# To find a symmetric operator that is not essentially self-adjoint under some conditions.

There is a theorem saying:

If a symmetric operator $$A$$ (in the Hilbert space $$h$$) satisfies $$Ran(A+z) = Ran(A+\bar z) = h,$$ for some $$z\in\mathbb{C}$$, then $$A$$ is self-adjoint.

However, this theorem implies that when symmetric operator $$A$$ satisfies merely that $$Ran(A+z)$$ and $$Ran(A+\bar z)$$ are $$\textit{dense}$$ in $$h$$, then $$A$$ is not necessarily self-adjoint. In fact, I suspect that $$A$$ is neither $$\textit{essentially self-adjoint}.$$ In that case, $$D(A)$$ cannot be dense.

To supplement my reasoning, I want to find proper $$\textit{example}$$ of it. Are there well-known examples? It would be nice if someone could help with this. Thanks.