Fredholm operator
From Wikipedia, the free encyclopedia
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.
A 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
such that
are compact operators on X and Y respectively.
The index of a Fredholm operator is
(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 of certain operators on manifolds.
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.
[edit] 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 on PlanetMath
- Eric W. Weisstein, Fredholm's Theorem at MathWorld.
- B.V. Khvedelidze (2001), “Fredholm theorems”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- 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.
