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| Auteur principal: | |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2507.09284 |
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Table des matières:
- Two vectors $x,y$ of a Banach space are said to form a parallel (resp. triangle equality attaining or TEA) pair if $\|x+λy\|=\|x\|+\|y\|$ holds for some scalar $λ$ with $|λ|=1$ (resp. $λ=1$). For $p\in \{1,\infty\},$ and $ m,n\geq 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.