Let be a compact Hausdorff space and a Banach space. A linear evolutionary system is a mapping , with the components given by a two- sided flow on and a family of linear selfmappings , , of that are strongly continuous in , continuous in , satisfying and .
Finite-dimensional examples arise if is the fundamental operator solution of a system of linear ordinary differential equations and is related to a space of time-dependent coefficients. For the finite-dimensional case, exponential dichotomies, i.e. continuous families of fibre-wise projectors onto the “stable” subbundle, satisfying and for all in an invariant subset , were constructed by the authors in a series of papers [R. J. Sacker and G. R. Sell, J. Differ. Equations 15, 429-458 (1974; Zbl 0294.58008); J. Differ. Equations 22, 478-496 (1976; Zbl 0338.58016 ); J. Differ. Equations 22, 497-522 (1976; Zbl 0339.58013)]. The present paper extends this theory to an infinite-dimensional setting, assuming that the are set contractions for large , in the weakly hyperbolic case (i.e. the only solution that exists and is bounded for is the zero solution). Criteria for the existence of exponential dichotomies over subsets of and over all with constant codimension of the stable subbundle are given; if such a global dichotomy does not exist, a Morse decomposition of , indexed by increasing codimensions of the stable subbundle is shown to exist. The main difficulty in extending the theory to the infinite-dimensional case comes from the fact that need not be defined or unique for . Applications to nonlinear evolution equations and in particular to Navier-Stokes equations are sketched; in these cases, is related to a compact attractor, and linearizing the equation gives rise to .