arXiv subjects

Helena del Río

Publications and source records attributed to Helena del Río.

3 recordsLinked to original sources

arXiv2026

On the Bollobás theorem for complex $C(K)$-spaces

Let $K$ and $S$ be compact Hausdorff spaces. We prove a Bollobás-type theorem for operators between complex spaces of continuous functions. More precisely, every operator that almost attains its norm at an initial function can be approximated by a norm-attaining operator whose norm-attaining function remains close to the original one. Our proof combines the iterative method developed previously in the real case with the recent construction for complex measures. In particular, our result provides a quantitative strengthening of the complex version of the Johnson-Wolfe density theorem. As a byproduct of our construction, the same conclusion can be obtained when considering only compact operators.

math.FA↗

arXiv2026

A group action approach to the Daugavet property

We introduce the $G$-Daugavet property ($G$-DPr, for short) for Banach spaces endowed with an action of a group $G$ by surjective linear isometries. This notion provides a common framework for the classical Daugavet property and the alternative Daugavet property, which correspond respectively to the trivial action and to the scalar action of $S_{\mathbb{K}}$. We establish several characterizations of the $G$-DPr in terms of $G$-slices and closed convex $G$-invariant hulls, recovering the usual slice descriptions of the DPr and the aDPr as particular cases. We show that the presence of a group action leads to new behavior in Daugavet theory. In particular, the $G$-DPr may hold on classical reflexive spaces in sharp contrast with the classical Daugavet property. We relate this phenomenon to convex transitivity, almost transitivity and finite-dimensional rotation problems. We also prove group-action versions of the classical characterizations for $L^1(μ, X)$- and $C(K,X)$-spaces. The paper also studies group separable determination, $G$-versions of numerical radius and numerical index, and connections between the $G$-DPr and strong Radon-Nikodým and SCD operators. Finally, we introduce a parameter which measures how far the $G$-DPr is from the classical DPr in a quantitative manner. As a consequence of these results, we obtain conditions under which the $G$-DPr recovers several classical implications, including the failure of the RNP for both $X$ and $X^*$, the presence of copies of $\ell_1$ and the failure of the unit ball to be an SCD set.

math.FA↗

arXiv2025

Range strongly exposing operators between Banach spaces

We introduce a new class of bounded linear operators, called range strongly exposing (RSE) operators, which form a natural intermediate class: weaker than Bourgain's absolutely strongly exposing operators, yet stronger than both uniquely quasi norm-attaining and classical norm-attaining operators. Several foundational results on norm-attaining operators are extended to the RSE setting. Among our main contributions, we establish that for every infinite-dimensional Banach space $Y$, there exists a Banach space $X$ such that the RSE operators from $X$ to $Y$ are not dense - an RSE analogue of a result by Acosta (1999) which applies only when $Y$ is strictly convex. We also show that the Radon-Nikodým property of $Y$ is sufficient to obtain that RSE operators from $L_1(μ)$ to $Y$ are dense and that this is also necessary if $μ$ is not purely atomic. This extends and sharpens classical results by Uhl (1976). As a consequence, we prove that the set of RSE operators between $L_1(μ)$ and $L_1 (ν)$ is dense if and only if at least one of the measures $μ$ or $ν$ is purely atomic, in contrast with the classical result by Iwanik (1979) which guarantees the denseness of norm-attaining operators for all measures $μ$ and $ν$. We also prove that weakly compact operators from any $C(K)$ space can always be approximated by (weakly compact) RSE operators, thereby strengthening a result of Schachermayer (1983). Additionally, we present several improvements of more recent results concerning finite-rank operators and $Γ$-flat operators which give, in particular, RSE versions of classical results on compact operators by Johnson-Wolfe (1979). Finally, we discuss RSE counterparts of results by Zizler and Lindenstrauss on the denseness of operators whose adjoints attain their norm.

math.FA↗