  We extend the notion of cointegration for time series taking values in a
potentially infinite dimensional Banach space. Examples of such time series
include stochastic processes in C[0,1] equipped with the supremum distance and
those in a finite dimensional vector space equipped with a non-Euclidean
distance. We then develop versions of the Granger-Johansen representation
theorems for I(1) and I(2) autoregressive (AR) processes taking values in such
a space. To achieve our goal, we first note that an AR(p) law of motion can be
characterized by a linear operator pencil via the companion form
representation, and then study the spectral properties of a linear operator
pencil to obtain a necessary and sufficient condition for a given AR(p) law of
motion to admit I(1) or I(2) solutions. These operator-theoretic results form a
fundamental basis for our representation theorems. Furthermore, it is shown
that our operator-theoretic approach is in fact a closely related extension of
the conventional approach taken in a Euclidean space setting. Our theoretical
results may be especially relevant in a recently growing literature on
functional time series analysis in Banach spaces.
