Real method of interpolation on subcouples of codimension one

Abstract

We find necessary and sufficient conditions under which the norms of the interpolation spaces (N0,N1)θ,q and (X0,X1)θ,q are equivalent on N, where N is the kernel of a nonzero functional ψ∈(X0∩X1)∗ and Ni is the normed space N with the norm inherited from Xi (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 application we completely resolve the problem of when the range of the operator Tθ=S−2θI (S denotes the shift operator and I the identity) is closed in any ℓp(μ), where the weight μ=(μn)n∈Z satisfies the inequalities μn≤μn+1≤2μn (n∈Z).