Nonreal resolvent estimate for a self-adjoint operator
= Nonreal resolvent estimate for a self-adjoint operator
{title2=$\|(A-z)^{-1}\|\leq|\operatorname{Im}z|^{-1}$}
Symmetry gives $\|(A-z)u\|\geq|\operatorname{Im}z|\|u\|$ 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.