  A first order differential equation with a periodic operator coefficient
acting in a pair of Hilbert spaces is considered. This setting models both
elliptic equations with periodic coefficients in a cylinder and parabolic
equations with time periodic coefficients. Our main results are a construction
of a pointwise projector and a spectral splitting of the system into a finite
dimensional system of ordinary differential equations with constant
coefficients and an infinite dimensional part whose solutions have better
properties in a certain sense. This complements the well-known asymptotic
results for periodic hypoelliptic problems in cylinders (Kuchment) and for
elliptic problems in quasicylinders (Nazarov).
  As an application we give a center manifold reduction for a class of
non-linear ordinary differential equations in Hilbert spaces with periodic
coefficients. This result generalizes the known case with constant coefficients
(Mielke).
