On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

A differentiation-invariant weighted Fréchet space ${\mathcal E}(\varphi)$ of infinitely differentiable functions in ${\mathbb R}^n$ generated by a countable family $\varphi$ of continuous real-valued functions in ${\mathbb R}^n$ is considered. It is shown that, under minimal restrictions on $\varphi$, any continuous linear operator on ${\mathcal E}(\varphi)$ that is not a scalar multiple of the identity mapping and commutes with the partial differentiation operators is hypercyclic. Examples of hypercyclic operators in ${\mathcal E}(\varphi)$ are presented for cases in which the space ${\mathcal E}(\varphi)$ is translation invariant.