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.

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

:$S:\; Y\; o\; X$

such that

:$mbox\{Id\}\_X\; -\; ST\; quadmbox\{and\}quad\; mbox\{Id\}\_Y\; -\; TS$

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 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.

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, " [http://www.math.ksu.edu/~ramm/papers/419amm.pdf A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators] ", "American Mathematical Monthly", 108 (2001) p. 855.
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* Bruce K. Driver, " [http://math.ucsd.edu/~driver/231-02-03/Lecture_Notes/compact.pdf Compact and Fredholm Operators and the Spectral Theorem] ", "Analysis Tools with Applications", Chapter 35, pp. 579-600.
* Robert C. McOwen, " [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102780323 Fredholm theory of partial differential equations on complete Riemannian manifolds] ", "Pacific J. Math." 87, no. 1 (1980), 169–185.