Linear preservers of parallel/TEA vectors in Lp(μ) spaces

Abstract

In normed vector spaces, two vectors $x,y$ are parallel (resp., triangle equality attaining (TEA)) if there is a scalar ξ with $\left|\xi \right|=1$ (resp., $\xi =1$) such that $\Vert x+\xi y\Vert =\Vert x\Vert +\Vert y\Vert$. This paper characterizes linear maps preserving these pairs in $L_1\left(\mu \right)$ and $L_{\infty }\left(\mu \right)$ spaces, where non-strict convexity enables rich geometric structures absent in $L_p$ spaces, with $p\in (0,1)\cup (1,\infty )$ (for which all linear maps trivially preserve such pairs). We first resolve finite-dimensional cases: ${\ell }_1$-norm TEA pair preservers are matrices with at most one nonzero entry per row. For ${\ell }_{\infty }$, TEA pair preservers are scalar multiples of isometries, except in $R^2$. These results extend to infinite dimensional spaces ${\ell }_1\left(\Lambda \right)$, $c_0\left(\Lambda \right)$, and ${\ell }_{\infty }\left(\Lambda \right)$, where TEA pair preservers are generalized permutation operators (for ${\ell }_1$) or scalar multiples of isometries (for $c_0$ and ${\ell }_{\infty }$). In all cases, parallel pair preservers are either TEA pair preservers or rank one maps. Crucially, we generalize to measure-theoretic settings. For $L_1\left(\mu \right)$, TEA pair preservers are automatically bounded and preserves disjointness; in many interesting cases, they are weighted compositions. Parallel pair preservers combine these with rank-one maps. For $L_{\infty }\left(\mu \right)$, bijective preservers are scalar isometries, establishing a dichotomy: $L_1$ preservers reflect sparsity, while $L_{\infty }$ preservers align with isometric symmetries. These results unify finite-dimensional, sequence-space, and general $L_p$ settings, advancing the classification of structure-preserving operators in Banach spaces.