The following is Exercise 9.iii on a write-up by Terence Tao, which I have been unable to fully crack:
Let $H$ be a (complex) Hilbert space with dense subspace $D$. Let $L:D\to H$ be a symmetric closed positive operator, and let $w\in\Bbb C\setminus\Bbb R_{\ge0}$. Then the resolvent $R(w)$ is defined on all of $H$ (i.e. $L-wI$ is surjective) if and only if $L$ is self adjoint.
Now, it is relatively straightforward to prove that the resolvent $R(w)$ exists for $L$, and the condition of $L$ being self-adjoint is equivalent to requiring that for any $g\in H$, if the functional $$D\to\Bbb C, \quad f\mapsto\langle Lf,g\rangle$$ is bounded, then $g\in D$.
However, no amount of finicky manipulation seems to get me closer to a proof that this is equivalent to $L-w$ being surjective.