Numerical-range spectral enclosure
= Numerical-range spectral enclosure
{title2=$\sigma(T)\subseteq\overline{W(T)}$}
For a bounded <self-adjoint operator>, $z$ outside the closed numerical range yields $\|(T-z)u\|\geq\operatorname{dist}(z,\overline{W(T)})\|u\|$. 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.