Additive mappings preserving orthogonality between complex inner product spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910648120639488 |
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| author | Li, Lei Liu, Siyu Peralta, Antonio M. |
| author_facet | Li, Lei Liu, Siyu Peralta, Antonio M. |
| contents | Let $H$ and $K$ be two complex inner product spaces with dim$(X)\geq 2$. We prove that for each non-zero additive mapping $A:H \to K$ with dense image the following statements are equivalent:
$(a)$ $A$ is (complex) linear or conjugate-linear mapping and there exists $γ>0$ such that $\| A (x) \| = γ\|x\|$, for all $x\in X$, that is, $A$ is a positive scalar multiple of a linear or a conjugate-linear isometry;
$(b)$ There exists $γ_1 >0$ such that one of the next properties holds for all $x,y \in H$:
$(b.1)$ $\langle A(x) |A(y)\rangle = γ_1 \langle x|y\rangle,$
$(b.2)$ $\langle A(x) |A(y)\rangle = γ_1 \langle y|x \rangle;$
$(c)$ $A$ is linear or conjugate-linear and preserves orthogonality in both directions;
$(d)$ $A$ is linear or conjugate-linear and preserves orthogonality;
$(e)$ $A$ is additive and preserves orthogonality in both directions;
$(f)$ $A$ is additive and preserves orthogonality.
This extends to the complex setting a recent generalization of the Koldobsky--Blanco--Turnšek theorem obtained by Wójcik for real normed spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive mappings preserving orthogonality between complex inner product spaces Li, Lei Liu, Siyu Peralta, Antonio M. Functional Analysis Let $H$ and $K$ be two complex inner product spaces with dim$(X)\geq 2$. We prove that for each non-zero additive mapping $A:H \to K$ with dense image the following statements are equivalent: $(a)$ $A$ is (complex) linear or conjugate-linear mapping and there exists $γ>0$ such that $\| A (x) \| = γ\|x\|$, for all $x\in X$, that is, $A$ is a positive scalar multiple of a linear or a conjugate-linear isometry; $(b)$ There exists $γ_1 >0$ such that one of the next properties holds for all $x,y \in H$: $(b.1)$ $\langle A(x) |A(y)\rangle = γ_1 \langle x|y\rangle,$ $(b.2)$ $\langle A(x) |A(y)\rangle = γ_1 \langle y|x \rangle;$ $(c)$ $A$ is linear or conjugate-linear and preserves orthogonality in both directions; $(d)$ $A$ is linear or conjugate-linear and preserves orthogonality; $(e)$ $A$ is additive and preserves orthogonality in both directions; $(f)$ $A$ is additive and preserves orthogonality. This extends to the complex setting a recent generalization of the Koldobsky--Blanco--Turnšek theorem obtained by Wójcik for real normed spaces. |
| title | Additive mappings preserving orthogonality between complex inner product spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2410.08101 |