Fredholm operator

In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel ${\displaystyle \ker T}$ and finite-dimensional (algebraic) cokernel ${\displaystyle \operatorname {coker} T=Y/\operatorname {ran} T}$, and with closed range ${\displaystyle \operatorname {ran} T}$. The last condition is actually redundant.^[1]

The index of a Fredholm operator is the integer

${\displaystyle \operatorname {ind} T:=\dim \ker T-\operatorname {codim} \operatorname {ran} T}$

or in other words,

${\displaystyle \operatorname {ind} T:=\dim \ker T-\operatorname {dim} \operatorname {coker} T.}$