Elliptic symbols and Fredholm operators

Let M be a smooth compact manifold without boundary and E,F smooth finite-rank complex Hermitian bundles. A differential operator P of positive order m extends continuously from H^s(M;E) to H^(s−m)(M;F). Our symbol convention replaces each derivative ∂_j by iξ_j in its highest-order terms. Ellipticity means this bundle map is invertible at every nonzero real covector. It concerns high-frequency directions and does not say that P itself has no kernel.

For example, the symbol of −i∂θ on a circle is ξ, which is nonzero away from the zero covector, but constant functions remain in the kernel. Lower-order terms can move these exceptional modes without changing the principal symbol. Conversely, checking the determinant of the coefficient at one point is insufficient on a manifold: invertibility must hold in every base point and every nonzero cotangent direction, with the bundles and coordinate transformations accounted for.

Elliptic pseudodifferential calculus constructs an operator Q of order −m such that QP and PQ equal the appropriate identities modulo smoothing operators. On a compact manifold those remainders act compactly on the relevant Sobolev spaces. Atkinson's theorem then makes P Fredholm: its kernel is finite dimensional, its image is closed, and its cokernel is finite dimensional. Closed range is part of the conclusion and cannot be omitted from a general definition of Fredholmness.

Elliptic regularity makes homogeneous solutions smooth, so the kernel and index do not depend on the chosen Sobolev exponent s. The cokernel is naturally identified with the smooth kernel of the formal adjoint using the Hermitian metrics, density and distributional pairing. At the convenient realization H^m to L² this is the L² adjoint equation, with the correct adjoint domain. One must not simply subtract dimensions of unrelated null spaces obtained from independently chosen boundary conditions.

On a noncompact manifold an invertible principal symbol alone gives no global Fredholm theorem. Spectrum can accumulate near zero through behavior at infinity, and compact Sobolev embeddings can fail. On a compact manifold with boundary, the interior symbol likewise does not specify a Fredholm problem. The trace space and an elliptic boundary condition supply additional data. Later lessons use spectral boundary conditions, whose global nature is precisely what permits certain Dirac problems to be Fredholm.

A formally self-adjoint elliptic operator on a closed manifold has equal kernel and cokernel dimensions and therefore index zero. Interesting nonzero indices usually arise by separating a graded operator into a map from its positive bundle to its negative bundle. Even when the numerical index vanishes, neither invertibility nor the absence of solutions follows. The circle example provides a calibration: a zero index can accompany either a one-dimensional kernel or no kernel at all.