Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

Abstract

We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space Θ acting on a Banach space X. Our main tool is the associated evolution semigroup on C0(Θ; X). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c0(Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation.