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Evolution semigroups, translation algebras, and exponential dichotomy of cocycles. (English) Zbl 0962.34035
The authors study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space ${\Theta }$ acting on a Banach space $X$. Their main tool is the associated evolution semigroup on ${C}_{0}\left({\Theta };X\right)$. They 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, they establish a spectral mapping/annular hull theorem for the evolution semigroup. In addition, the dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on ${c}_{0}\left(ℤ;X\right)$. Here, they develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that the dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, the exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, they extend the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous, mild solutions to the inhomogeneous equation.
##### MSC:
 34D09 Dichotomy, trichotomy 34G20 Nonlinear ODE in abstract spaces 47H20 Semigroups of nonlinear operators