Real method of interpolation on subcouples of codimension one

We find necessary and sufficient conditions under which the norms of the interpolation spaces (N₀, N₁)_θ,q and (X₀ X₁) _θq are equivalent on N, where N is the kernel of a nonzero functional ψ ε (X₀ ∩ X ₁)* and N_i is the normed space N with the norm inherited from X_i (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an applicatin we completely resolve the problem of when the range of the operator T_θ = S - 2 ^θ (S denotes the shift operator and I the identity) is closed in any ℓ_p(μ), where the weight μ; = (μ_n) _nεℤ satisfies the inequalities μ_n ≤ μ_n+1 ≤ 2μ_n (n ε ℤ).