Minimality of $T_z^{\ast }$-Invariant Subspaces of $H^2({\mathbb{D}}^2)$

Abstract

In this paper, we establish necessary conditions for the minimality of infinite-dimensional $T_z^{\ast }$-invariant subspaces of $H^2({\mathbb{D}}^2)$. More precisely, we prove that if $M\subset H^2({\mathbb{D}}^2)$ is an infinite-dimensional, minimal $T_z^{\ast }$-invariant subspace, then, for every $n\in {\mathbb{N}}_0$, either $\overline{P_n(M)}=H^2(\mathbb{D})$ or ${\overline{P_n(M)}}^{\perp }={\varphi }_{a,\theta }{H}^2(\mathbb{D})$ where ${\varphi }_{a,\theta }$ is an atomic singular inner function and $P_n$ denotes the orthogonal projection of $H^2({\mathbb{D}}^2)$ onto $H^2(z)w^n$. As $T_z^{\ast }$ is a universal operator on $H^2({\mathbb{D}}^2)$, this result reduces the Invariant Subspace Problem to the analysis of these two alternatives for infinite-dimensional minimal $T_z^{\ast }$-invariant subspaces.

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