An algebraic characterization of linearity for additive maps preserving orthogonality

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 $$\hbox{dim}(H)\ge 2$$ dim ( H ) ≥ 2 , and let $$A: H\rightarrow K$$ A : H → K be an a...

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Published inAnnals of functional analysis Vol. 16; no. 4
Main Authors Li, Lei, Liu, Siyu, Peralta, Antonio M.
Format Journal Article
LanguageEnglish
Published 01.10.2025
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ISSN2639-7390
2008-8752
DOI10.1007/s43034-025-00454-0

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Summary: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 $$\hbox{dim}(H)\ge 2$$ dim ( H ) ≥ 2 , and let $$A: H\rightarrow K$$ A : H → 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 $$z\in H$$ z ∈ H we have $$A(i z) \in \{\pm i A(z)\}$$ A ( i z ) ∈ { ± i A ( z ) } . (c) There exists a non-zero point $$z\in H$$ z ∈ H such that $$A(i z) \in \{\pm i A(z)\}$$ A ( i z ) ∈ { ± i A ( z ) } . (d) There exists a non-zero point $$z\in H$$ z ∈ H such that $$i A(z) \in A(H)$$ i A ( z ) ∈ A ( H ) . The mapping A is neither complex-linear nor conjugate-linear if, and only if, there exists a non-zero $$x\in H$$ x ∈ H such that $$i A(x)\notin A(H)$$ i A ( x ) ∉ A ( H ) (equivalently, for every non-zero $$x\in H$$ x ∈ H , $$i A(x)\notin A(H)$$ i A ( x ) ∉ 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 $$\hbox{dim}(K)< 2\hbox{dim}(H)$$ dim ( K ) < 2 dim ( H ) .
ISSN:2639-7390
2008-8752
DOI:10.1007/s43034-025-00454-0