The Natural Components of an Autoregressive Time Series on Banach Space

We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future or at a nominated initial time – depending on whether the corresponding spectral point is outside, inside or on the unit circle.