Abstract
Let \(\{\varphi _n(z)\}_{n\ge 0}\) be a sequence of inner functions satisfying that \(\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)\) for every \(n\ge 0\) and \(\{\varphi _n(z)\}_{n\ge 0}\) has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \(\mathcal{M }\) of \(H^2(\mathbb{D }^2)\). The ranks of \(\mathcal{M }\ominus w\mathcal{M }\) for \(\mathcal{F }_z\) and \(\mathcal{F }^*_z\) respectively are determined, where \(\mathcal{F }_z\) is the fringe operator on \(\mathcal{M }\ominus w\mathcal{M }\). Let \(\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }\). It is also proved that the rank of \(\mathcal{M }\ominus w\mathcal{M }\) for \(\mathcal{F }^*_z\) equals to the rank of \(\mathcal{N }\) for \(T^*_z\) and \(T^*_w\).
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The first author is partially supported by Grant-in-Aid for Scientific Research (No.21540166), Japan Society for the Promotion of Science.
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Izuchi, K.J., Izuchi, K.H. & Izuchi, Y. Ranks of backward shift invariant subspaces of the Hardy space over the bidisk. Math. Z. 274, 885–903 (2013). https://doi.org/10.1007/s00209-012-1100-2
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DOI: https://doi.org/10.1007/s00209-012-1100-2
Keywords
- Hardy space over the bidisk
- Rank of backward shift invariant subspace
- Sequence of inner functions
- Fringe operator