A Fredholm theory for a class of first-order elliptic partial differential operators in ⁿ

The objects of interest are linear first-order elliptic partial differential operators with domain //i(/?" ; C*) in L2(Rn;Ck), the first-order coefficients of which become constant and the zero-order coefficient of which vanishes outside a compact set in R". 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. 0. Introduction. As usual, let L2(Rn; Ck) denote the Hubert space of equivalence classes of C-valued functions on Rn whose absolute values are Lebesguesquare-integrable over Rn. Let 7/i(Än; Ck) denote the Hubert space consisting of those elements of L2(Rn; Ck) which have (strong) first partial derivatives in L2(Rn; Ck). Denote the usual norms on L2(Rn; Ck) and H1(Rn; Ck) by || || and ||i, respectively. Consider a linear first-order partial differential operator " d A0u(x) = 2 ^i äF "(*) with domain H^R"; Ck) which has constant coefficients and no zero-order term. Suppose that A0 is elliptic in the sense that