 Numerical-range spectral enclosure

For a bounded self-adjoint operator, $z$ outside the closed numerical range yields $\Vert (T-z)u\Vert \ge \mathrm{dist}(z,\overline{W(T)})\Vert u\Vert$. The same inequality for $T-\overline{z}$ makes the range dense, and the first makes it closed. Thus $z$ is in the resolvent. The enclosure remains valid for general bounded Hilbert-space operators by the analogous adjoint numerical-range argument.