  We prove novel results on interpolation of Fredholm operators including an
abstract factorization theorem. The main result of this paper provides
sufficient conditions on the parameters $\theta \in (0,1)$ and $q\in \lbrack
1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real
interpolation space $(X_{0},X_{1})_{\theta ,q}$ to $(Y_{0},Y_{1})_{\theta ,q} $
for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between
compatible pairs of Banach spaces such that its restrictions to the endpoint
spaces are Fredholm operators. These conditions are expressed in terms of the
corresponding indices generated by the $K$-functional of elements from the
kernel of the operator $A$ in the interpolation sum $X_{0}+X_{1}$. If in
addition $q\in \lbrack 1,\infty )$ and $A$ is invertible operator on endpoint
spaces, then these conditions are also necessary. We apply these results to
obtain and present an affirmative solution of the famous Lions-Magenes problem
on the real interpolation of closed subspaces. We also discuss some
applications to the spectral theory of operators as well as to perturbation of
the Hardy operator by identity on weighted $L_{p}$-spaces.
