On orthogonally additive mappings

Abstract

The conditional Cauchy functional equationF: (X, +, ⊥) → (Y, +), F(x + y) = F(x) + F(y) x, y ∈ X, x ⊥ y, has first been studied under regularity (mainly continuity and boundedness) conditions and by referring to the inner product and the Birkhoff—James orthogonalities (A. Pinsker 1938, K. Sundaresan 1972, S. Gudder and D. Strawther 1975). The latter authors proposed an axiomatic framework for the space (X, +, ⊥), and it then became possible to modify their axioms so that it could be proved without any regularity condition that the odd solutions of (*) are additive and the even ones are quadratic (cf., e.g., ([8], [12]). The results obtained included the classical case of the inner product orthogonality as well as the three following generalizations thereof: (i) Birkhoff—James orthogonality on a normed space, (ii) orthogonality induced by a non-isotropic sesquilinear functional, (iii) semi-inner product orthogonality.