On the non-hypercyclicity of normal operators, their exponentials, and symmetric operators

Marat V. Markin, Edward S. Sichel

Published 2019-08-06Version 1

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 $\left\{e^{tA}\right\}_{t\ge 0}$ 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.