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'' → ''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.
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.

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References

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.