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Abstract. Let P be a classical pseudodifferential operator of order m ∈ C on an ndimensional C ∞ manifold Ω 1 . For the truncation P Ω to a smooth subset Ω there is a wellknown theory of boundary value problems when P Ω has the transmission property (preserves C ∞ (Ω)) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian (−∆) µ with µ / ∈ Z, are not covered. They have instead the µ-transmission property defined in Hörmander's books, mapping x µ n C ∞ (Ω) into C ∞ (Ω). In an unpublished lecture note from 1965, Hörmander described an L 2 -solvability theory for µ-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in L p Sobolev spaces (1 < p < ∞) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces (s → ∞). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Hölder spaces, which radically improve recent regularity results for fractional Laplacians.

This paper introduces a class of pseudodifferential operators depending on a parameter in a particular way. The main application is a complete expansion of the trace of the resolvent of a Dirac-type operator with nonlocal boundary conditions of the kind introduced by Atiyah, Patodi, and Singer [APS]. This extends the partial expansion in [G2] to a complete one, and extends the complete expansion in [GS 1 ] to the case where the Dirac operator does not have a product structure near the boundary. A secondary application is to obtain a complete expansion of the resolvent of a ~bdo on a compact manifold, essentially reproving a result of Agranovich [Agr]. The resolvent expansion yields immediately an expansion of the trace of the heat kernel, and determines the singularities of the zeta function; moreover, a pseudodifferential factor can be allowed.A major motive for these expansions is to obtain index formulas for elliptic operators; there are many such applications in the physics and geometry literature. The index formula comes from one particular term in the expansion, but each term is a spectral invariant, and they have been used for other purposes as well as for the index. In particular, Branson and Gilkey have a number of papers (e.g.[BG] and [Gi]) analyzing these invariants, and drawing geometric consequences.Interest in the asymptotic behavior of the resolvent goes back to Carleman [C]. More recently, Agmon [Agm] developed it extensively for analytic applications; he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable. This idea was developed in [S1] to analyze the singularities of the zeta function of an elliptic Odo on a compact manifold, and in [$3] to analyze the resolvent of a differential operator with differential boundary conditions. The technique works smoothly for differential operators, producing so-called local invariants, integrals over the underlying * Work partially supported by NSF grant DMS-9004655. 482G. Grubb, R.T. Seeley manifold of densities defined locally by the symbol of the operator involved. But in the pseudodifferential case it only goes so far. In particular, [APS] identifies an interesting nonlocal term in the expansion, contributing to the index formula. Grubb [G2] has studied carefully how far the Agmon approach does go, in a framework of pseudodifferential boundary problems, obtaining an expansion up to and including the first nonlocal term. The present paper modifies the technique, thus yielding the complete expansion, with a full sequence of nonlocal terms, and logarithmic terms as noticed by [DG] in the case of a tpdo on a compact manifold.A simple example illustrates the modification. Suppose that a(x, ~) is the leading symbol of a first-order elliptic system A. Then the first term in the Sdo expansion of the resolvent (A -2) -1 has symbol (a -2) -1. If a is a polynomial in ~, then each ~-derivative of (a-2) -1 increases the rate of decay as 2 ~ oo; this is an important property of the class of Sdo's with parameter ...

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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.