Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

TL;DR

This paper demonstrates that complex Banach spaces with unconditional Schauder decompositions can support chaotic and frequently hypercyclic operators, while some real Banach spaces with unconditional bases cannot support any chaotic operators.

Contribution

It establishes the existence of chaotic and frequently hypercyclic operators on complex Banach spaces with unconditional Schauder decompositions, contrasting with real spaces.

Findings

Complex spaces with unconditional Schauder decompositions support chaotic operators.

Some real Banach spaces with unconditional bases do not support chaotic operators.

The results highlight differences between real and complex Banach spaces regarding chaos.

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. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.