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 $\Vert x+y\Vert =\Vert x\Vert +\Vert y\Vert ,$ and a parallel pair if $\Vert x+\hat{I}»y\Vert =\Vert x\Vert +\Vert y\Vert$ holds for some unimodular scalar $\hat{I}».$ 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 $\ker (T)\ne \{0\},$ then $T$ preserves parallel pairs if and only if $rank(T)\le 1$. %In particular, $T:{\ell }_p\to {\ell }_1$ preserves parallel pairs if and only if $rank(T)\le 1,$ 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 $rank(T)\le 1$ (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 $(0\ne )T:{\ell }_1\to \mathcal{Y}$ preserves TEA pairs if and only if $\hat{I}›=\{i\in \mathbb{N}:T{e}_i\ne 0\}$ 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.