Fredholm operator
|
|
In mathematics, 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 → 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>
The name is for Erik Ivar Fredholm. He worked on integral equations. A continuous kernel function on a closed interval on the real line gives rise to a compact integral operator, as follows from basic results on equicontinuity. The abstract structure of Fredholm's theory (if not the computational aspects) can now be derived in terms of the spectral theory of Fredholm operators on Hilbert space. This reverses history, in the sense that David Hilbert abstracted 'Hilbert space' in association with research on integral equations prompted by Fredholm's (amongst other things).
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.
See also
References
- D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2.