# 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 $Ax=b$.

#### Comments

The index defined above is also called the analytic index of $A$, 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) |

**How to Cite This Entry:**

Index of an operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Index_of_an_operator&oldid=43452