Interpolation of Fredholm operators

Abstract

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.