A Fredholm theory for a class of first-order elliptic partial differential operators in $\textbf {R}^{n}$

Abstract:

The objects of interest are linear first-order elliptic partial differential operators with domain ${H_1}({R^n};{C^k})$ in ${L_2}({R^n};{C^k})$, the first-order coefficients of which become constant and the zero-order coefficient of which vanishes outside a compact set in ${R^n}$. It is shown that operators of this type are “practically” Fredholm in the following way: Such an operator has a finite index which is invariant under small perturbations, and its range can be characterized in terms of the range of an operator with constant coefficients and a finite index-related number of orthogonality conditions.