A spectral theory for linear differential systems

This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.