The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in the mid-70s, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced operators adapted to the boundary taking centre stage in formulating and understanding non-local boundary conditions.

That being said, much of this analysis has been confined to the situation when adapted boundary operators can be chosen self-adjoint. Dirac-type operators are the quintessential example. Nevertheless, natural geometric operators such as the Rarita-Schwinger operator on 3/2-spiniors, arising from physics in the study of the so-called Delta baryon, falls outside of this class. Analytically, this requires analysis beyond self-adjoint operators. In recent work with Bär, the compact boundary case is handled for general first-order elliptic operators, using spectral theory to choose adapted boundary operators to be invertible bi-sectorial. The Fourier circle methods present in the self-adjoint analysis are replaced by the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory and semi-group techniques. This allows for a full understanding of the maximal domain of the interior operator as a bounded surjection to a space on the boundary of mixed Sobolev regularity, constructed from spectral projectors associated to the adapted boundary operator. Regularity and Fredholm extensions are also studied.

For the noncompact case, a preliminary trace theorem as well as regularity theory are handed by resorting to the case with compact boundary. This necessitates deforming the coefficients of the interior operator in a compact neighbourhood. Therefore, even for Dirac-type operators, allowing for fully general symbols in the compact boundary case is paramount. Under slightly stronger geometric assumptions near the noncompact boundary (automatic for the compact case) and when the interior operator admits a self-adjoint adapted boundary operator, an upgraded trace theorem mirroring the compact setting is obtained. Importantly, there is no spectral assumptions other than self-adjointness on the adapted boundary operator. This, in particular, means that the spectrum of this operator can be the entire real line. Again, the primarily tool that is used in the analysis is the bounded holomorphic functional calculus.