Additive mappings preserving orthogonality between complex inner product spaces

Abstract

Let $H$ and $K$ be two complex inner product spaces with dim$(X)\ge 2$. We prove that for each non-zero additive mapping $A:H\to K$ with dense image the following statements are equivalent: $(a)$ $A$ is (complex) linear or conjugate-linear mapping and there exists ${\hat{I}}^3>0$ such that $\Vert A(x)\Vert ={\hat{I}}^3\Vert x\Vert$, for all $x\in X$, that is, $A$ is a positive scalar multiple of a linear or a conjugate-linear isometry; $(b)$ There exists ${\hat{I}}_1^3>0$ such that one of the next properties holds for all $x,y\in H$: $(b.1)$ $⟨A(x)|A(y)⟩={\hat{I}}_1^3⟨x|y⟩,$ $(b.2)$ $⟨A(x)|A(y)⟩={\hat{I}}_1^3⟨y|x⟩;$ $(c)$ $A$ is linear or conjugate-linear and preserves orthogonality in both directions; $(d)$ $A$ is linear or conjugate-linear and preserves orthogonality; $(e)$ $A$ is additive and preserves orthogonality in both directions; $(f)$ $A$ is additive and preserves orthogonality. This extends to the complex setting a recent generalization of the Koldobsky--Blanco--TurnÅ¡ek theorem obtained by Wójcik for real normed spaces.