Hille Yosida theorem application
Solution 1:
If $A : \mathcal{D}(A)\subseteq X\rightarrow X$ is a closed densely-defined linear operator on a Hilbert space $X$, then $A^{\star}A$ is a closed densely-defined positive selfadjoint linear operator. So start with $\Delta$ on $H^{2}_{0}$ and define $\Delta^{\star}\Delta$, which has the same domain and action as your proposed $\Delta^{2}$.