Isomorphisms on interpolation spaces generated by the method of means

We investigate the stability of isomorphisms acting between interpolation spaces generated by the method of means. We focus on the methods which are determined by balanced sequences for non-degenerate quasi-concave functions. The key point for our investigation is that these methods have orbital description by a single element generated by a special ideal of operators between Banach couples. We prove that if an operator is invertible in one orbit it is also invertible by nearby orbits provided that the corresponding indices of quasi-concave functions generated these orbits are close to each other. In particular, these results apply to the real method of interpolation