I am reading a text at the moment and I am not sure what the authors mean by $D(A^{-1})$. In this situation $A: D(A) \subset H \longrightarrow H$ is an unbounded operator on a hilbert space $H$ which is also positive selfadjoint. It has an orthonormal eigenbasis $\{e_k\}$ with eigenvectors $\{\lambda_k\}$. The domain $D(A)$ is, as far as I can tell, equipped with the graph norm.
Now I would by default say that the domain of the inverse operator is then: $$D(A^{-1}) = \left\{x \in H: \sum_{k}^\infty \lambda^{-k} \langle x, e_k \rangle e_k \in H\right\}.$$ They seem to use $D(A^{-1})$ in a dual sense however, explicitly writing $$D(A) \subset H = H' \subset D(A^{-1}).$$ I can see that there is a natural embedding $$D(A^{-1}) \rightarrow D(A)'$$ by using the Riesz representation isomorphism. Then for $x \in D(A^{-1})$ and $y \in D(A)$: $$ |\langle x, y\rangle| = |\langle A A^{-1}x, y \rangle| = |\langle A^{-1}, Ay \rangle| \leq \|A^{-1} x\|_H \|y\|_{D(A)}. $$ But I do not see why this would be surjective.
The authors also explicitly apply $A^{-1}$ to elements of $D(A^{-1})$ and get something in $H$. If the authors just used the notation $D(A^{-1})$ for $D(A)'$ then it is not clear to me what operator $A^{-1}$ represents on dual objects.
Could someone with more experience in functional analysis figure out how this is to be understood? Thanks a lot!