Before revision:  To accomplish this we obtain necessary and sufficient conditions for a pole in the inverse of a holomorphic index-zero Fredholm operator pencil to be simple, and a formula for its residue. In the case of first order autoregressive dynamics with a unit root our results take a particularly simple form , with the residue associated with the simple pole at one proportional to a Riesz projection .
After revision:  To accomplish this we obtain necessary and sufficient conditions for a pole in the inverse of a holomorphic index-zero Fredholm operator pencil to be of first or second order. Those conditions form the basis for our development of I(1) and I(2) representations of autoregressive Hilbertian processes. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one .