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Register a new userHilbert-Schmidtness of some finitely generated submodules in $H^2(mathbb{D}^2)$
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A closed subspace $mathcal{M}$ of the Hardy space $H^2(mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $mathcal{M}$ containing $z_1 - varphi(z_2)$ is Hilbert-Schmidt, where $varphi$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.
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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, Andos dilation theorem of commuting operator pairs, Rudins function theory on $H^2(mathbb D^n)$, and Douglas-Paulsens 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 authors 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.
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