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 $\delta ⩽\Vert f(z){\Vert }_E⩽1$ for all $z$ from the unit disk $\mathbb{D}$ there exists some solution $g=\{g_j{\}}_{j\in \mathbb{Z}}\in H_{\infty }(E^{\prime })$, $\Vert g{\Vert }_{H_{\infty }(E^{\prime })}⩽C_{\delta }$ of the Bézout equation ${\sum }_j{f}_j{g}_j=1$, also known as the vector-valued corona problem with data in $H_{\infty }(E)$.