I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb R)$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{T : H \to H, T \space \text {is non-bounded and linear operator} \}$.
We say An operator $T \in B(H)$ is chaotic if $T$ is hypercyclic and has a dense set of periodic points.
I would like to know if $\frac{\Bbb d}{\Bbb d x}$ is a chaotic operator in $B(H)$. where $\frac{\Bbb d}{\Bbb d x}$ act on functions $u \in H $ such that defined in region $S$ and vanish at the boundary of $S$
Thank you for any help