Uniform algebras generated by holomorphic and close-to-harmonic functions

Abstract:

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc $\overline {\mathbb {D}}$ generated by $z$ and $h$, where $h$ is a nowhere-holomorphic harmonic function on $\mathbb {D}$ that is continuous up to $\partial {\mathbb {D}}$, equals $\mathcal {C}(\overline {\mathbb {D}})$. The abstract tools used by Axler and Shields make harmonicity of $h$ an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if $h$ is replaced by $h+R$, where $R$ is a non-harmonic perturbation whose Laplacian is “small” in a certain sense.