On sufficient and necessary conditions for linear hypercyclicity and chaos

TL;DR

This paper refines conditions for linear hypercyclicity and chaos in Banach space operators, establishing new criteria and inheritance properties for bounded and unbounded operators, with implications for spectral analysis.

Contribution

It strengthens a known hypercyclicity condition to derive sufficient conditions for chaos and extends these results to unbounded operators, including inheritance and spectral properties.

Findings

Strengthened hypercyclicity condition implies chaos.

Hypercyclicity is inherited by inverses, powers, and multiples.

Necessary conditions extend to unbounded operators.

Abstract

By strengthening one of the hypotheses of a well-known sufficient condition for the hypercyclicity of linear operators in Banach spaces, we arrive at a sufficient condition for linear chaos and reveal consequences of the latter for inverses, powers, multiples, and spectral properties. Extending the results, familiar for bounded linear operators, we also show that the hypercyclicity of unbounded linear operators subject to the sufficient condition for hypercyclicity is inherited by their bounded inverses, powers, and unimodular multiples and that necessary conditions for linear hypercyclicity stretch to the unbounded case.