TITLE: Why do we distinguish the continuous spectrum and the residual spectrum?
QUESTION [23 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. 
I want to know why do we care about whether $R(aI-T)$ is dense, thanks.

REPLY [9 votes]: As pointed out by M. Reed, B. Simon in the section VI.3 of their "Methods of Modern Mathematical Physics: Functional Analysis":

The reason that we single out the residual spectrum is that it does not occur for a large class of operators, for example, for self-adjoint operators.