every $x^*\in X^*$ the support map $s(x^*,M):Σ\to \mathbb{R}$ defined by $s(x^*,M)(A):=\sup \{x^*(x):x\in M(A)\}$ is countably additive. (iii) $s(x^*,M)$ is countably additive for every $x^*\in Y$. As an application, we give a result on the factorization of multimeasures through reflexive Banach spaces.

math.FA↗

arXiv2024

A note on summability in Banach spaces

Let $Z$ and $X$ be Banach spaces. Suppose that $X$ is Asplund. Let $\mathcal{M}$ be a bounded set of operators from $Z$ to $X$ with the following property: a bounded sequence $(z_n)_{n\in \mathbb{N}}$ in $Z$ is weakly null if, for each $M \in \mathcal{M}$, the sequence $(M(z_n))_{n\in \mathbb{N}}$ is weakly null. Let $(z_n)_{n\in \mathbb{N}}$ be a sequence in $Z$ such that: (a) for each $n\in \mathbb{N}$, the set $\{M(z_n):M\in \mathcal{M}\}$ is relatively norm compact; (b) for each sequence $(M_n)_{n\in \mathbb{N}}$ in $\mathcal{M}$, the series $\sum_{n=1}^\infty M_n(z_n)$ is weakly unconditionally Cauchy. We prove that if $T\in \mathcal{M}$ is Dunford-Pettis and $\inf_{n\in \mathbb{N}}\|T(z_n)\|\|z_n\|^{-1}>0$, then the series $\sum_{n=1}^\infty T(z_n)$ is absolutely convergent. As an application, we provide another proof of the fact that a countably additive vector measure taking values in an Asplund Banach space has finite variation whenever its integration operator is Dunford-Pettis.

math.FA↗

arXiv2024

On vector measures with values in $c_0(κ)$

Let $ν$ be a vector measure defined on a $σ$-algebra $Σ$ and taking values in a Banach space. We prove that if $ν$ is homogeneous and $L_1(ν)$ is non-separable, then there is a vector measure $\tildeν:Σ\to c_0(κ)$ such that $L_1(ν)=L_1(\tildeν)$ with equal norms, where $κ$ is the density character of $L_1(ν)$. This is a non-separable version of a result of [G.P. Curbera, Pacific J. Math. 162 (1994), no. 2, 287--303].

math.FA↗

arXiv2024

Dunford-Pettis type properties in $L_1$ of a vector measure

Let $ν$ be a countably additive vector measure defined on a $σ$-algebra and taking values in a Banach space. In this paper we deal with the following three properties for the Banach lattice $L_1(ν)$ of all $ν$-integrable real-valued functions: the Dunford-Pettis property, the positive Schur property and being lattice-isomorphic to an AL-space. We give new results and we also provide alternative proofs of some already known ones.

math.FA↗

arXiv2023

Weak precompactness in projective tensor products

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the projective tensor product $X \widehat{\otimes}_πY$, then either $W_1$ or $W_2$ is relatively norm compact. For instance, such a property holds for the pair $(\ell_p,\ell_q)$ if $1<p,q<\infty$ satisfy $1/p+1/q\geq 1$. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of $\ell_1$ into $X \widehat{\otimes}_πY$ for arbitrary Banach spaces $X$ and $Y$, in connection with the compactness of all operators from $X$ to $Y^*$.

math.FA↗

arXiv2023

On weakly almost square Banach spaces

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Ω,Σ)$ be a measurable space, let $E$ be a Banach lattice and let $ν:Σ\to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(ν)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue-Bochner space $L_1(μ,Y)$ is weakly almost square for any Banach space~$Y$ and for any non-atomic finite measure~$μ$. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space which fails the diameter two property. In this line we prove that if $X$ is any Banach space containing a complemented isomorphic copy of~$c_0$, then for every $0<\varepsilon<1$ there exists an equivalent norm $|\cdot|$ on~$X$ satisfying: (i)~every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter~$2$; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; and (iii)~$(X,|\cdot|)$ is $(r,s)$-SQ for all $0<r,s < \frac{1-\varepsilon}{1+\varepsilon}$.

math.FA↗

arXiv2022

On weak compactness in projective tensor products

We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower $p$-estimate (resp., $q$-estimate). If $X$ and $Y$ are strongly weakly compactly generated, then so is its projective tensor product $X \widehat{\otimes}_πY$.

math.FA↗

arXiv2022

Topological properties in tensor products of Banach spaces

Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $Y^*$ is compact; (iii) $\ell_p(Γ)\widehat{\otimes}_π\ell_q(Γ)$ is subspace of a Hilbert generated space for any $1< p,q<\infty$ such that $1/p+1/q<1$ and for any infinite set $Γ$. We also pay attention to the injective tensor product $X\widehat{\otimes}_\varepsilon Y$. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if $K$ is a compact space such that $C(K\times K)$ has property (C), then all regular Borel probability measures on $K$ have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the $w^*$-angelicity of the dual unit ball under injective tensor products.

math.FA↗

arXiv2021

On control measures of multimeasures

Let $M$ be a multimeasure defined on a $σ$-algebra and taking values in the family of bounded non-empty subsets of a Banach space $X$. We prove that $M$ admits a control measure whenever $X$ contains no subspace isomorphic to $c_0(ω_1)$. The additional assumption on $X$ is shown to be essential.

