Two problems on submodules of $H^2(\mathbb{D}^n)$
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| Formato: | Preprint |
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2024
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| _version_ | 1866909223329202176 |
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| author | Debnath, Ramlal Sarkar, Srijan |
| author_facet | Debnath, Ramlal Sarkar, Srijan |
| contents | Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2(\mathbb{D}^n)$ (where $n \geq 2$), let $R_{z_j}:=M_{z_j}|_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ ev_{z_j}$, for each $j \in \{1,\ldots,n\}$. Here, $ev_{z_j}$ is the operator evaluating at $0$ in the $z_j$-th variable. In this article, we prove that given any subset $Λ\subseteq \{1,\ldots,n\}$, there exists a collection of one-variable inner functions $\{ϕ_λ(z_λ)\}_{λ\in Λ}$ on $\mathbb{D}^n$, such that \[ \mathcal{S} = \sum_{λ\in Λ} ϕ_λ(z_λ)H^2(\mathbb{D}^n), \] if and only if the conditions $ (I_{\mathcal{S}}-E_{z_k}E_{z_k}^*)(I_{\mathcal{S}}-R_{z_k}R_{z_k}^*)=0$ for all $k \in \{1,\dots,n\} \setminus Λ$, and $(I_{\mathcal{S}}-E_{z_{i}}E_{z_{i}}^*)(I_{\mathcal{S}}-R_{z_{i}}R_{z_{i}}^*)(I_{\mathcal{S}}-E_{z_{j}}E_{z_{j}}^*)(I_{\mathcal{S}}-R_{z_{j}}R_{z_{j}}^*)=0$ for all distinct $i,j \in Λ$ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two problems on submodules of $H^2(\mathbb{D}^n)$ Debnath, Ramlal Sarkar, Srijan Functional Analysis Complex Variables 47A13, 47A15, 47A20, 32A10, 30J05, 30J10, 30H05, 30H10 Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2(\mathbb{D}^n)$ (where $n \geq 2$), let $R_{z_j}:=M_{z_j}|_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ ev_{z_j}$, for each $j \in \{1,\ldots,n\}$. Here, $ev_{z_j}$ is the operator evaluating at $0$ in the $z_j$-th variable. In this article, we prove that given any subset $Λ\subseteq \{1,\ldots,n\}$, there exists a collection of one-variable inner functions $\{ϕ_λ(z_λ)\}_{λ\in Λ}$ on $\mathbb{D}^n$, such that \[ \mathcal{S} = \sum_{λ\in Λ} ϕ_λ(z_λ)H^2(\mathbb{D}^n), \] if and only if the conditions $ (I_{\mathcal{S}}-E_{z_k}E_{z_k}^*)(I_{\mathcal{S}}-R_{z_k}R_{z_k}^*)=0$ for all $k \in \{1,\dots,n\} \setminus Λ$, and $(I_{\mathcal{S}}-E_{z_{i}}E_{z_{i}}^*)(I_{\mathcal{S}}-R_{z_{i}}R_{z_{i}}^*)(I_{\mathcal{S}}-E_{z_{j}}E_{z_{j}}^*)(I_{\mathcal{S}}-R_{z_{j}}R_{z_{j}}^*)=0$ for all distinct $i,j \in Λ$ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules. |
| title | Two problems on submodules of $H^2(\mathbb{D}^n)$ |
| topic | Functional Analysis Complex Variables 47A13, 47A15, 47A20, 32A10, 30J05, 30J10, 30H05, 30H10 |
| url | https://arxiv.org/abs/2406.09245 |