The invertible double of elliptic operators

TL;DR

This paper reviews the properties of Dirac operators on manifolds, introduces a new construction of invertible doubles for elliptic operators, and explores implications for spectral flow and boundary projections.

Contribution

It presents a novel construction of a canonical invertible double for general elliptic operators and derives formulas for the Calderon projection, extending classical results.

Findings

Generalization of the Cobordism Theorem

New spectral flow theorems for elliptic operators

Continuity results for Calderon projections under coefficient variation

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 Calderon 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 Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon 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 re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case.