Evolution semigroups, translation algebras, and exponential dichotomy of cocycles

We study the exponential dichotomy of an exponentially bounded, strongly co ntinuous cocycle over a continuous flow on a locally compact metric space T heta acting on a Banach space X. Our main tool is the associated evolution semigroup on C-0(Theta; X). We prove that the cocycle has exponential dicho tomy if and only if the evolution semigroup is hyperbolic if and only if th e imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spect ral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of wei ghted shift operators defined on c(0)(Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolut ion operators. These results imply that dichotomy persists under small pert urbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with unif orm constants. Finally, we extend to our situation the classical Perron the orem which says that dichotomy is equivalent to the existence and uniquenes s of bounded, continuous, mild solutions to the inhomogeneous equation. (C) 1999 Academic Press.