Fredholm operator Guide, Meaning , Facts, Information and Description

In mathematics, a Fredholm operator is a bounded linear operator between two Hilbert spaces whose range is closed and whose kernel and cokernel are finite-dimensional. Equivalently, an operator T: H₁→H₂ is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

S: H₂→H₁

such that

Id_H1 − ST and Id_H2 − TS

are compact operators on H₁ and H₂ respectively.

A Fredholm operator has a well-defined index, which remains constant under continuous deformation of the operator itself. An elliptic differential operator can be extended to a Fredholm operator. The Atiyah-Singer index theorem gives a topological characterization of the index. The use of Fredholm operators in PDE theory is an abstract form of the parametrix method.

See also

• Integral equation
• Fredholm integral equation

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