Rudin's Submodules of $H^2(\mathbb{D}^2)$

Let $\{α_n\}_{n\geq 0}$ be a sequence of scalars in the open unit disc of $\mathbb{C}$, and let $\{l_n\}_{n\geq 0}$ be a sequence of natural numbers satisfying $\sum_{n=0}^\infty (1 - l_n|α_n|) <\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace $\mathcal{S}_Φ = \vee_{n=0}^\infty \Big( z_1^n \prod_{k=n}^\infty \left(\frac{-\barα_k}{|α_k|} \frac{z_2 - α_k}{1 - \barα_k z_2}\right)^{l_k} H^2(\mathbb{D}^2)\Big),$ is called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that $\text{dim} (\mathcal{S}_Φ\ominus (z_1 \mathcal{S}_Φ+ z_2\mathcal{S}_Φ))= 1+\#\{n\ge 0: α_n=0\}<\infty.$In particular, this answer a question earlier raised by Douglas and Yang (2000).

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