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
$\mu_n\leq\mu_{n+1}\leq 2\mu_n$ $(n\in{\mathbb Z}).$

1. Department of Mathematics and Mechanics
   Samara State University
   Acad. Pavlov Street
   443011 Samara, Russia
   
2. NICTA
   Locked Bag 8001
   Canberra, 2601 ACT, Australia