Exponential dichotomy of difference equations and applications to evolution equations on the half-line

For a sequence of bounded linear operator {An}∞n=0 on a Banach space X, we investigate the characterization of exponential dichotomy of the difference equations vn+1 = Anvn. We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn+1 = Anvn + fn in l∞ space. Then we apply the results to study the exponential dichotomy of evolution families generated by evolution equations.