Exponential separation, exponential dichotomy and spectral theory for linear systems of ordinary differential equations

Abstract

This paper is concerned with linear nonautonomous systems of ordinary differential equations. A criterion for exponential separation in terms of exponential dichotomy is given. As corollaries we obtain the roughness theorem for exponential separation and the new result that an upper triangular system on a half-line is exponentially separated if and only if the system corresponding to its diagonal is. A minimal decomposition into exponentially separated subspaces is defined. It turns out that it is, in general, finer than the Sacker-Sell spectral decomposition but that the two decompositions coincide for almost periodic systems.