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.

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Abstract. This paper discusses the inverse problem of how much information on an operator can be determined/detected from 'measurements on the boundary'. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of the operator 'visible' from 'boundary measurements').We show results in an abstract setting, where we consider the relation between the Mfunction (the abstract Dirichlet to Neumann map or the transfer matrix in system theory) and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum.The abstract results are illustrated by examples of Schrödinger operators, matrixdifferential operators and, mostly, by multiplication operators perturbed by integral operators (the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator.1. Introduction. In this paper we consider inverse problems in a boundary triple setting involving a formally adjoint pair of operators A and A [35]. We define, and develop formulae for, the detectable subspaces associated with the information available from the abstract Dirichlet to Neumann maps or Titchmarsh-Weyl functions M (λ) (see Definitions 2.2 and 2.7). Our focus is on non-selfadjoint operators, but some of the results are new even in the symmetric case.In the formally symmetric case V. Derkach and M. Malamud [21] (see also Ryzhov [44]) show that if the reducing subspace corresponding to the simple part of the operator (which is a special case of the detectable subspace) is the whole Hilbert space, then the operator can be reconstructed up to unitary equivalence. In terms of the Q-function, this result was proved earlier by

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a twodimensional periodic medium; the defect is infinitely extended and aligned with one of the coordinate axes. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that guided mode spectrum can be created by arbitrarily 'small' perturbations. Secondly, we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum k x , the perturbation generates at most finitely many new eigenvalues inside the gap.