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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, Fredholm theory is a theory of integral equations. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Erik Ivar Fredholm (April 7, 1866 - August 17, 1927) was a Swedish mathematician who established the modern theory of integral equations. ...

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 In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, the range of a function is the set of all output values produced by that function. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... The word modulo is the Latin ablative of modulus. ... In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a...

S: Y → X

such that

are compact operators on X and Y respectively.

The index of a Fredholm operator is

(see dimension, kernel, codimension, and range). :For other senses of this word, see dimension (disambiguation). ... It has been suggested that this article or section be merged into kernel (mathematics). ... In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

The index of T remains constant under compact perturbations of T. The Atiyah-Singer index theorem gives a topological characterization of the index. In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. ...

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. In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

There is "an index interpretation" for the number of closed trajectories of a Lienard vector field as in this arxived note.It suggests a possible relation between the second part of the Hilbert 16th problem and index theory:

http://arxiv.org/abs/math.DS/0408037

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
• Weisstein, Eric W., Fredholm's Theorem at MathWorld.
• B.V. Khvedelidze, "Fredholm theorems" SpringerLink Encyclopaedia of Mathematics (2001)
• 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.