  We prove the stability of isomorphisms between Banach spaces generated by
interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which
includes, as special cases, the real and complex methods up to equivalence of
norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre
and Gustavsson-Peetre. This result is used to show the existence of solution of
certain operator analytic equation. A by product of these results is a more
general variant of the Albrecht-M\"uller result which states that the
interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation
spaces. We show applications for positive operators between Calder\'on function
lattices. We also derive connections between the spectrum of interpolated
operators.
