A remark on orthogonally additive bijections (Q2350274)

We say that a mapping \(f\colon H\to Y\), where \(H\) is an inner product space of dimension at least \(2\) and \(Y\) is an abelian group, is orthogonally additive if \(f(x+y)=f(x)+f(y)\) for all orthogonal vectors \(x,y\in H\). \textit{K. Baron} and \textit{J. Rätz} [Bull. Pol. Acad. Sci., Math. 43, No. 3, 187--189 (1995; Zbl 0840.39011)] showed that if the space \(H\) is a real inner product space, then each orthogonally additive mapping \(f\colon H\to Y\) is of the form \(f(x)=a(\|x\|^2)+b(x)\), where \(a\colon\mathbb{R}\to Y\) and \(b\colon H\to Y\) are additive mappings. This result was used by K. Baron to prove that an orthogonally additive bijection defined on a real inner product space is additive. The author generalizes the mentioned two results to orthogonally additive mappings defined on complex inner product spaces and inner product spaces over quaternions.