Continuous and residual spectrum

Summary

The continuous and residual spectra are the parts of the spectrum of an operator that consist of numbers which are not eigenvalues. At such a $\lambda$, the operator $T - \lambda I$ is injective but not invertible. The spectrum is continuous at $\lambda$ when the range is dense and residual when it is not. Together with the point spectrum, they split the spectrum into three disjoint pieces.