Real method of interpolation on subcouples of codimension one

S. V. Astashkin ¹   ; P. Sunehag ²

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

Studia Mathematica, Tome 185 (2008) no. 2, pp. 151-168

Résumé

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

Détail

DOI : 10.4064/sm185-2-4

Keywords: necessary sufficient conditions under which norms interpolation spaces theta theta equivalent where kernel nonzero functional psi cap * normed space norm inherited proof based reducing problem its partial studied ivanov kalton where psi bounded endpoint spaces application completely resolve problem range operator theta s theta denotes shift operator identity closed ell where weight mathbb satisfies inequalities leq leq mathbb