# Index of an operator

The difference between the dimensions of the deficiency subspaces (cf. Deficiency subspace) of a linear operator $A\colon L_0\to L_1$, that is, between those of its kernel $\operatorname{Ker}A=A^{-1}(0)$ and its cokernel $\operatorname{Coker}A=L_1/A(L_0)$, if these spaces are finite-dimensional. The index of an operator is a homotopy invariant that characterizes the solvability of the equation $Ax=b$.

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