Numerical-range spectral enclosure ($\sigma (T)\subseteq \overline{W(T)}$)

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.

For bounded self-adjoint $T$, put $B=\beta I-T\ge 0$ with $\beta =\sup W(T)$. Unit vectors with $⟨B{u}_n,u_n⟩\to 0$ satisfy $\Vert B{u}_n{\Vert }^2\le \Vert B\Vert ⟨B{u}_n,u_n⟩$ by positive-form Cauchy-Schwarz inequality. Thus $\beta$ is an approximate eigenvalue. Applying the same argument to $T-\alpha I$ proves the result at $\alpha =\inf W(T)$.