Hilbert-schmidtness of some finitely generated submodules in \(H^2(\mathbb{D}^2)\). (English) Zbl 06879567
Summary: 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.
Keywords:
Hardy space over the bidisk; submodule; core operator; Hilbert-Schmidt submodule; fringe operator; Fredholm indexReferences:
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