Runge-Type Approximation Theorem for Banach-valued ${\mathbf H^\infty}$ Functions on a Polydisk

Abstract

Let $\mathbb D^n\subset\mathbb C^n$ be the open unit polydisk, $K\subset\mathbb D^n$ be an $n$-ary Cartesian product of planar sets, and $\hat U\subset \mathfrak M^n$ be an open neighbourhood of the closure $\bar K$ of $K$ in $\mathfrak M^n$, where $\mathfrak M$ is the maximal ideal space of the algebra $H^\infty$ of bounded holomorphic functions on $\mathbb D$. Let $X$ be a complex Banach space and $H^\infty(V,X)$ be the space of bounded $X$-valued holomorphic functions on an open set $V\subset\mathbb D^n$. We prove that any $f\in H^\infty(U,X)$, where $U=\hat U\cap\mathbb D^n$, can be uniformly approximated on $K$ by ratios $h/b$, where $h\in H^\infty(\mathbb D^n,X)$ and $b$ is the product of interpolating Blaschke products such that $\inf_K |b|>0$. Moreover, if $\bar K$ is contained in a compact holomorphically convex subset of $\hat U$, then $h/b$ above can be replaced by $h$ for any $f$. The results follow from a new constructive Runge-type approximation theorem for Banach-valued holomorphic functions on open subsets of $\mathbb D$ and extend the fundamental results of Su\'{a}rez on Runge-type approximation for analytic germs on compact subsets of $\mathfrak M$. They can also be applied to the long-standing corona problem which asks whether $\mathbb D^n$ is dense in the maximal ideal space of $H^\infty(\mathbb D^n)$ for all $n\ge 2$.