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

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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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