Linear maps preserving ${\ell }_p$-norm parallel vectors

Abstract

Two vectors $x,y$ in a normed vector space are parallel if there is a scalar $\mu$ with $|\mu |=1$ such that $\Vert x+\mu y\Vert =\Vert x\Vert +\Vert y\Vert$; they form a triangle equality attaining (TEA) pair if $\Vert x+y\Vert =\Vert x\Vert +\Vert y\Vert$. 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 ${\ell }_{\infty }(\Lambda )$.