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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.18958 |
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| _version_ | 1866913707664080896 |
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| author | Zu, Chao Lu, Yufeng |
| author_facet | Zu, Chao Lu, Yufeng |
| contents | Let $θ(z),φ(w)$ be two nonconstant inner functions and $M$ be a submodule in $H^2(\mathbb{D}^2)$. Let $C_{θ,φ}$ denote the composition operator on $H^2(\mathbb{D}^2)$ defined by $C_{θ,φ}f(z,w)=f(θ(z),φ(w))$, and $M_{θ,φ}$ denote the submodule $[C_{θ,φ}M]$, that is, the smallest submodule containing $C_{θ,φ}M$. Let $K^M_{λ,μ}(z,w)$ and $K^{M_{θ,φ}}_{λ,μ}(z,w)$ be the reproducing kernel of $M$ and $M_{θ,φ}$, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that \[K^{M_{θ,φ}}= K^M \circ B~ \cdot R,\] where $B=(θ,φ)$, $R_{λ,μ}(z,w)=\frac{1-\overline{θ(λ)}θ(z)}{1-\barλz} \frac{1-\overline{φ(μ)}φ(w)}{1-\barμw}$. This implies that $M_{θ,φ}$ is a Hilbert-Schmidt submodule if and only if $M$ is. Moreover, as an application, we prove that the Hilbert-Schmidt norms of submodules $[θ(z)-φ(w)]$ are uniformly bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert-Schmidtness of the $M_{θ,φ}$-type submodules Zu, Chao Lu, Yufeng Functional Analysis Operator Algebras Primary 46E20 Secondary 46E22 Let $θ(z),φ(w)$ be two nonconstant inner functions and $M$ be a submodule in $H^2(\mathbb{D}^2)$. Let $C_{θ,φ}$ denote the composition operator on $H^2(\mathbb{D}^2)$ defined by $C_{θ,φ}f(z,w)=f(θ(z),φ(w))$, and $M_{θ,φ}$ denote the submodule $[C_{θ,φ}M]$, that is, the smallest submodule containing $C_{θ,φ}M$. Let $K^M_{λ,μ}(z,w)$ and $K^{M_{θ,φ}}_{λ,μ}(z,w)$ be the reproducing kernel of $M$ and $M_{θ,φ}$, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that \[K^{M_{θ,φ}}= K^M \circ B~ \cdot R,\] where $B=(θ,φ)$, $R_{λ,μ}(z,w)=\frac{1-\overline{θ(λ)}θ(z)}{1-\barλz} \frac{1-\overline{φ(μ)}φ(w)}{1-\barμw}$. This implies that $M_{θ,φ}$ is a Hilbert-Schmidt submodule if and only if $M$ is. Moreover, as an application, we prove that the Hilbert-Schmidt norms of submodules $[θ(z)-φ(w)]$ are uniformly bounded. |
| title | Hilbert-Schmidtness of the $M_{θ,φ}$-type submodules |
| topic | Functional Analysis Operator Algebras Primary 46E20 Secondary 46E22 |
| url | https://arxiv.org/abs/2502.18958 |