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 Suárez 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 $n\ge 2$.

23 pages; under review