Fredholm Operator

The Fredholm operator is a '' Compact Operator s, i.e., if there exists a bounded linear operator

S

such that

:

\mbox{Id}_X - ST \quad\mbox{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 Equation s 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.

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.

	CATEGORIES ABOUT FREDHOLM OPERATOR

	fredholm theory

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