Is intersection of a dense subspace and a closed subspace of a Hilbert space also Dense?

I have a Hilbert space $H$ and a closed operator $T$ defined on its domain $D(T)$ which is dense in H. Also $M = \text{range} \ T^n$, for some $n$, is given to be closed. Consider the restriction of $T$ to $M$, that is, $T_n = T \big| _M$. To be able to define the adjoint of $T_n$, I need $T_n$ to be densely defined. That is I need to show domain $D(T_n)$ = $D(T) \cap M$ is densely defined. Is it true?

In other words, is restrcition to a closed subspace of a densely defined closed linear operator also densely defined?

The answer to the question in the title is no.

Let $H=\ell^2(\mathbb N)$, $$ N=\{x\in H:\ \exists m:\ x(n)=0,\ \forall n\geq m\} $$ and $$ M=\{\lambda z:\ \lambda\in\mathbb C\}, $$ where $$ z=\left(1,\frac12,\frac13,\frac14,\ldots\right). $$ Then $N$ is dense, $M$ is closed, and $N\cap M=\{0\}$.