  In this paper, we combine results on extensions of operators with recent
results on the relation between the M-function and the spectrum, to examine the
spectral behaviour of boundary value problems. M-functions are defined for
general closed extensions, and associated with realisations of elliptic
operators. In particular, we consider both ODE and PDE examples where it is
possible for the operator to possess spectral points that can not be detected
by the M-function.
