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

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