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 $X$ and $Y$ , then there exists a bounded operator $G\colon Y \to X$ such that \begin{equation} GF-\identity_X \in \Kset(X), \quad FG-\identity_Y \in \Kset(Y), \end{equation}where $\Kset(X)$ 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 $\{F\colon X \to X\}$ is an open subset of the Banach algebra of bounded operators $\{T\colon X \to X\}$ .

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