Before revision: 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, , d, each one associated with components of the process integrated of order h.
After revision: 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, and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration 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, , d, each one associated with components of the process integrated of order h.