Real method of interpolation on subcouples of codimension one
Volume 185 / 2008
Abstract
We find necessary and sufficient conditions under which the norms of the interpolation spaces $(N_0,N_1)_{\theta,q}$ and $(X_0,X_1)_{\theta,q}$ are equivalent on $N,$ where $N$ is the kernel of a nonzero functional $\psi\in (X_0\cap X_1)^*$ 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 $\psi$ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_\theta=S-2^\theta I$ ($S$ denotes the shift operator and $I$ the identity) is closed in any $\ell_p(\mu),$ where the weight $\mu=(\mu_n)_{n\in{\mathbb Z}}$ satisfies the inequalities $\mu_n\leq\mu_{n+1}\leq 2\mu_n$ $(n\in{\mathbb Z}).$
