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Abstract. We study the adjointness problem for the closed extensions of a general b-elliptic operatorThe case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.

Abstract. We study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent.

Let Y be a smooth connected manifold, Σ ⊂ C an open set and (σ, y) → Py(σ) a family of unbounded Fredholm operators D ⊂ H 1 → H 2 of index 0 depending smoothly on (y, σ) ∈ Y × Σ and holomorphically on σ. We show how to associate to P, under mild hypotheses, a smooth vector bundle K → Y whose fiber over a given y ∈ Y consists of classes, modulo holomorphic elements, of meromorphic elements φ with Pyφ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.2010 Mathematics Subject Classification. Primary: 58J32; Secondary: 58J05,35J48,35J58.