Definition:Fredholm Operator
Jump to navigation
Jump to search
Definition
Let $U, V$ be vector spaces.
Let $T: U \to V$ be a linear transformation.
$T$ is a Fredholm operator if and only if:
- $(1): \quad \map \ker T$ is finite-dimensional
- $(2): \quad$ the quotient space $V / \Img T$ is finite-dimensional
where:
Also known as
A Fredholm operator is also known as a linear transform of finite index.
Also see
- Definition:Index of Fredholm Operator
- Definition:Pseudoinverse of Linear Transformation
- Linear Transformation is Fredholm Operator iff Pseudoinverse exists
- Results about Fredholm operators can be found here.
Source of Name
This entry was named for Erik Ivar Fredholm.
Sources
- 2002: Peter D. Lax: Functional Analysis: Chapter $27$: Index Theory
- 2006: Bruce Blackadar: Operator Algebras: $\text I.8.3$ Fredholm Theory