Normal spectrum equals approximate point spectrum
= Normal spectrum equals approximate point spectrum
{title2=$\sigma(T)=\sigma_{\mathrm{ap}}(T)$}
For a normal bounded operator, $\sigma(T)=\sigma_{\mathrm{ap}}(T)$. If $T-\lambda I$ is bounded below, its range is closed; normality makes its adjoint injective, so the range is also dense and the operator is invertible.