TITLE: Isolated point in spectrum
QUESTION [12 upvotes]: "Any isolated point in the spectrum of a self-adjoint operator must be an eigenvalue".
Is there an easy way to see this?
The spectral theorem tells us that any self-adjoint operator is unitarily equivalent to a multiplication operator in $L^2$ for suitable measure space. And I think unitary equivalence preserves the spectrum. So this reduces to studying isolated points in the spectrum of a multiplication operator...but does it help?

REPLY [3 votes]: You may use the characterization of the spectrum set using spectral measure. 
A point on the complex plane belong to then spectrum if and only if any neibourhood of that  point has nonzero spectral measure. 
A point $\lambda$ on the complex plane is an eigenvalue if and only if the set $\{\lambda\}$ has nonzero spectral measure.
So your claim is correct.