I proposed [this](https://math.stackexchange.com/q/1339628/156150) 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



Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space?
If I interpret $H$ to mean $L^2(\mathbb R)$,
and if I interpret $\frac{\mathbb d}{\mathbb d x}$ to mean differentiation by $x$,
then the set of periodic vectors is not dense in $H$.


Excluding functions that grow exponentially, a function is periodic w.r.t. differentiation by $x$ if and only if it is of the form $a \cos(x)+b \sin(x)$. Indeed, $(a \cos(x)+b \sin(x))''''=a \cos(x)+b \sin(x)$.


Such functions are not dense in $H$.


Actually, such functions are not even in $H$. 


