Additive mappings preserving orthogonality between complex inner product spaces

Abstract

Let H and K be two complex inner product spaces with dim(H) ≥ 2. We prove that for each non-zero mapping A : H → 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 ∈ H, that is, A is a positive scalar multiple of a linear or a conjugate-linear isometry; (b) There exists γ1 > 0 such that one of the next properties holds for all x, y ∈ H: (b.1) 〈A(x)|A(y)〉 = γ1〈x|y〉, (b.2) 〈A(x)|A(y)〉 = γ1〈y|x〉; (c) A is linear or conjugate-linear and preserves orthogonal- ity; (d) A is additive and preserves orthogonality in both direc- tions; (e) A is additive and preserves orthogonality.