Index of an operator

The difference between the dimensions of the deficiency subspaces (cf. Deficiency subspace) of a linear operator , that is, between those of its kernel and its cokernel , if these spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation .

Comments

The index defined above is also called the analytic index of , cf. Index formulas.

An important case, in which the index is well defined and is a homotopy invariant, is that of elliptic partial differential operators acting on sections of vector bundles over compact manifolds.

One can also define the index of, e.g., a linear Fredholm operator between Banach spaces, of an elliptic boundary value problem and of an "almost" pseudo-differential operator (cf. also [a1]).

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)