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A first order differential equation with a periodic operator coefficientacting in a pair of Hilbert spaces is considered. This setting models bothelliptic equations with periodic coefficients in a cylinder and parabolicequations with time periodic coefficients. Our main results are a constructionof a pointwise projector and a spectral splitting of the system into a finitedimensional system of ordinary differential equations with constantcoefficients and an infinite dimensional part whose solutions have betterproperties in a certain sense. This complements the well-known asymptoticresults for periodic hypoelliptic problems in cylinders (Kuchment) and forelliptic problems in quasicylinders (Nazarov).  As an application we give a center manifold reduction for a class ofnon-linear ordinary differential equations in Hilbert spaces with periodiccoefficients. This result generalizes the known case with constant coefficients(Mielke).

Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces