Journal of the
Korean Mathematical Society JKMS

For a nonautonomous dynamics defined by a sequence of bounded linear operators on a Banach space, we give a characterization of the existence of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of a certain linear operator between general admissible spaces. This notion of an exponential dichotomy contains as very special cases the notions of uniform, nonuniform and tempered exponential dichotomies. As applications, we detail the consequences of our results for the class of tempered exponential dichotomies, which are ubiquitous in the context of ergodic theory, and we show that the notion of an exponential dichotomy under sufficiently small parameterized perturbations persists and that their stable and unstable spaces are as regular as the perturbation.

Keywords: Tempered exponential dichotomies, admissible spaces

MSC numbers: Primary 37D99

Supported by: L. Barreira and C. Valls were supported by FCT/Portugal through UID/MAT/04459/2013. J. Rijo was supported by FCT/Portugal through the grant PD/BD/128413/2017.