This is not an accurate question. I am not so sure about unbounded operators on a Hilbert space.
Let $D$ be the differential operator on $L^2(0,1)$. Well, to somewhat, we can extend $D$ to a normal operator, still denoted by $D$. So, it has a spectral representation, $D:=\int \lambda dE_\lambda$. My question is, are there any sequences of bounded linear operators on $L^2(0,1)$ converging to $D$ in some topology? and what is this topology?