Fredholm operators and nonuniform exponential dichotomies

We show that the existence of a nonuniform exponential dichotomy for a one-sided sequence (Am)m ≥ 0 of invertible d × d   matrices is equivalent to the Fredholm property of a certain linear operator between spaces of bounded sequences. Moreover, for a two-sided sequence (Am)m∈Z(Am)m∈Z we show that the existence of a nonuniform exponential dichotomy implies that a related operator S   is Fredholm and that if it is Fredholm, then the sequence admits nonuniform exponential dichotomies on Z0+ and Z0−. We also give conditions on S   so that the sequence admits a nonuniform exponential dichotomy on ZZ. Finally, we use the former characterizations to establish the robustness of the notion of a nonuniform exponential dichotomy.