A brief survey on operator theory in $H^2(\mathbb D^2)$

TL;DR

This survey introduces the operator theory in the Hardy space over the bidisc, highlighting its connections to invariant subspaces, operator models, and related theories, while showcasing recent growth and open questions in the field.

Contribution

It provides an organized overview of operator theory in $H^2(\mathbb D^2)$, connecting key concepts and recent developments to inspire further research.

Findings

Growth of operator theory in $H^2(\mathbb D^2)$ over the past two decades

Connections between operator properties and invariant subspace structure

Compilation of results and references to guide new researchers

Abstract

This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(\mathbb D^2)$. As an important component of multivariable operator theory, the theory in $H^2(\mathbb D^2)$ focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in $H^2(\mathbb D^2)$. Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in $H^2(\mathbb D^2)$ is motivated by and still tightly related to several other influential theories, namely Nagy-Foias theory on operator models, Ando's dilation theorem of commuting operator pairs, Rudin's function theory on $H^2(\mathbb D^n)$, and Douglas-Paulsen's framework of Hilbert modules. Due to the simplicity of the setting, a great supply of examples in particular, the operator theory in $H^2(\mathbb D^2)$ has seen remarkable growth in the past two decades. This survey is far from a full account of this development but rather a glimpse from the author's perspective. Its goal is to show an organized structure of this theory, to bring together some results and references and to inspire curiosity on new researchers.