Multiplication operator ($M_{f}u=fu$)

For a finite almost-everywhere measurable function $f$, the multiplication operator on L2 space has maximal operator domain $D(M_f)=\{u\in L^2:fu\in L^2\}$. It is a closed operator: if $u_n\to u$ and $f{u}_n\to v$ in L2 space, subsequences converge almost everywhere, giving $v=fu$. Its domain is dense, since truncating a function to the sets $|f|\le n$ approximates it in L2 space. If $f$ is bounded, this is a bounded linear operator of norm equal to the essential supremum of $|f|$.

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 $\{|m|\le n\}$ 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.