Hilbert-Schmidtness of the Mθ,φ-type submodules

Abstract

Let $\theta \left(z\right),\phi \left(w\right)$ be two nonconstant inner functions and M be a submodule in $H^2\left(D^2\right)$. Let $C_{\theta ,\phi }$ denote the composition operator on $H^2\left(D^2\right)$ defined by $C_{\theta ,\phi }f(z,w)=f\left(\theta \right(z),\phi (w\left)\right)$, and $M_{\theta ,\phi }$ denote the submodule $\left[C_{\theta ,\phi }M\right]$, 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 $B=(\theta ,\phi )$, $R_{\lambda ,\mu }(z,w)=\frac{1-\overline{\theta \left(\lambda \right)}\theta \left(z\right)}{1-\overline{\lambda }z}\frac{1-\overline{\phi \left(\mu \right)}\phi \left(w\right)}{1-\overline{\mu }w}$. 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 $\left[\theta \right(z)-\phi (w\left)\right]$ are uniformly bounded.