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

## TL;DR

This paper provides a simple proof that normal and symmetric operators, as well as their exponentials under certain spectral conditions, are not hypercyclic in complex Hilbert spaces.

## Contribution

It offers a straightforward proof of non-hypercyclicity for normal and symmetric operators and their exponentials, extending known results with minimal complexity.

## Key findings

- Normal operators are not hypercyclic.
- Exponentials of normal operators are not hypercyclic under certain spectral conditions.
- Symmetric operators are not hypercyclic.

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