A new class of frequently hypercyclic operators

TL;DR

This paper introduces a new class of operators on Banach spaces that are frequently hypercyclic, characterized by having a rich set of eigenvectors on the unit circle, expanding understanding of linear dynamical systems.

Contribution

It establishes that operators with a dense set of eigenvectors on the unit circle are automatically frequently hypercyclic, a novel criterion in the study of hypercyclic operators.

Findings

Operators with sufficiently many eigenvectors on the unit circle are frequently hypercyclic.

The paper provides a new criterion for frequent hypercyclicity based on eigenvector properties.

This advances the classification of hypercyclic operators in linear dynamics.

Abstract

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive lower density. We prove that if T is a bounded operator on X which has "sufficiently many" eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is automatically frequently hypercyclic.