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.


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