Cross commutators of Rudin's submodules

Abstract

Let $b(z)={\prod }_{n=1}^{\infty }\frac{-{\bar{\alpha }}_n}{|{\alpha }_n|}\frac{z-{\alpha }_n}{1-{\bar{\alpha }}_{n}z}$, where ${\sum }_{n=1}^{\infty }(1-|{\alpha }_n|)<\infty$, be the Blaschke product with zeros at ${\alpha }_n\in \mathbb{D}\setminus \{0\}$. Then $\cls = \vee_{n=1}^\infty \big(z^n H^2(\mathbb{D})\big) \otimes \big(\prod_{k=n}^\infty \frac{-\barα_n}{|α_n|} \frac{z - α_n}{1 - \barα_n z} H^2(\mathbb{D})\big)$ is a joint $(M_{z_1},M_{z_2})$ invariant subspace of the Hardy space $H^2({\mathbb{D}}^2)\cong H^2(\mathbb{D})\otimes H^2(\mathbb{D})$. This class of subspaces was originally introduced by Rudin in the context of infinite cardinality of generating sets of shift invariant subspaces of $H^2({\mathbb{D}}^2)$. \noindent In this paper we prove that for a Rudin invariant subspace $\cls$ of $H^2({\mathbb{D}}^2)$, the cross commutator $[(P_{\cls} M_{z_1}|_{\cls})^*, M_{z_2}|_{\cls}] = (P_{\cls} M_{z_1} |_{\cls})^* (M_{z_2}|_{\cls}) - (M_{z_2}|_{\cls}) (P_{\cls} M_{z_1}|_{\cls})^*$ is not compact. Consequently, Rudin's invariant subspaces are both infinitely generated and not essentially doubly commuting.

The paper has been withdrawn due to a logical gap in the main result