This paper derives a generalization of the Granger-Johansen Representation Theorem valid for H-valued autoregressive (AR) processes , where H is an infinite dimensional separable Hilbert space, under the assumption that 1 is an eigenvalue of finite type of the AR operator function and that no other non-zero eigenvalue lies within or on the unit circle. A necessary and sufficient condition for integration of order d=1,2,\dots is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0,%DIFDELCMD < \dots%%% , d, each one associated with components of the process integrated of order h. These results mirror the ones recently obtained in the finite dimensional case, with the only difference that the number of cointegrating relations  of order 0 is infinite  .
This paper defines the class of \mathcal{H autoregressive (AR) processes with a unit root of finite type, where \mathcal{H is an infinite dimensional separable Hilbert space, and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration order d=1,2,\dots %DIFDELCMD < \dots%%% . An existence theorem shows that the solution of an AR with a unit root of finite type is necessarily integrated of some finite integer d and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the Triangular representation of the process . Except for the fact that the number of cointegrating relations that are integrated of order 0 is infinite , the representation of \mathcal{H .
R
derives a generalization of the Granger-Johansen Representation Theorem valid for H-valued
defines the class of \mathcal{H
R
, where H
with a unit root of finite type, where \mathcal{H
R
under the assumption that 1 is an eigenvalue of finite type of the AR operator function and that no other non-zero eigenvalue lies within or on the unit circle. A necessary and sufficient condition for integration of
and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration
D
is given in terms of the decomposition of the space H into the direct sum of d+1 closed subspaces \tau_h, h=0,
R
, d, each one associated with components
. An existence theorem shows that the solution of an AR with a unit root of finite type is necessarily integrated of some finite integer d and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the Triangular representation
R
integrated of order h. These results mirror the ones recently obtained in the finite dimensional case, with the only difference
. Except for the fact
A
that are integrated
A
, the representation of \mathcal{H