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

Abstract

Let {α n } n≥0 be a sequence of scalars in the open unit disc of C, and let {l n } n≥0 be a sequence of natural numbers satisfyingis called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that dim(S Φ ⊖ (z 1 S Φ + z 2 S Φ )) = 1 + #{n ≥ 0 : α n = 0} < ∞. In particular, this answer a question earlier raised by Douglas and Yang (2000) [4].