An algebraic characterization of linearity for additive maps preserving orthogonality

Abstract

Abstract 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(iz) ∈ {±iA(z)}. (c) There exists a non-zero point z ∈ H such that A(iz) ∈ {±iA(z)}. (d) There exists a non-zero point z ∈ H such that iA(z) ∈ A(H). The mapping A is neither complex-linear nor conjugate-linear if, and only if, there exists a non-zero x ∈ H such that iA(x) ∉ A(H) (equivalently, for every non-zero x ∈ H, iA(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 fnite dimensional with dim(K) < 2dim(H).