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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

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    $\begingroup$ I'm voting to close this question because it has been already asked on MSE and 20 hours is not a sufficient time to ask it again here. Also even the question does not seem to be clear if I read through the comments over at MSE. $\endgroup$ Jun 26, 2015 at 22:20
  • $\begingroup$ @JohannesHahn, since it's not clear i see to delete it from MO $\endgroup$ Jun 26, 2015 at 22:37
  • $\begingroup$ [deleted over-hasty comment] I still feel this question should have been left on MSE and is not appropriate for MO $\endgroup$
    – Yemon Choi
    Jun 27, 2015 at 13:54

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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 $H$.

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  • $\begingroup$ @ Andre henriques, thank you for your answer, since the question answered why the downvote for it ? $\endgroup$ Jun 26, 2015 at 23:34
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    $\begingroup$ And actually the differentiation is not even an operator as OP defined, hence the question does not really make sense. $\endgroup$ Jun 27, 2015 at 9:22
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    $\begingroup$ @AndrásBátkai perhaps the OP had something like link.springer.com/article/10.1007%2FBF01299846 in mind? $\endgroup$
    – Yemon Choi
    Jun 27, 2015 at 13:54

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