Editing Atiyah–Singer index theorem (section)

==Symbol of a differential operator==
If ''D'' is a differential operator on a Euclidean space of order ''n'' in ''k'' variables <math>x_1, \dots, x_k</math>, then its [[Symbol of a differential operator|symbol]] is the function of 2''k'' variables
<math>x_1, \dots, x_k, y_1, \dots, y_k</math>, given by dropping all terms of order less than ''n'' and replacing <math>\partial/\partial x_i</math> by <math>y_i</math>. So the symbol is homogeneous in the variables ''y'', of degree ''n''. The symbol is well defined even though <math>\partial/\partial x_i</math> does not commute with <math>x_i</math> because we keep only the highest order terms and differential operators commute "up to lower-order terms".  The operator is called '''elliptic''' if the symbol is nonzero whenever at least one ''y'' is nonzero.

Example: The Laplace operator in ''k'' variables has symbol <math>y_1^2 + \cdots + y_k^2</math>, and so is elliptic as this is nonzero whenever any of the <math>y_i</math>'s are nonzero. The wave operator has symbol <math>-y_1^2 + \cdots + y_k^2</math>, which is not elliptic if <math>k\ge 2</math>, as the symbol vanishes for some non-zero values of the ''y''s.

The symbol of a differential operator  of order ''n'' on a smooth manifold ''X'' is defined in much the same way using local coordinate charts, and is a function on the [[cotangent bundle]] of ''X'', homogeneous of degree ''n'' on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (see [[jet bundle]]); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundles ''E'' and ''F'' is a section of the pullback of the bundle Hom(''E'', ''F'') to the cotangent space of ''X''. The differential operator is called ''elliptic'' if the element of Hom(''E<sub>x</sub>'', ''F<sub>x</sub>'') is invertible for all non-zero cotangent vectors at any point ''x'' of ''X''.

A key property of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible.  More precisely, an elliptic operator ''D'' on a compact manifold has a (non-unique) '''[[parametrix]]''' (or '''pseudoinverse''') ''D''′ such that  ''DD′ -1 ''and ''D′D -1'' are both compact operators.  An important consequence is  that the kernel of ''D'' is finite-dimensional, because all eigenspaces of compact operators, other than the kernel, are finite-dimensional.  (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic [[pseudodifferential operator]].)