  We establish spectral convergence results of approximations of unbounded
non-selfadjoint linear operators with compact resolvents by operators that
converge in generalized strong resolvent sense. The aim is to establish general
assumptions that ensure spectral exactness, i.e. that every true eigenvalue is
approximated and no spurious eigenvalues occur. A main ingredient is the
discrete compactness of the sequence of resolvents of the approximating
operators. We establish sufficient conditions and perturbation results for
strong convergence and for discrete compactness of the resolvents.
