A generalization of a theorem of Chernoff on standard operator algebras

Let X be a real or complex Banach space, let A be a standard operator algebra on X, let n be a positive integer, and let D : A -> L( X) be a linear mapping such that the equality D(A(2n)) = D(A(n)) A(n) + A(n)D(A(n)) holds for every A. A. We prove that D can be written in a unique way as D = D-1+ D-0 where D-1 : A -> L( X) is of the form A -> AB - BA for some B. L(X), and D-0 : A -> L( X) is a linear mapping such that D-0(A(n)) = 0 for every A -> A. The case n = 1 of this result refines a theorem of Chernoff.