This is almost a copy of https://math.stackexchange.com/questions/3931318/when-the-adjoint-of-an-unbounded-operator-on-a-hilbert-space-coincides-with-the

I am trying to work with infinite matrices in a Hilbert space. I want to consider these as unbounded operators, but I have some troubles understanding how the domain of the adjoint operator is defined in this case.

Namely, suppose we have a closed and densely defined operator $A$ with a domain $D(A)$ which is a subspace of a Hilbert space $\mathcal{H}$. Let $\mathcal{H}$ have an orthonormal basis $\{e_n\}_{n=1}^\infty$. Suppose $\{e_n\} \in D(A)$. Then for the operator $A$ there exists an infinite matrix $A_{ij} = \{\langle Ae_j, e_i\rangle\}_{ij}$.

We know that there is a usual procedure to define $A^*$ with its domain $D(A^*)$. Suppose $\{e_n\} \in D(A^*)$. Now consider the formal adjoint operator $A_* = \{\overline{A_{ji}}\}$ with the domain $D(A_*)$ consisting of those $\zeta$ such that $\eta_j = \sum {\overline{A_{ji}}} \zeta_i$ is in $\ell^2$. Are there some simple conditions on $A$ and on $D(A)$ for these domains to coincide: $D(A^*) = D(A_*)$?

What can be said on this matter if $A_{ij}$ is a finite-band matrix? Or when $A$ is formally self-adjoint ($A_{ij} = \overline{A_{ji}})$?

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