Advanced and Retarded Green Functions (Hyperbolic PDEs)

Question

I have often come across the statement the advanced and retarded Green functions are unique for linear hyperbolic PDEs. Now if I consider the linear PDE operator, it is not really invertible given that the PDE has a solution. So one has to say that we are speaking of the resolvent of the PDE operator when we speak of advanced and retarded Green functions. As an inverse linear operator, the resolvent would be unique. However, we end up having two of them now (depending upon the support of the solution being to the "future" or "past of the support of the inhomogeneity function or source on the right side of the equation) Is the issue simply with the scalar we have in the resolvent?