Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

Autores UPV
Año
Revista ISRAEL JOURNAL OF MATHEMATICS

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 $\mathbb{C}^\mathbb{N}$. 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.