A generalization of a theorem of Chernoff on standard operator algebras

Let X be a real or complex Banach space, let $${\mathcal {A}}$$ be a standard operator algebra on X, let n be a positive integer, and let $$D:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)$$ be a linear mapping such that the equality $$D(A^{2n})=D(A^n)A^n+A^nD(A^n)$$ holds for every $$A\in {\mathcal {A}}$$ . We prove that D can be written in a unique way as $$D=D_1+D_0$$ where $$D_1:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)$$ is of the form $$A\rightarrow AB-BA$$ for some $$B\in {\mathcal {L}} (X)$$ , and $$D_0:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)$$ is a linear mapping such that $$D_0(A^n)=0$$ for every $$A\in {\mathcal {A}}$$ . The case $$n=1$$ of this result refines a theorem of Chernoff.