Linear maps preserving $\ell_p$-norm parallel vectors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914889902063616 |
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| author | Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching |
| author_facet | Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching |
| contents | Two vectors $x, y$ in a normed vector space are parallel if there is a scalar $μ$ with $|μ| = 1$ such that $\|x+μy\| = \|x\| + \|y\|$; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. 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 $\ell_1(Λ)$, $c_0(Λ)$ and $\ell_\infty(Λ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_19276 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear maps preserving $\ell_p$-norm parallel vectors Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching Functional Analysis Rings and Algebras 15A86, 15A60 Two vectors $x, y$ in a normed vector space are parallel if there is a scalar $μ$ with $|μ| = 1$ such that $\|x+μy\| = \|x\| + \|y\|$; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. 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 $\ell_1(Λ)$, $c_0(Λ)$ and $\ell_\infty(Λ)$. |
| title | Linear maps preserving $\ell_p$-norm parallel vectors |
| topic | Functional Analysis Rings and Algebras 15A86, 15A60 |
| url | https://arxiv.org/abs/2407.19276 |