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 $\cls = \vee_{n=1}^\infty \big(z^n H^2(\mathbb{D})\big) \otimes \big(\prod_{k=n}^\infty \frac{-\bar{\alpha}n}{|\alpha_n|} \frac{z - \alpha_n}{1 - \bar{\alpha}_n z} H^2(\mathbb{D})\big)$ 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.