Linear maps on $\mathcal{L}({\ell }_p^n,{\ell }_p^m)$, $(p\in \{1,\infty \})$ preserving parallel pairs

Abstract

Two vectors $x,y$ of a Banach space are said to form a parallel (resp. triangle equality attaining or TEA) pair if $\Vert x+\lambda y\Vert =\Vert x\Vert +\Vert y\Vert$ holds for some scalar $\lambda$ with $|\lambda |=1$ (resp. $\lambda =1$). For $p\in \{1,\infty \},$ and $m,n\ge 2,$ we study the linear maps $T:\mathcal{L}({\ell }_p^n,{\ell }_p^m)\to \mathcal{L}({\ell }_p^n,{\ell }_p^m)$ that preserve parallel (resp. TEA) pairs, that is, those linear maps $T$ for which $T(A),T(B)$ form a parallel (resp. TEA) pair whenever $A,B$ form a parallel (resp. TEA) pair of $\mathcal{L}({\ell }_p^n,{\ell }_p^m).$ We prove that if $T$ is non-zero, then the following are equivalent: (1) $T$ preserves TEA pairs. (2) $T$ preserves parallel pairs and rank$(T)>1$. (3) $T$ preserves parallel pairs and $T$ is invertible. (4) $T$ is a scalar multiple of an isometry.