The ideals of the algebra $H^{\infty}$: interpolation and Bézout equation

We will discuss two problems connected with the properties of ideals of the algebra $H^{\infty}$. The first part of the talk will be based on the joint work with S.V. Kislyakov and will be dedicated to the real interpolation for the spaces formed by the intersections of ideals or in a more general situation of modules under $w^{\ast}$-closed subalgebras of $L^{\infty}(\mu)$. One can take $K^p_{\theta}$ и $H^p(\mathbb{T}^2)$ as an examples of such spaces.
During the second part of the talk we will discuss the metric aspects of the problem of ideals. Let $X$ be the Banach lattice of sequences and let $X'$ be order dual to $X$. For any functions $h \in H^\infty(\mathbb{D})$ and $f \in H^\infty(\mathbb{D}; X)$ the task is to find a function $g \in H^\infty(\mathbb{D}; X')$ such that
$$ h(z) = \sum\limits_{i=1}^\infty f(z,i) g(z,i), \,\,\, z \in \mathbb{D}. $$
Also we need to control the value $\|g\|_{H^\infty(\mathbb{D};X')}$. V.A. Tolokonnikov solved the problem of ideals for the classical case $X=l^2$. We will establish that the problem of ideals could be solved for $q$-concave Banach lattices by applying the method (based on fixed point theorem) due to D.V.Rutsky. In particular we will show that the problem of ideals could be solved for $X = l^p, 1 \le p < \infty$.