Suppose we have a closable and densely defined operator $A$ with a domain $dom(A)$ which is a subspace of a Hilbert space $\mathcal{H}$. Let $\mathcal{H}$ have an orthonormal basis $\{e_n\}_{n=1}^\infty$. So the operator $A$ can be viewed as an infinite matrix $A_{ij}$.
We know that there is a usual procedure to define $A^*$ with its domain $dom(A^*)$. Now consider the formal adjoint operator $A_* = \{\overline{A_{ji}}\}$ with the domain $dom(A_*)$. Are there some simple conditions on $A$ when these domains coincide: $dom(A^*) = dom(A_*)$?
What can be said on this matter if $A_{ij}$ is a finite-band matrix? Or when $A$ is formally self-adjoint ($A_{ij} = \overline{A_{ji}})$?