Hadamard expansions for powers of causal Green's operators and "resolvents''

We derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated to a normally hyperbolic operator $P$, as well as expansions for advanced/retarded Green’s operators associated to $P-z$ for $z\in \mathbb{C}$. These expansions involve the same Hadamard coefficients as the original Hadamard expansion.