arXiv subjects

B. Randrianantoanina

Publications and source records attributed to B. Randrianantoanina.

5 recordsLinked to original sources

arXiv2026

Nonlinear type and metric embeddings of lamplighter spaces

We prove that for all metric spaces $X$ the following properties of the lamplighter space $\mathsf{La}(X)$ are equivalent: (1) every snowflake of $\mathsf{La}(X)$ admits a biLipschitz embedding into a finite product of $\mathbb{R}$-trees, (2) every snowflake of $\mathsf{La}(X)$ admits a biLipschitz embedding into a Hilbert space, (3) $\mathsf{La}(X)$ has finite Nagata dimension, (4) $\mathsf{La}(X)$ has Markov type 2, (5) $\mathsf{La}(X)$ has nontrivial Enflo type, (6) $\mathsf{La}(X)$ does not contain the Hamming cubes with uniform distortion. We characterize metric spaces $X$ for which $\mathsf{La}(X)$ satisfies properties (1)-(6) as those that are ``TSP-efficient" - a new condition that we introduce - which roughly means that the traveling salesman problem in $X$ can be solved as ``efficiently" as the traveling salesman problem in $\mathbb{R}$, up to a constant multiplicative factor. We also prove that if such metric spaces $X$ admit a biLipschitz embedding into $\mathbb{R}^n$, then $\mathsf{La}(X)$ admits a biLipschitz embedding into a finite product of $\mathbb{R}$-trees, and therefore into every nonsuperreflexive Banach space. Finally, we give a full characterization of metric spaces $X$ such that the lamplighter space $\mathsf{La}(X)$ biLipschitz embeds into a Hilbert space.

math.FA↗

arXiv2015

On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces

We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions. Further, we prove that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than $\ell_2$ has at least a countable number of such renormings. On the other hand we show that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of $\ell_p$.

math.FA↗

arXiv2012

Narrow and $\ell_2$-strictly singular operators from $L_p$

In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and function Lebesgue spaces $X$ with the property that every operator $T:L_p \to X$ is narrow. Next, using similar methods we prove that every $\ell_2$-strictly singular operator from $L_p$, $1<p<\infty$, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator $T$ on $L_1[0,1]$ satisfies the assumption that for each measurable set $A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic embedding, then $T$ is narrow. (Here $L_1(A) = \{x \in L_1: {\rm supp} \, x \subseteq A\}$.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being $\ell_2$-strictly singular, for operators on $L_p[0,1]$, $1<p<2$, to be narrow. We define a notion of a "gentle" growth of a function and we prove that for $1 < p < 2$ every operator $T$ on $L_p$ which, for every $A\subseteq[0,1]$, sends a function of "gentle" growth supported on $A$ to a function of arbitrarily small norm is narrow.

math.FA↗

arXiv2012

On Enflo and narrow operators acting on $L_p$

The first part of the paper is inspired by a theorem of H. Rosenthal, that if an operator on $L_1[0,1]$ satisfies the assumption that for each measurable set $A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic embedding, then the operator is narrow. (Here $L_1(A) = \bigl\{x \in L_1: \,\, {\rm supp} \, x \subseteq A \bigr\}$.) This leads to a natural question of finding mildest possible assumptions for operators on a given space $X$, which will imply that the operator is narrow. We find a partial answer to this question for operators on $L_p(0,1)$ with $1<p<2$. Namely we define a notion of a "gentle" growth of a function and we prove that for $1 < p < 2$ every operator $T$ on $L_p$ which is unbounded from below on $L_p(A)$, $A \subseteq [0,1]$, by means of function having a "gentle" growth, is narrow. In the second part of the paper we consider the question for what Banach spaces $X$, every operator $T:L_p \lra X$ is narrow. We prove that for $2 < p, r < \infty$ every operator $T: L_p\rightarrow\ell_r$ is narrow, which completes the list of results for operators from $L_p$ to sequence and function Lebesgue spaces.

math.FA↗

arXiv2008

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \fpp if there exists $ε>0$ such that, for every infinite subset $A$ of the unit sphere of the dual space, $A\cup (-A)$ fails to be $(2-ε)$-separated. In particular, $E$-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.

math.FA↗