Fredholm operator
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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 → X
such that
- <math> \mbox{Id}_X - ST \quad\mbox{and}\quad \mbox{Id}_Y - TS </math>
are compact operators on X and Y respectively.
The index of a Fredholm operator is
- <math> \mbox{ind}\,T = \dim \ker T - \mbox{codim}\,\mbox{ran}\,T </math>
(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.
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- 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.de:Fredholm-Operator