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 $X$ ?

Added (from a comment Sep. 30 '15): If $X$ is a Hilbert space, and the dense subspace $Y$ is also a Hilbert space under some norm, then is the existence of such closed operator true?