Nonreal resolvent estimate for a self-adjoint operator ($\Vert (A-z{)}^{-1}\Vert \le |\mathrm{Im}z{|}^{-1}$)

Symmetry gives $\Vert (A-z)u\Vert \ge |\mathrm{Im}z|\Vert u\Vert$ when $z$ is nonreal. Closedness makes the range closed; its orthogonal complement is $\ker (A-\overline{z})=0$, so it is also dense and hence surjective. This proves the resolvent estimate and reality of the spectrum. Equality of the adjoint domains is needed in the range argument.