Interpolation problem for \(\ell^1\) and an \(F\)-space (Q1025758)

For an \(F\)-space \(B\) with an invariant metric \(d\), let \(B^*_1\) be the unit ball of the dual space. A sequence \((\varphi_n)\subset B^*_1\) is called an \(\ell^1\)-interpolating sequence if for every sequence \((w_n)\subset\ell^1(\mathbb C)\) there exists an element \(f\in B\) such that \(\varphi_n(f)=w_n\) for all \(n\). Given \((\varphi_n)\), let \(\rho_n= \sup\{|\varphi_n(f)|: f=0\;\text{on}\;(\varphi_k)_{k \neq n}, d(f, 0)\leq 1\}\) and \(\sigma(\varphi_n, \varphi_k)=\sup\{|\varphi_n(f)|: \phi_k(f)=0, d(f, 0)\leq 1\}\). The author first proves a couple of general results: (i) For an arbitrary Banach space \(B\), \((\varphi_n)\) is an \(\ell^1\)-interpolating sequence if and only if \(\inf_n \rho_n>0\); (ii) For a Banach space \(B\) having a predual, if \(\inf_n\prod_{k\neq n} \sigma(\varphi_n, \varphi_k)>0\) and \((\varphi_n)\) is in the predual, then \((\varphi_n)\) is an \(\ell^1\)-interpolating sequence. Applying these general results the author shows that, when \((\varphi_n)\) is embedded in the unit disk \(D\), each of the two conditions \(\inf_n \rho_n>0\) and \(\inf_n\prod_{k\neq n} \sigma(\varphi_n, \varphi_k)>0\) is equivalent to \((\varphi_n)\) being an \(\ell^1\)-interpolating sequence for the Smirnov class \(N_+(D)\). The \(\ell^p\)-analogues for the Hardy spaces \(H^p(D)\) \((0<p<1\)) are also proved in the course of the proofs. In conjunction with such results, the author asks whether the \(\ell^1\)-analogues for the Hardy spaces \(H^p(D)\) \((0<p<1\)) are true.