Corona problem with data in ideal spaces of sequences

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 $H_{\mathrm{\infty }}(E)$ such that $\delta ⩽\mathrm{\parallel }f(z){\mathrm{\parallel }}_E⩽1$ 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 $H_{\mathrm{\infty }}(E)$.