Additive mappings preserving orthogonality between complex inner product spaces

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 γ 1 > 0 such that one of the next properties holds for all x , y ∈ H : ( b .1) 〈 A ( x )| A ( y )〉 = γ 1 〈 x | y 〉, ( b .2) 〈 A ( x )| A ( y )〉 = γ 1 〈 y | x 〉; ( c ) A is linear or conjugate-linear and preserves orthogonality; ( d ) A is additive and preserves orthogonality in both directions; ( e ) 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.