Additive mappings preserving orthogonality between complex inner product spaces

Abstract

Let H and K be two complex inner product spaces with dim$\left(H\right)\ge 2$. We prove that for each non-zero 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 $\gamma >0$ such that $\Vert A\left(x\right)\Vert =\gamma \Vert x\Vert$, for all $x\in H$, that is, A is a positive scalar multiple of a linear or a conjugate-linear isometry;
• (a)
  There exists ${\gamma }_1>0$ such that one of the next properties holds for all $x,y\in H$:

  • $(b.1)$
    $〈A\left(x\right)\left|A\right(y)〉={\gamma }_1〈x|y〉$,
  • $(b.1)$
    $〈A\left(x\right)\left|A\right(y)〉={\gamma }_1〈y|x〉$;
• (a)
  A is linear or conjugate-linear and preserves orthogonality;
• (a)
  A is additive and preserves orthogonality in both directions;
• (a)
  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.