On sufficient and necessary conditions for linear hypercyclicity and chaos

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.