Invertibility of operators in spaces of real interpolation

Let A be a linear bounded operator from a couple (X) over right arrow = (X-0, X-1) to a couple (Y) over right arrow = (Y-0, Y-1) such that the restrictions of A on the spaces X-0 and X-1 have bounded inverses. This condition does not imply that the restriction of A on the real interpolation space (X-0, X-1)(theta,q) has a bounded inverse for all values of the parameters theta and q. In this paper under some conditions on the kernel of A we describe all spaces (X-0, X-1)(theta,q) such that the operator A : (x(0), X-1)(theta,q) -> (Y-0, Y-1) has a bounded inverse.