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

Abstract

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

Theorems & Definitions (49)

• Theorem 1.1: Berger-Shaw Theorem, Berger
• Conjecture 1.2: R. Yang, 1999, Ya1999
• Conjecture 1.3
• Lemma 2.1
• Theorem 2.1
• Theorem 2.2: Theorem 3.8, Cowen
• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Definition 2.2