  This paper consider inverting a holomorphic Fredholm operator pencil.
Specifically, we provide necessary and sufficient conditions for the inverse of
a holomorphic Fredholm operator pencil to have a simple pole and a second order
pole. Based on those results, a closed-form expression of the Laurent expansion
of the inverse around an isolated singularity is obtained in each case. As an
application of the results, we obtain a suitable extension of the
Granger-Johansen representation theory for random sequences taking values in a
separable Banach space. Due to our closed-form expression of the inverse, we
may fully characterize I(1) and I(2) solutions except a term that depends on
initial values.
