Interpolation of subspaces of infinite codimension

Given three normed spaces $X\mathrm{\_}0,X\mathrm{\_}1,\widetilde{X\mathrm{\_}1}$, with $\widetilde{X\mathrm{\_}1}$ a closed subspace of $X\mathrm{\_}1$ with same norm, how does $[X\mathrm{\_}0,X\mathrm{\_}1]\mathrm{\_}\theta$ compare to $[X_0 , X_1 ]_θ,? This classical problem of interpolation theory has been mostly considered in the case of finite codimensional subspaces. We show here that quotient J norms are a useful tool which allows to tackle several relevant cases.