Equivalence between two differential operators

Question

I try to understand the relation between these two operators ${(I - \partial _x^2)^{ - 1}}\partial _x^4$ and $\partial _x^2$. the first one acts on $H_0^2 \cap {H^4}$ to $H_0^1 \cap {H^2}$, and the second acts on $H_0^1 \cap {H^2}$ to ${L^2}$, do we can say that the two operators are equivalent ? or they have the same role?can I say that the first one acts on $H_0^1 \cap {H^2}$ to $L^2$? thanks