Linear maps in $\mathcal{L}(\ell_{\MakeLowercase{p}},\mathcal{Y}) $ preserving parallel and TEA pairs

Abstract

A pair $(x,y)$ of vectors in a Banach space $\mathcal{Y}$ is said to be a triangle equality attaining (or TEA) pair if $\|x+ y\|=\|x\|+\|y\|,$ and a parallel pair if $\|x+λy\|=\|x\|+\|y\|$ holds for some unimodular scalar $λ.$ In this article, we explore bounded linear maps $T:\ell_p\to \mathcal{Y}$ preserving parallel and TEA pairs. For $p\in(1,\infty)$ all linear maps trivially preserve parallel pairs. We prove that for $p=\infty,$ if $\ker(T)\neq \{0\},$ then $T$ preserves parallel pairs if and only if $rank(T)\leq 1$. %In particular, $T:\ell_p\to \ell_1$ preserves parallel pairs if and only if $rank(T)\leq 1,$ and TEA pairs if and only if $T=0$. In particular, $T:\ell_\infty\to \ell_1$ preserves parallel (resp. TEA) pairs if and only if $rank(T)\leq 1$ (resp. $T=0$). Analogous characterizations hold if $T$ is defined from $\ell_\infty^n,$ except when $n=2$ and the field is real. In this specific setting, we further characterize such maps $T:\ell_\infty^2\to \ell_1.$ \\ Focusing on $p=1$, we establish a necessary condition for the preservation of parallel pairs. Specifically, we characterize invertible parallel pair preservers $T:\ell_1^n\to\ell_\infty^n$, as well as the general class of such maps $T:\ell_1^2 \to \ell_\infty^m,$ revealing the intricate structure inherent to these mappings. Furthermore, we prove that $(0\neq)~T:\ell_1\to \mathcal{Y}$ preserves TEA pairs if and only if $Λ=\{i\in \mathbb{N}:Te_i\neq 0\}$ is singleton, where $\mathcal{Y}$ is either strictly convex or $\ell_\infty^m$ over the complex field. Finally we characterize the TEA pair preservers $T:\ell_1^2\to \ell_\infty^m$ over the real field.