Ranks of backward shift invariant subspaces of the Hardy space over the bidisk |
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Authors: | Kei Ji Izuchi Kou Hei Izuchi Yuko Izuchi |
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Institution: | 1. Department of Mathematics, Niigata University, 950-2181, Niigata, Japan 2. Department of Mathematics, Faculty of Education, Yamaguchi University, 753-8511, Yamaguchi, Japan 3. Aoyama-shinmachi 18-6-301, Nishi-ku, 950-2006, Niigata, Japan
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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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