An algebraic characterization of linearity for additive maps preserving orthogonality
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913748214611968 |
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| author | Li, Lei Liu, Siyu Peralta, Antonio M. |
| author_facet | Li, Lei Liu, Siyu Peralta, Antonio M. |
| contents | We study when an additive mapping preserving orthogonality between two complex inner product spaces is automatically complex-linear or conjugate-linear. Concretely, let $H$ and $K$ be complex inner product spaces with dim$(H)\geq 2$, and let $A: H\to K$ be an additive map preserving orthogonality. We obtain that $A$ is zero or a positive scalar multiple of a real-linear isometry from $H$ into $K$. We further prove that the following statements are equivalent:
$(a)$ $A$ is complex-linear or conjugate-linear.
$(b)$ For every $z\in H$ we have $A(i z) \in \{\pm i A(z)\}$.
$(c)$ There exists a non-zero point $z\in H$ such that $A(i z) \in \{\pm i A(z)\}$.
$(d)$ There exists a non-zero point $z\in H$ such that $i A(z) \in A(H)$.
The mapping $A$ neither is complex-linear nor conjugate-linear if, and only if, there exists a non-zero $x\in H$ such that $i A(x)\notin A(H)$ (equivalently, for every non-zero $x\in H$, $i A(x)\notin A(H)$).
Among the consequences we show that, under the hypothesis above, the mapping $A$ is automatically complex-linear or conjugate-linear if $A$ has dense range, or if $H$ and $K$ are finite dimensional with dim$(K)< 2\hbox{dim}(H)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An algebraic characterization of linearity for additive maps preserving orthogonality Li, Lei Liu, Siyu Peralta, Antonio M. Functional Analysis We study when an additive mapping preserving orthogonality between two complex inner product spaces is automatically complex-linear or conjugate-linear. Concretely, let $H$ and $K$ be complex inner product spaces with dim$(H)\geq 2$, and let $A: H\to K$ be an additive map preserving orthogonality. We obtain that $A$ is zero or a positive scalar multiple of a real-linear isometry from $H$ into $K$. We further prove that the following statements are equivalent: $(a)$ $A$ is complex-linear or conjugate-linear. $(b)$ For every $z\in H$ we have $A(i z) \in \{\pm i A(z)\}$. $(c)$ There exists a non-zero point $z\in H$ such that $A(i z) \in \{\pm i A(z)\}$. $(d)$ There exists a non-zero point $z\in H$ such that $i A(z) \in A(H)$. The mapping $A$ neither is complex-linear nor conjugate-linear if, and only if, there exists a non-zero $x\in H$ such that $i A(x)\notin A(H)$ (equivalently, for every non-zero $x\in H$, $i A(x)\notin A(H)$). Among the consequences we show that, under the hypothesis above, the mapping $A$ is automatically complex-linear or conjugate-linear if $A$ has dense range, or if $H$ and $K$ are finite dimensional with dim$(K)< 2\hbox{dim}(H)$. |
| title | An algebraic characterization of linearity for additive maps preserving orthogonality |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2503.16341 |