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oa.operator algebras – When the adjoint of an unbounded operator on a Hilbert house coincides with the formal adjoint on its unaffected province?

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

I’m attempting to labor with innumerable matrices in a Hilbert house.

I wish to deem these as unbounded operators, however I’ve some troubles judgement how the province of the adjoint operator is outlined on this illustration.

Namely, suppose we now have a closed and densely outlined operator $A$ with a province $D(A)$ which is a subspace of a Hilbert house $mathcal{H}$.

Let $mathcal{H}$ have an orthonormal foundation ${e_n}_{n=1}^infty$.

Suppose ${e_n} in D(A)$.

Then for the operator $A$ there exists an innumerable matrix $A_{ij} = {langle Ae_j, e_irangle}_{ij}$.

We know that there’s a habitual process to outline $A^*$ with its province $D(A^*)$. Suppose ${e_n} in D(A^*)$.

Now deem the formal adjoint operator $A_* = {overline{A_{ji}}}$ with the province $D(A_*)$ consisting of these $zeta$ such that $eta_j = sum {overline{A_{ji}}} zeta_i$ is in $ell^2$.

Are there some unostentatious situations on $A$ and on $D(A)$ for these domains to coincide: $D(A^*) = D(A_*)$?

What can breathe mentioned on this signify if $A_{ij}$ is a finite-band matrix?

Or when $A$ is formally self-adjoint ($A_{ij} = overline{A_{ji}})$?

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