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.

Key words Closed extension, M -function, abstract boundary spaces, boundary triplets, elliptic PDEs, pseudodifferential boundary operators, essential spectrum MSC (2000) 35J25, 35J30, 35J55, 35P05, 47A10, 47A11 Dedicated to the memory of Leonid R. VolevichIn 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 cannot be detected by the M -function.

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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.

Abstract. We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following two domains of definition:, where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positve analytic contraction semigroups on L p (Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and 1 < p ≤ 2, the Laplacian with domain D 2 (∆) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that proves that, in general, the Dirichlet-Laplacian with domain D 1 (∆) is not even a closed operator.