Fredholm solvability

In mathematics, Fredholm solvability encompasses results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved. The concept is named after Erik Ivar Fredholm.

Let A be a real n × n-matrix and ${\displaystyle b\in \mathbb {R} ^{n}}$ a vector.

The Fredholm alternative in ${\displaystyle \mathbb {R} ^{n}}$ states that the equation ${\displaystyle Ax=b}$ has a solution if and only if ${\displaystyle b^{T}v=0}$ for every vector ${\displaystyle v\in \mathbb {R} ^{n}}$ satisfying ${\displaystyle A^{T}v=0}$. This alternative has many applications, for example, in bifurcation theory. It can be generalized to abstract spaces. So, let ${\displaystyle E}$ and ${\displaystyle F}$ be Banach spaces and let ${\displaystyle T:E\rightarrow F}$ be a continuous linear operator. Let ${\displaystyle E^{*}}$, respectively ${\displaystyle F^{*}}$, denote the topological dual of ${\displaystyle E}$, respectively ${\displaystyle F}$, and let ${\displaystyle T^{*}}$ denote the adjoint of ${\displaystyle T}$ (cf. also Duality; Adjoint operator). Define

${\displaystyle (\ker T^{*})^{\perp }=\{y\in F:(y,y^{*})=0{\text{ for every }}y^{*}\in \ker T^{*}\}}$

An equation ${\displaystyle Tx=y}$ is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever ${\displaystyle y\in (\ker T^{*})^{\perp }}$. A classical result states that ${\displaystyle Tx=y}$ is normally solvable if and only if ${\displaystyle T(E)}$ is closed in ${\displaystyle F}$.

In non-linear analysis, this latter result is used as definition of normal solvability for non-linear operators.