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

Abstract

Let $\{{\alpha }_n{\}}_{n\ge 0}$ be a sequence of scalars in the open unit disc of $\mathbb{C}$, and let $\{l_n{\}}_{n\ge 0}$ 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 \[\mathcal{S}_Φ = \vee_{n=0}^\infty \Big( z_1^n \prod_{k=n}^\infty \left(\frac{-\barα_k}{|α_k|} \frac{z_2 - α_k}{1 - \barα_k z_2}\right)^{l_k} H^2(\mathbb{D}^2)\Big),\] 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).

6 pages. Revised. To appear in C. R. Acad. Sci. Paris