= Spectrum of a real multiplication operator
{title2=$\sigma(M_m)=\operatorname{essran}_\mu m$}
For a real <measurable function> $m$ on a <sigma-finite measure> space, multiplication on $D(M_m)=\{f\in L^2:mf\in L^2\}$ is an <unbounded self-adjoint operator>. Testing its adjoint relation on $\{|m|\leq n\}$ identifies the same maximal domain. Outside the <essential range>, multiplication by $(m-z)^{-1}$ is a bounded resolvent. Inside it, normalized indicators of finite-positive-measure sets where $m$ is close to $z$ are <approximate eigenvectors>, so the inverse cannot be bounded.
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