math.FA↗

arXiv2021

$\varepsilon$-weakly precompact sets in Banach spaces

A bounded subset $M$ of a Banach space $X$ is said to be $\varepsilon$-weakly precompact, for a given $\varepsilon\geq 0$, if every sequence $(x_n)_{n\in \mathbb{N}}$ in $M$ admits a subsequence $(x_{n_k})_{k\in \mathbb{N}}$ such that $$ \limsup_{k\to \infty}x^*(x_{n_k})-\liminf_{k\to\infty}x^*(x_{n_k}) \leq \varepsilon $$ for all $x^*\in B_{X^*}$. In this paper we discuss several aspects of $\varepsilon$-weakly precompact sets. On the one hand, we give quantitative versions of the following known results: (a) the absolutely convex hull of a weakly precompact set is weakly precompact (Stegall), and (b) for any probability measure $μ$, the set of all Bochner $μ$-integrable functions taking values in a weakly precompact subset of $X$ is weakly precompact in $L_1(μ,X)$ (Bourgain, Maurey, Pisier). On the other hand, we introduce a relative of a Banach space property considered by Kampoukos and Mercourakis when studying subspaces of strongly weakly compactly generated spaces. We say that a Banach space $X$ has property $\mathfrak{KM}_w$ if there is a family $\{M_{n,p}:n,p\in \mathbb{N}\}$ of subsets of $X$ such that: (i) $M_{n,p}$ is $\frac{1}{p}$-weakly precompact for all $n,p\in \mathbb{N}$, and (ii) for each weakly precompact set $C \subseteq X$ and for each $p\in \mathbb{N}$ there is $n\in \mathbb{N}$ such that $C \subseteq M_{n,p}$. All subspaces of strongly weakly precompactly generated spaces have property $\mathfrak{KM}_w$. Among other things, we study the three-space problem and the stability under unconditional sums of property $\mathfrak{KM}_w$.

math.FA↗

arXiv2020

A Godefroy-Kalton principle for free Banach lattices

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting property if every lattice homomorphism to $X$ having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a $1$-unconditional basis as well as free Banach lattices. The case of $C(K)$ spaces will also be analyzed.

math.FA↗

arXiv2020

Isometric factorization of vector measures and applications to spaces of integrable functions

Let $X$ be a Banach space, $Σ$ be a $σ$-algebra, and $m:Σ\to X$ be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pelczýnski factorization procedure that there exist a reflexive Banach space $Y$, a vector measure $\tilde{m}:Σ\to Y$ and an injective operator $J:Y \to X$ such that $m$ factors as $m=J\circ \tilde{m}$. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pelczýnski factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on $L_1(m)$ is weakly compact, then $L_1(m)$ is equal, up to equivalence of norms, to some $L_1(\tilde m)$ where $Y$ is reflexive; here we prove that the above equality can be taken to be isometric.

math.FA↗

arXiv2020

Almost everywhere convergent sequences of weak$^*$-to-norm continuous operators

Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $μ$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $μ$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $λ$-bounded approximation property for some $λ\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq λ\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the approximation property (resp., $λ$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.

math.FA↗

arXiv2019

On the range of a vector measure

Let $(Ω,Σ,μ)$ be a finite measure space, $Z$ be a Banach space and $ν:Σ\to Z^*$ be a countably additive $μ$-continuous vector measure. Let $X \subseteq Z^*$ be a norm-closed subspace which is norming for $Z$. Write $σ(Z,X)$ (resp. $μ(X,Z)$) to denote the weak (resp. Mackey) topology on $Z$ (resp. $X$) associated to the dual pair $\langle X,Z\rangle$. Suppose that, either $(Z,σ(Z,X))$ has the Mazur property, or $(B_{X^*},w^*)$ is convex block compact and $(X,μ(X,Z))$ is complete. We prove that the range of $ν$ is contained in $X$ if, for each $A\in Σ$ with $μ(A)>0$, the $w^*$-closed convex hull of $\{\frac{ν(B)}{μ(B)}: \, B\in Σ, \, B \subseteq A, \, μ(B)>0\}$ intersects $X$. This extends results obtained by Freniche [Proc. Amer. Math. Soc. 107 (1989), no. 1, 119--124] when $Z=X^*$.

math.FA↗

arXiv2018

Cesàro convergent sequences in the Mackey topology

A Banach space $X$ is said to have property ($μ^s$) if every weak$^*$-null sequence in $X^*$ admits a subsequence such that all of its subsequences are Cesàro convergent to $0$ with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property $(μ^s)$ holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property $(μ^s)$ under $\ell^p$-sums is discussed. For a family $\mathcal{A}$ of relatively weakly compact subsets of $X$, we consider the weaker property $(μ_\mathcal{A}^s)$ which only requires uniform convergence on the elements of $\mathcal{A}$, and we give some applications to Banach lattices and Lebesgue-Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property $(μ_\mathcal{A}^s)$ for the family of all $L$-weakly compact sets. This sharpens a result of de Pagter, Dodds and Sukochev. On the other hand, we prove that $L^1(ν,X)$ (for a finite measure $ν$) has property $(μ_\mathcal{A}^s)$ for the family of all $δ\mathcal{S}$-sets whenever $X$ is a subspace of a strongly super weakly compactly generated space.

math.FA↗