An algebraic characterization of linearity for additive maps preserving orthogonality

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 \(\hbox{dim}(H)\ge 2\), and let \(A: H\rightarrow K\) 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 \(z\in H\) we have \(A(i z) \in \{\pm i A(z)\}\).

(c):

There exists a non-zero point \(z\in H\) such that \(A(i z) \in \{\pm i A(z)\}\).

(d):

There exists a non-zero point \(z\in H\) such that \(i A(z) \in A(H)\).

The mapping A is neither complex-linear nor conjugate-linear if, and only if, there exists a non-zero \(x\in H\) such that \(i A(x)\notin A(H)\) (equivalently, for every non-zero \(x\in H\), \(i A(x)\notin A(H)\)). 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 finite dimensional with \(\hbox{dim}(K)< 2\hbox{dim}(H)\).