Rudin's Submodules of $H^2(\mathbb{D}^2)$ (1405.1388v2)

Abstract: Let ${\alpha_n}{n\geq 0}$ 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 $\sum_{n=0}^\infty (1 - l_n|\alpha_n|) <\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace [\mathcal{S}{\Phi} = \vee{n=0}^\infty \Big( z_1^n \prod_{k=n}^\infty \left(\frac{-\bar{\alpha}k}{|\alpha_k|} \frac{z_2 - \alpha_k}{1 - \bar{\alpha}_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}{\Phi}\ominus (z_1 \mathcal{S}{\Phi}+ z_2\mathcal{S}{\Phi}))= 1+#{n\ge 0: \alpha_n=0}<\infty. ]In particular, this answer a question earlier raised by Douglas and Yang (2000).