Abstract. In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M -function see the same singularities as the resolvent of a certain restriction A B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S andS such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M -function is analytic. We present three examples -one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -which together indicate that the abstract results are probably best possible.

We consider the problem of how to compute eigenvalues of a selfadjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires noá priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.