Interpolation problem for $\ell^1$ and an $F$-space

Abstract

Let $B$ be an $F$-space and $B^\ast _1$ the unit ball of the dual space. A sequence $(\phi _n)$ in $B^\ast _1$ is called $\ell ^1$-interpolating if for every sequence $(w_n)$ in $\ell ^1$ there exists an element $f$ in $B$ such that $\phi _n(f)=w_n$ for all $n$. In order to study an interpolation problem for $\ell ^1$, we introduce two quantities $\rho _n$ and ${\prod_{k\ne n}}\sigma (\phi _n,\phi _k)$. For arbitrary Banach space, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n}\rho _n>0$. Moreover, when a Banach space has a predual, we show that if ${\inf_n\prod_{k\ne n}}\sigma(\phi_n, \phi_k)>0$ then $(\phi_n)$ is an $\ell^1$-interpolating sequence. When $(\phi _n)$ is embeded in the open unit disc in the complex plane, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n \prod_{k\ne n}}\sigma (\phi _n,\phi _k)>0$, for a Hardy space $H^p(D)(1\leq p\leq \infty )$ and the Smirnov class $N_+(D)$.