Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$? Perhaps this looks not like a problem of research level, however, I can't solve it myself and can't find any reference on google.

Furthermore, does it exist a closed linear operator $T$ on $X$, such that $D(T)=Y$ and $T$ is also the generator of some $C_0$ semigroup on $X$ ?

If any of this is not true, What about if $X$ is a Hilbert space ?

Thanks for any comment.