The Friedrichs extension for elliptic wedge operators of second order

Abstract

Let M be a smooth compact manifold whose boundary is the total space of a fibration N → Y with compact fibers, let E → M be a vector bundle. Let be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of A, the trace bundle of A relative to ν splits as a direct sum I = IF ⊕ IaF and there is a natural map P: C∞(Y;IF) → C∞(M; E) such that CIF∞(M; E) = P(C∞(Y;IF)) + Ċ∞(M;E) ⊂ Dmax(A). It is shown that the closure of A when given the domain CIF∞(M;E) is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.