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A brief survey on operator theory in $H^2(\mathbb D^2)$
Functional Analysis · Submitted 29 Sep 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(\mathbb D^2)$. | contribution 0.84 | 0.43 | 0.05 | asserted 0.93 |
| s1 | 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 $H^2(\mathbb D^2)$. | background 0.94 | 0.04 | 0.08 | asserted 1.00 |
| s2 | Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. | contribution 0.61 | 0.05 | 0.19 | asserted 1.00 |
| s3 | 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. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s4 | 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. | background 0.98 | 0.07 | 0.04 | asserted 1.00 |
| s5 | This survey is far from a full account of this development but rather a glimpse from the author's perspective. | limitation 1.00 | 0.41 | 0.05 | asserted 1.00 |
| s6 | 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. | contribution 0.81 | 0.32 | 0.11 | asserted 1.00 |
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Original 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.
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References
How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Cites it as background 2 papers
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The defect spaces of the two restricted coordinate shifts, their cross-commutator, and the associated core operator therefore play an important role in the structure theory of bidisc submodules; see [ref] .
Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules · Introduction · 0.93
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There has been by now much work on the problem leading to deeper appreciation as to how complicated the structure is in the bidisk setting: a small sample of such work is [ref] .
Dilation and Model Theory for Pairs of Commuting Contractions · Admissible triples · 0.95
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