Corona problem with data in ideal spaces of sequences

TL;DR

This paper proves the solvability of the vector-valued corona problem for functions in Hardy spaces with data in ideal sequence spaces, extending classical results to Banach lattices with order continuous norms.

Contribution

It establishes the existence of solutions to the corona problem in Hardy spaces for data in ideal sequence spaces of Banach lattices with order continuous norms.

Findings

Existence of solutions to the vector-valued corona problem in Hardy spaces.

Extension of corona problem results to Banach lattices with order continuous norms.

Bounded solutions in $H_ (E')$ for data in $H_ (E)$.

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 $H_\infty (E)$.