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.
Categories: math.FA math.OA