Fredholm operator

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 $F:X\to Y$ is a Fredholm operator between two vector spaces $X$ and $Y$,then there exists a bounded operator $G:Y\to X$ such that

$$GF-{1\mathrm{I}}_X\in 𝕂(X),FG-{1\mathrm{I}}_Y\in 𝕂(Y),$$(1)

where $𝕂(X)$ denotes the space of compact operators on $X$.(Another way to say this is that $F$ is invertible in the Calkin algebra).The set of Fredholm operators $\{F:X\to X\}$ is an open subset of the Banach algebra of bounded operators $\{T:X\to X\}$.

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