Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$. Assume also that $D\subset H$ is a core domain for $A$; that is, $D$ is dense in $\text{Dom(A)}$ with respect to the graph norm.

Is it true that we can pick an eigenvector basis for $H$ to consist of elements in $D$ instead of $\text{Dom(A)}$?

Thank you in advance.