Ellipticity with Global Projection Conditions

Abstract

We give an overview of elliptic operators on a compact smooth manifold with boundary or edge, with elliptic boundary or edge conditions in global projection approach, introduced in Schulze (J Funct Anal 179:374–408, 2001) and then continued in Schulze and Seiler (J Funct Anal 206(2):449–498, 2004; J Inst Math Jussieu 5(1):101–123, 2006). Such conditions are motivated by the fact that important elliptic operators do not admit Shapiro-Lopatinskii elliptic conditions, though they always admit global projection conditions. Basically Shapiro-Lopatinskii conditions are a special case, such that the global projection idea unifies different concepts. There is a similarity to Toeplitz operators whence it also makes sense to talk about boundary or edge problems of Toeplitz type. Another stream of investigations goes back to Atiyah-Patodi-Singer (Math Proc Camb Philos Soc 77/78/79:43–69/405–432/315–330, 1975/1976/1976), though the analytical ideas and intentions from there are quite different. In our approach we focus on the aspect of operator algebras in scales of Sobolev spaces or subspaces induced by pseudo-differential projections, on parametrices within those structures, and on the role of principal symbolic hierarchies coming from the singular analysis. Ellipticity with global projection conditions in the singular analysis is also a source of new challenges. A few of them are indicated in the final part of this exposition.