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 θ ∈ ( 0 , 1 ) and q ∈ [ 1 , ∞ ] under which an operator A is a Fredholm operator from the real interpolation space ( X 0 , X 1 ) θ , q to ( Y 0 , Y 1 ) θ , q for a given operator A : ( X 0 , X 1 ) → ( 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 A is invertible operator on endpoint spaces, then these conditions are also necessary. We apply these results to present a solution of the variant of a long-standing problem of Lions–Magenes for the real interpolation method. 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.