Corona problem with data in ideal spaces of sequences

Abstract

Let $E$ be a Banach lattice on $\mathbb Z$ having order continuous norm. We show that for any function $f = \{f_j\}_{j \in \mathbb Z}$ from the Hardy space $H_\infty (E)$ such that $\delta \leqslant \|f (z)\|_E \leqslant 1$ for all $z$ from the unit disk $\mathbb D$ there exists some solution $g = \{g_j\}_{j \in \mathbb Z} \in H_\infty (E')$, $\|g\|_{H_\infty (E')} \leqslant C_\delta$ of the B\'ezout equation $\sum_j f_j g_j = 1$, also known as the vector-valued corona problem with data in $H_\infty (E)$.