This is an announcement for the paper "Frequent hypercyclicity, chaos, and unconditional Schauder decompositions" by Manuel De la Rosa, Leonhard Frerick, Sophie Grivaux, and Alfredo Peris.
Abstract: 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. This result is extended to complex Frechet spaces with a continuous norm and an unconditional Schauder decomposition, and also to complex Frechet spaces with an unconditional basis, which gives a partial positive answer to a problem posed by Bonet. We also solve a problem of Bes and Chan in the negative by presenting hypercyclic, but non-chaotic operators on \C^\N. We extend the main result to C_0-semigroups of operators. Finally, in contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.
Archive classification: math.FA
Submitted from: grivaux@math.univ-lille1.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1005.1416
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