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Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl  M ‐function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the  M ‐function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the  M ‐function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the  M ‐function corresponds to the Dirichlet to Neumann map.

Boundary triplets and M ‐functions for non‐selfadjoint operators, with applications to elliptic PDEs and block operator matrices