The Friedrichs extension for elliptic wedge operators of second order

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 = I_F ⊕ I_aF and there is a natural map P: C^∞(Y;I_F) → C^∞(M; E) such that C_IF^∞(M; E) = P(C^∞(Y;I_F)) + Ċ^∞(M;E) ⊂ D_max(A). It is shown that the closure of A when given the domain C_IF^∞(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.