The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Let T : A(X) → L(X) be a linear mapping satisfying the relation 2T(AA∗A) = T(A)A∗A + AA∗T(A) for all A ∈ A(X). In this case T is of the form T(A) = λA for all A ∈ A(X), where λ is some fixed complex number.