Definition:Index of Fredholm Operator

Definition

Let $U, V$ be vector spaces.

Let $T: U \to V$ be a Fredholm operator.

The index of $T$ is defined as:

$\map {\mathrm{ind} } T := \map \dim {\map \ker T} - \map {\mathrm {codim}} {\Img T}$

where:

$\map \dim {\map \ker T}$ denotes the dimension of the kernel of $T$

$\map {\mathrm {codim}} {\Img T}$ denotes the codimension of the image of $T$

Also see

• Linear Transformation is Fredholm Operator iff Pseudoinverse exists
• Definition:Pseudoinverse of Linear Transformation

Sources

• 2002: Peter D. Lax: Functional Analysis: Chapter $27$: Index Theory