$A - \lambda I$ surjective for self adjoint $A: D(A) \rightarrow H $, $\lambda \in \mathbb C \setminus \mathbb R$

Question

I want an argument for why this is true, implying that such lambda are in the resolvent set.

I have tried to construct pathological examples in order to understand this fact, but I still don't get it.

I suspect the solution is embarrassingly simple.

Answer 1

If $A$ is self-adjoint, then then the spectrum of $A$ is contained in $ \mathbb R$. You can find this result in nearly every book on operator theory.