Hilbert-Schmidtness of the $M_{\theta,\varphi}$-type submodules

TL;DR

This paper investigates the Hilbert-Schmidtness of certain submodules in the bidisk Hardy space, establishing conditions under which these submodules are Hilbert-Schmidt and analyzing their norm bounds.

Contribution

It provides a new characterization of Hilbert-Schmidt submodules via reproducing kernels and composition operators, extending understanding of their structure in multivariable Hardy spaces.

Findings

Characterizes Hilbert-Schmidt submodules using kernel composition

Shows equivalence of Hilbert-Schmidtness between submodules and their generating modules

Establishes uniform boundedness of Hilbert-Schmidt norms for specific submodules

Abstract

Let $\theta(z),\varphi(w)$ be two nonconstant inner functions and $M$ be a submodule in $H^2(\mathbb{D}^2)$. Let $C_{\theta,\varphi}$ denote the composition operator on $H^2(\mathbb{D}^2)$ defined by $C_{\theta,\varphi}f(z,w)=f(\theta(z),\varphi(w))$, and $M_{\theta,\varphi}$ denote the submodule $[C_{\theta,\varphi}M]$, that is, the smallest submodule containing $C_{\theta,\varphi}M$. Let $K^M_{\lambda,\mu}(z,w)$ and $K^{M_{\theta,\varphi}}_{\lambda,\mu}(z,w)$ be the reproducing kernel of $M$ and $M_{\theta,\varphi}$, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that \[K^{M_{\theta,\varphi}}= K^M \circ B~ \cdot R,\] where $B=(\theta,\varphi)$, $R_{\lambda,\mu}(z,w)=\frac{1-\overline{\theta(\lambda)}\theta(z)}{1-\bar{\lambda}z} \frac{1-\overline{\varphi(\mu)}\varphi(w)}{1-\bar{\mu}w}$. This implies that $M_{\theta,\varphi}$ is a Hilbert-Schmidt submodule if and only if $M$ is. Moreover, as an application, we prove that the Hilbert-Schmidt norms of submodules $[\theta(z)-\varphi(w)]$ are uniformly bounded.