Corona problem with data in ideal spaces of sequences

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