In this book our main objective is to characterize asymptotic properties (stability, hyperbolicity, exponential dichotomy) of linear differential equations on Banach spaces and infinite dimensional dynamical systems in terms of spectral properties of a special type of associated semigroup that we call an evolution semigroup. We use methods from the theory of strongly continuous semigroups of linear operators, the theory of nonautonomous abstract Cauchy problems on Banach spaces, the theory of C^{*}- and Banach algebras, ergodic theory, the theory of hyperbolic dynamical systems and Lyapunov exponents. Applications to linear control theory, magnetohydrodynamics, and to the theory of transfer operators are given. |