Consider a finite number $A_1,\dots,A_n$ of densely defined linear operators on a Hilbert space $\mathcal{H}$; we suppose that there exists a dense subspace $\mathcal{D}$, contained in the domain $\operatorname{dom} A_j$ of all $A_j$'s, such that \begin{equation} \overline{\operatorname{ran}\,\left(A_j\restriction\mathcal{D}\right)}=\mathcal{H}\qquad\text{for all $j=1,\dots,n$}, \end{equation} and let $A=\sum_{j=1}^nA_j$ with domain $\mathcal{D}$.
In general, despite the range of each operator being dense, nothing similar can be said about the range $\operatorname{ran}A$ of $A$: take $n=2$ and $A_2=-A_1$ as a trivial counterexample (the range of the sum is trivial). However, I wonder whether one can find sufficient conditions for $\operatorname{ran}A$ to be dense, for instance some sort of linear independence between the $A_j$'s.