PlanetMath: Fredholm operator

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Fredholm operator

(Definition)

A Fredholm operator is a bounded operator between Banach spaces that has a finite dimensional kernel and cokernel (and closed range). Equivalently, it is invertible modulo compact operators. That is, if $F\colon X \to Y$ is a Fredholm operator between two vector spaces and , then there exists a bounded operator $G\colon Y \to X$ such that

$\displaystyle GF-\mathord{\mathrm{1\!\!\!\:I}}_X \in \mathbb{K}(X), \quad FG-\mathord{\mathrm{1\!\!\!\:I}}_Y \in \mathbb{K}(Y),$

(1)

where $\mathbb{K}(X)$ denotes the space of compact operators on

. (Another way to say this is that

is invertible in the Calkin algebra). The set of Fredholm operators $\{F\colon X \to X\}$ is an open subset of the Banach algebra of bounded operators $\{T\colon X \to X\}$ .

If is Fredholm then so is its adjoint, . If $T \in \mathbb{K}(X,Y)$ is a compact operator then is also Fredholm.

"Fredholm operator" is owned by mhale.

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