On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

Abstract

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

Keywords

• hypercyclicity
• scalar type spectral operator
• normal operator
• <i>c</i>₀-semigroup