This short review and work program is dedicated to the memory of Krzysztof P. Wojciechowski (1953Wojciechowski ( -2008, who was a leader of the investigation of spectral invariants of Dirac type operators for almost 30 years.Abstract. First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderón projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderón projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderón projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.
IntroductionThis paper reviews our recent results, obtained jointly with Chaofeng Zhu [7], about basic analytical properties of elliptic operators on compact manifolds with smooth boundary. Furthermore, we outline a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.Our main results are − to develop the basic elliptic analysis in full generality, and not only for the generic case of operators of Dirac type in product metrics (i.e., we assume neither constant coefficients in normal direction nor symmetry of the tangential operator); − to give an analytical proof of the cobordism invariance of the index in greatest generality; and − to prove the continuity of the Calderón projection and of related families of global elliptic boundary value problems under parameter variation.