The author considers an evolution family on and the integral equation
He characterizes the exponential dichotomy of the evolution family through the solvability of this integral equation in admissible function spaces which contain function spaces of type, the Lorentz spaces , and many other function spaces occurring in interpolation theory.
The paper applies the technique of choosing test functions related to integral equations. This technique allows to use the Banach isomorphism theorem applied to an abstract differential operator for obtaining explicit dichotomy estimates.
Using the characterization of exponential dichotomy, the author proves the robustness of exponential dichotomy of evolution families on under small perturbations.