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

Rongwei Yang

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.