Problem of ideals in the algebra $H^{\infty }$ for some spaces of sequences

Abstract:

Metric aspects of the problem of ideals are studied. Let $h$ be a function in the class $H^{\infty }(\mathbb {D})$ and $f$ a vector-valued function in the class $H^{\infty }(\mathbb {D};E)$, i.e., $f$ takes values in some lattice of sequences $E$. Suppose that $|h(z)| \le \|f(z)\|^{\alpha }_{E} \le 1$ for some parameter $\alpha$. The task is to find a function $g$ in $H^{\infty }(\mathbb {D};E’)$, where $E’$ is the order dual of $E$, such that $\sum f_j g_j = h$. Also it is necessary to control the value of $\|g\|_{H^{\infty }(E’)}$. The classical case with $E = l^2$ was investigated by V. A. Tolokonnikov in 1981. Recently, the author managed to obtain a similar result for the space $E = l^1$. In this paper it is shown that the problem of ideals can be solved for any $q$-concave Banach lattice $E$ with finite $q$; in particular, $E=l^p$ with $p \in [1,\infty )$ fits.