arXiv subjects

Antonio J. Guirao

Publications and source records attributed to Antonio J. Guirao.

6 recordsLinked to original sources

arXiv2023

On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ containing $c_0$ is an $r$-Lipschitz retract for $r < 2$, which implies in particular that $\mathcal{F}(\ell_\infty)$ is not $r$-Plichko for $r < 2$.

math.FA↗

arXiv2017

On the preserved extremal structure of Lipschitz-free spaces

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points.

math.FA↗

arXiv2017

$Γ$-flatness and Bishop-Phelps-Bollobás type theorems for operators

The Bishop-Phelps-Bollobás property deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T$ nearly attains its norm by an operator $T_0$ and a vector $x_0$, respectively, such that $T_0$ attains its norm at $x_0$. In this note we extend the already known results about {the} Bishop-Phelps-Bollobás property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: $Γ$-flat operators and Banach spaces with ACK$_ρ$ structure. In particular, we prove a general BPB-type theorem for $Γ$-flat operators acting to a space with ACK$_ρ$ structure and show that uniform algebras and spaces with the property $β$ have ACK$_ρ$ structure. We also study the stability of the ACK$_ρ$ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces $Y$ such that the Bishop-Phelps-Bollobás property for Asplund operators is valid for all pairs of the form ($X,Y$).

math.FA↗

arXiv2015

Weakly compactly generated Banach lattices

We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice $X$, the existence of a weakly compact set $K \subset X$ such that $X$ coincides with the band generated by $K$, implies that $X$ is WCG.

math.FA↗

arXiv2013

On the Bishop-Phelps-Bollobas property for numerical radius in C(K)-spaces

We study the Bishop-Phelps-Bollobas property for numerical radius within the framework of C(K) spaces. We present several sufficient conditions on a compact space K ensuring that C(K) has the Bishop-Phelps-Bollobas property for numerical radius. In particular, we show that C(K) has such property whenever K is metrizable.

math.FA↗

arXiv2013

The Bishop-Phelps-Bollobás property for numerical radius in $\ell_1(\mathbb{C})$

We show that the set of bounded linear operators from $X$ to $X$ admits a Bishop-Phelps-Bollobás type theorem for numerical radius whenever $X$ is $\ell_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollobás theorem for $\ell_1(\mathbb{C})$.

math.FA↗