Dynamical Spectrum for time dependent linear systems in Banach spaces

Abstract

This paper is primarily concerned with the Dynamical Spectrum for time dependent linear differential equations in Banach spaces. We give a characterization of the Dynamical Spectrum which is an extension of the Sacker-Sell Theorem. Also we define the Lyapunov exponents, who measure the decay rate of the solutions of a linear differential equations; we investigate the relation between the Dynamical Spectrum, the Spectral Subbundles associated with the corresponding spectral intervals and the Lyapunov exponents. These problems are treated in the unified setting of a Linear Skew-Product Semiflow. Finally we present some examples of Linear Skew-Product Semiflow arising from time dependent functional differential equations and parabolic partial differential equations.