Connections of the Corona Problem with Operator Theory and Complex Geometry

TL;DR

This paper explores the connections between the corona problem, operator theory, and complex geometry using Hilbert modules and reproducing kernel Hilbert spaces, offering new insights and interpretations of existing approaches.

Contribution

It introduces a Hilbert module framework to interpret various approaches to the corona problem and presents new observations linking complex geometry and operator theory.

Findings

Revealed new relationships between the corona problem and operator theory.

Provided a reinterpretation of the corona problem within the Hilbert module framework.

Made several novel observations connecting complex geometry to the corona problem.

Abstract

The corona problem was motivated by the question of the density of the open unit disk D in the maximal ideal space of the algebra, H1(D), of bounded holomorphic functions on D. In this note we study relationships of the problem with questions in operator theory and complex geometry. We use the framework of Hilbert modules focusing on reproducing kernel Hilbert spaces of holomorphic functions on a domain, in Cm. We interpret several of the approaches to the corona problem from this point of view. A few new observations are made along the way. 2012 MSC: 46515, 32A36, 32A70, 30H80, 30H10, 32A65, 32A35, 32A38 Keywords: corona problem, Hilbert modules, reproducing kernel Hilbert space, commutant lifting theorem 1