A Brief Survey of Operator Theory in H 2(𝔻2)

This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc H ² (𝔻²). As an important component of multivariable operator theory, the theory in H ²(𝔻²) focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in H ²(𝔻²). The connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in H ²(𝔻²) is motivated by and still tightly related to several other influential theories, namely, the Nagy-Foias theory on operator models, Ando’s dilation theorem of commuting operator pairs, Rudin’s function theory on H ²(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(𝔻²) 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 in new researchers.