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.
After revision: This paper derives a generalization of the Granger-Johansen Representation Theorem valid for H-valued autoregressive (AR) processes with a unit root of finite type, where \mathcal{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.