Let A be a Banach algebra with unity 1, and M be a unital Banach left A-module. Let δ : A → M be a continuous linear map with the property that ab + ba = z ⇒ 2aδ(b) + 2bδ(a) = δ(z), a, b ∈ A, where z ∈ A. In this article, we first characterize the continuous linear maps δ satisfying the above property for z = 1. Then we consider the case A = M = Alg L, where Alg L is a reflexive algebra on a Hilbert space H, and z = P is a non-trivial idempotent in A with P (H) ∈ L, and then we describe δ. Finally, we apply the main results to CSL-algebras, irreducible CDC-algebras and nest algebras on a Hilbert space H.
