  We extend the Granger-Johansen representation theorems for I(1) and I(2)
vector autoregressive processes to accommodate processes that take values in an
arbitrary complex separable Hilbert space. This more general setting is of
central relevance for statistical applications involving functional time
series. We first obtain a range of 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.
