TITLE: Why do we distinguish the continuous spectrum and the residual spectrum?
QUESTION [6 upvotes]: As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension. 
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ is injective, and $R(aI-T)$ is not $X$. If $R(aI-T)$ is dense in $X$, then $a$ belongs to the continuous specturm, it not, $a$ belongs to the residual spectrum. 
1) I want to know why do we care about whether $R(aI-T)$ is dense.
2) It seems as we just emphasize this difference for the operator, but not for the Banach algebra, why?

REPLY [4 votes]: For $\lambda$ in the continuous spectrum, while there may not be genuine eigenvectors, there are approximate eigenvectors.
That point is related to the fact that (as Bob Israel mentions) normal operators on Hilbert spaces, which do have a spectral theorem, do not have residual spectrum. It is only slightly frivolous to observe that an operator on Hilbert space with residual spectrum could not admit a reasonable spectral theorem. 
(By "reasonable spectral theorem" we need not require decomposition into actual eigenvectors, but something much weaker, than can be described in various ways.)