Linear maps preserving $\ell_p$-norm parallel vectors

TL;DR

This paper characterizes linear maps on finite and infinite-dimensional spaces that preserve vectors with specific norm-related parallelism properties, revealing different structures depending on the $ ext{p}$-norm used.

Contribution

It provides a complete description of linear maps preserving parallel and TEA pairs in $ ext{l}_p$-normed spaces for all $p$, including special cases like $ ext{l}_1$ and $ ext{l}_$.

Findings

Linear maps preserve parallel and TEA pairs for $1<p<$.

TEA preservers in $ ext{l}_1$ form a semigroup with specific structure.

Characterization of preservers in $ ext{l}_$ spaces, including the exceptional case.

Abstract

Two vectors $x, y$ in a normed vector space are parallel if there is a scalar $\mu$ with $|\mu| = 1$ such that $\|x+\mu y\| = \|x\| + \|y\|$; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. In this paper, we characterize linear maps on $F^n=R^n$ or $C^n$, equipped with the $\ell_p$-norm for $p \in [1, \infty]$, preserving parallel pairs or preserving TEA pairs. Indeed, any linear map will preserve parallel pairs and TEA pairs when $1< p <\infty$. For the $\ell_1$-norm, TEA preservers form a semigroup of matrices in which each row has at most one nonzero entries; adding rank one matrices to this semigroup will be the semigroup of parallel preserves. For the $\ell_\infty$-norm, a nonzero TEA preserver, or a parallel preserver of rank greater than one, is always a multiple of an $\ell_\infty$-norm isometry, except when $F^n = R^2$. We also have a characterization for the exceptional case. The results are extended to linear maps of the infinite dimensional spaces $\ell_1(\Lambda)$, $c_0(\Lambda)$ and $\ell_\infty(\Lambda)$.