Let A be an algebra. A linear mapping d :Given two derivations d and d ′ on a C * -algebra A , we prove that there exists a derivation D on A such thatand only if d and d ′ are linearly dependent.

Let (X, ⊥) be a real vector space of dimension at least 3, with the orthogonality defined on it by:(i) for all x ∈ X, x ⊥ 0 and 0 ⊥ x, (ii) for all x, y ∈ X \ {0}, x ⊥ y if and only if x, y are linearly independent.We show that any orthogonally quadratic mapping on X is a quadratic mapping. Also we prove the Hyers-Ulam stability of orthogonally quadratic functional equation and the Hyers-Ulam stability of orthogonally pexiderized quadratic functional equation.

Let H be a Hilbert C * -module over a unital C * -algebra A. In this paper, we find the general form of the mappings T : H → H satisfing 2 T (x), T (y) = T (x), y + x, T (y) (x, y ∈ H), as adjointable (bounded) A-linear operators. The generalized Hyers-Ulam stability of the functional equation is discussed.