Identities related to derivations and centralizers on standard operator algebras

In this paper identities related to derivations and centralizers on. operator algebras are considered. We prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) and F(X) be the algebra of all bounded linear operators and the ideal of all finite rank operators on X, respectively. Suppose there exist linear mappings D, G : F(X) -> L(X) such that D(A(2)) = D(A)A + AG(A) and G(A(2)) = G(A)A + AD(A) is fulfilled for all A is an element of F(X). In this case there exists B is an element of L(X) such that D(A) = G(A) = [A, B] holds for all A is an element of F(X).