engleski

Fredholm operators and nonuniform exponential dichotomies

We show that the existence of a nonuniform exponential dichotomy for a one-sided sequence $(A_m)_{; ; ; ; m\ge0}; ; ; ; $ of invertible $d\times d$ matrices is equivalent to the Fredholm property of a certain linear operator between spaces of bounded sequences. Moreover, for a two-sided sequence $(A_m)_{; ; ; ; m\in\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 $\Z^+_0$ and~$\Z^-_0$. We also give conditions on $S$ so that the sequence admits a nonuniform exponential dichotomy on~$\Z$. Finally, we use the former characterizations to establish the robustness of the notion of a nonuniform exponential dichotomy.

Exponential dichotomies; robustness; Fredholm operators

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