Sobolev-like Hilbert spaces induced by elliptic operators

We investigate properties of function spaces induced by the inner product Sobolev spaces $H^s(\mathrm{\Omega })$ over a bounded Euclidean domain $\mathrm{\Omega }$ and by an elliptic differential operator $A$ on $\bar{\mathrm{\Omega }}$. The domain and the coefficients of $A$ are of the class $C^{\mathrm{\infty }}$. These spaces consist of all distributions $u\in H^s(\mathrm{\Omega })$ such that $Au\in H^{\lambda }(\mathrm{\Omega })$ and are endowed with the corresponding graph norm, with $s,\lambda \in \mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.