Identity related to additive mappings on standard operator algebras

Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X and let A(X)⊆L(X) be a standard operator algebra. Suppose there exists a linear mapping T:A(X)→L(X) satisfying the relation T(An) = T(A)An−1 − AT(An−2)A − An−1T(A) for all A∈A(X), where n > 2 is some fixed integer. Then T is of the form: (i)T(A) = 0 for all A∈F(X) and (ii) T(A) = BA, for all A∈A(X) and some B∈L(X).