The point spectrum is the set of eigenvalues of . If is an eigenvalue, with eigenfunction f, then . So for almost all t in X. Thus almost everywhere. But f is an eigenfunction, so it cannot be 0 almost everywhere. Therefore there is a set of positive measure on which . Hence .
The approximate point spectrum is probably harder to describe. I don't offhand know what it is, but I would guess that it is the set of complex numbers such that for every . See if you can prove that!