In this paper, a general approximation theory for the eigenvalues and corresponding subspaces of generalized eigenfunctions of a certain class of compact operators is developed. This theory is then used to obtain rate of convergence estimates for the errors which arise when the eigenvalues of nonselfadjoint elliptic partial differential operators are approximated by Rayleigh-Ritz-Galerkin type methods using finite-dimensional spaces of trial functions, e.g. spline functions. The approximation methods include several in which the functions in the space of trial functions are not required to satisfy any boundary conditions.