Boundary value problems for general first-order elliptic differential operators

The relative index theorem for general first-order elliptic operators

“…A key result from [7], generalising the results of [6], is the following. Theorem 2.4 (Theorem 2.9 [7]).…”

“…A key result from [7], generalising the results of [6], is the following. Theorem 2.4 (Theorem 2.9 [7]). A subspace B is an elliptically regular boundary condition, i.e., B ⊂ H 1 2 (∂M, E) and B † ⊂ H 1 2 (∂M, F) if and only if for some (and hence all) invertible bisectorial adapted boundary operators A, B is L 2 -graphically decomposable with respect to χ + (A).…”

“…This automatically self-improves in regularity: for complete D, we obtain C ∞ c (M, E) is dense in dom(D max ). See Theorem 2.3 (i) in [7].…”

“…In [7], two important assertions are made. The first is that is that A exists, and can always be chosen invertible ω-bisectorial (the spectrum sits in a bisector in the complex plane containing the real line, and there are resolvent estimates of the form (ζ − A) −1 ≤ C outside of this bisector).…”

“…It is an operator which fails to be of Dirac-type and is perhaps the quintessential example of a non Dirac-type operator. In [7], the authors generalised the equivalence between elliptic regularity for a boundary condition and the admission of a graphical decomposition. This set the stage to consider relative index theorems for operators beyond those that are Diractype.…”