2025/03/20 by Lei Li, Siyu Liu, Li, Lei +3
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2503.16341
openalex publication_date 2025/03/20 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
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)≥ 2, and let 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∈ H we have A(i z) ∈ \± i A(z)\. (c) There exists a non-zero point z∈ H such that A(i z) ∈ \± i A(z)\. (d) There exists a non-zero point z∈ H such that i A(z) ∈ A(H). The mapping A neither is complex-linear nor conjugate-linear if, and only if, there exists a non-zero x∈ H such that i A(x)∉ A(H) (equivalently, for every non-zero x∈ 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 dim(K)< 2\hboxdim(H).