arXiv subjects
Publications and source records attributed to Audrey Fovelle.
In their papers, Leonetti, Russo, Somaglia, Menet and Papathanasiou posed the question of whether there exists an infinite dimensional vector space of sequences in $\ell_\infty$ having (except for the zero sequence) an even amount of accumulation points. Here we answer this question in the negative, by showing that this previous set of sequences is not even $3$-lineable and, therefore, not lineable.
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.
We prove an optimal result of stability under $\ell_p$-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.
In this note, we introduce and study the notions of asymptotic B-convexity and asymptotic infratype $p$, and we prove asymptotic analogs of a series of results due to Giesy \cite{Giesy66} and Pisier \cite{Pisier74}. In particular, we give a simplified proof of an asymptotic version of Pisier's $\ell_1$-theorem that was originally proven by Causey, Draga, and Kochanek in \cite{CDK19}. We also briefly discuss the notion of asymptotic stable type.
We prove that if $Y$ is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in $\ell_1$, or a normalized sequence with upper p-estimates for some $p>1$, then $Y$ does not satisfy Lindenstrauss' property B.
In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of quasi-reflexive spaces satisfying upper $\ell_p$ tree estimates to the setting of $\ell_p$-sums of such spaces. Our result provides us with a tool for constructing the first examples of Banach spaces that are not quasi-reflexive but nevertheless admit some concentration inequality. We also give a sufficient condition for a space to be asymptotic-$c_0$ in terms of a concentration property, as well as relevant counterexamples.