Functions preserving the biadditivity (Q2187827)

If \(H\) and \(K\) are two Hilbert spaces and \(f,\,g:H\to K\) are two unknown functions, \textit{R. Łukasik} and \textit{P. Wójcik} [Aequationes Math. 90, No. 3, 495--499 (2016; Zbl 1346.39033)] proved under some conditions that \(f\) and \(g\) satisfy the functional equation \(\langle f(x), g(y)\rangle=\langle x, y\rangle\) for each \(x, y\in H\). In this paper, the authors give a generalization of such a functional equation in the case of abelian groups. Suppose that \(S\) is a semigroup, \(H\) and \(X\) are two abelian groups, and \(A: S^2 \to X,\, B: H^2 \to X\) are two biadditive functions. For two unknown mappings \(f, g : S \to H\), the authors prove \( (f, g)\) satisfies the functional equation \(B(f(x), g(y)) = A(x, y)\) under some assumptions. This is an extension of the well-known characterization of the linear isometry.