On a theorem of Bishop and commutants of Toeplitz operators in $\mathbb{C}^n$

TL;DR

This paper proves an approximation theorem for certain domains in complex n-space where the ar problem is solvable in bounded functions, extending classical results to higher dimensions and linking to Toeplitz operator commutants.

Contribution

It introduces a new approximation theorem for domains in omplex^n with solvable ar problem in L-infinity, and generalizes the Axler-ukovi
0d-Rao theorem to higher dimensions.

Findings

Established an approximation theorem for specific omplex domains.

Extended the Axler-ukovi
0d-Rao theorem to omplex^n.

Connected ar problem solvability with Toeplitz operator commutants.

Abstract

We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-\v{C}u\v{c}kovi\'c-Rao Theorem in higher dimensions.