Runge-type approximation theorem for Banach-valued H∞ functions on a polydisk

Abstract

Let $D^n\subset {ℂ}^n$ denote the open unit polydisk, and let $K\subset D^n$ be a Cartesian product of planar compacta. Let $\hat{U}\subset M^n$ be an open neighborhood of the closure $\bar{K}$ in $M^n$, where $M$ is the maximal ideal space of the algebra $H^{\infty }$ of bounded holomorphic functions on $D$. Given a complex Banach space $X$, denote by $H^{\infty }(V,X)$ the Banach space of bounded $X$-valued holomorphic functions on an open set $V\subset D^n$.

We prove that any $f\in H^{\infty }(U,X)$, where $U=\hat{U}\cap D^n$, can be uniformly approximated on $K$ by functions of the form $h/b$, where $h\in H^{\infty }(D^n,X)$ and $b$ is a finite product of interpolating Blaschke products satisfying ${inf}_K\left|b\right|>0$. Moreover, if $\bar{K}$ is contained in a compact holomorphically convex subset of $\hat{U}$, then such approximations can be achieved without denominators: that is, $f$ can be approximated uniformly on $K$ by elements of $H^{\infty }(D^n,X)$.

These results follow, essentially by reduction, from the main contribution of this paper: a novel constructive Runge-type approximation theorem for Banach-valued holomorphic functions on open subsets of the unit disk $D$. Our work extends foundational results of Suárez concerning approximation of analytic germs on compact subsets of $M$, and it offers new perspectives on the classical corona problem, which asks whether $D^n$ is dense in the maximal ideal space of $H^{\infty }\left(D^n\right)$ for all $n\ge 2$.