Fredholm operator

In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm.

The Fredholm operator is a bounded linear operator between two Banach spaces whose range is closed and whose kernel and cokernel are finite-dimensional. Equivalently, an operator T : X → Y is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

$S:Y\to X$

such that

${\mbox{Id}}_{X}-ST\quad {\mbox{and}}\quad {\mbox{Id}}_{Y}-TS$

are compact operators on X and Y respectively.

The index of a Fredholm operator is

${\mbox{ind}}\,T=\dim \ker T-{\mbox{codim}}\,{\mbox{ran}}\,T$

(see dimension, kernel, codimension, and range).

The index of T remains constant under compact perturbations of T. The Atiyah-Singer index theorem gives a topological characterization of the index.

An elliptic operator can be extended to a Fredholm operator. The use of Fredholm operators in partial differential equations is an abstract form of the parametrix method.

References

• D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2.
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855.
• "Fredholm operator". PlanetMath.
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.
• B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics, EMS Press
• Bruce K. Driver, "Compact and Fredholm Operators and the Spectral Theorem", Analysis Tools with Applications, Chapter 35, pp. 579-600.
• Robert C. McOwen, "Fredholm theory of partial differential equations on complete Riemannian manifolds", Pacific J. Math. 87, no. 1 (1980), 169–185.