Necessary and sufficient conditions for invertibility of operators in spaces of real interpolation

Let A be a bounded linear operator from a couple (X0,X1) to a couple (Y0,Y1) such that the restrictions of A on the end spaces X0 and X1 have bounded inverses defined on Y0 and Y1, respectively. We are interested in the problem of how to determine if the restriction of A on the space (X0,X1)θ,q has a bounded inverse defined on the space (Y0,Y1)θ,q. In this paper, we show that a solution to this problem can be given in terms of indices of two subspaces of the kernel of the operator A on the space X0+X1.