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Fredholm operators and nonuniform exponential dichotomies

Author

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  • Barreira, Luis
  • Dragičević, Davor
  • Valls, Claudia

Abstract

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 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 Z. Finally, we use the former characterizations to establish the robustness of the notion of a nonuniform exponential dichotomy.

Suggested Citation

  • Barreira, Luis & Dragičević, Davor & Valls, Claudia, 2016. "Fredholm operators and nonuniform exponential dichotomies," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 120-127.
    • Handle: RePEc:eee:chsofr:v:85:y:2016:i:c:p:120-127
    DOI: 10.1016/j.chaos.2016.01.021
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