Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

@article{Rosa2010FrequentHC,
  title={Frequent hypercyclicity, chaos, and unconditional Schauder decompositions},
  author={M. A. Cruz de la Rosa and Leonhard Frerick and Sophie Grivaux and Alfredo Peris},
  journal={Israel Journal of Mathematics},
  year={2010},
  volume={190},
  pages={389-399}
}
We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator. 
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