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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.09284 |
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| _version_ | 1866909688055988224 |
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| author | Mal, Arpita |
| author_facet | Mal, Arpita |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear maps on $\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ preserving parallel pairs Mal, Arpita Functional Analysis 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. |
| title | Linear maps on $\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ preserving parallel pairs |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.09284 |