Unitary parts of Toeplitz operators with operator-valued symbols
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916112105472000 |
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| author | Narayanan, E. K. Sarkar, Srijan |
| author_facet | Narayanan, E. K. Sarkar, Srijan |
| contents | Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is unitary. We show that for a Hilbert space $\mathcal{E}$ and operator-valued symbol $Φ\in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T})$, the Toeplitz operator $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ has such a unitary subspace if and only if there exists a Hilbert space $\mathcal{F}$, an inner function $Θ(z) \in H_{\mathcal{B}(\mathcal{F}, \mathcal{E})}^{\infty}(\mathbb{D})$, and a unitary $U:\mathcal{F} \rightarrow \mathcal{F}$ such that \[ Φ(e^{it}) Θ(e^{it}) = Θ(e^{it}) U \quad \text{and} \quad Φ(e^{it})^* Θ(e^{it}) = Θ(e^{it}) U^* \quad (\text{ a.e. on }\mathbb{T}). \] This result can be seen as a generalization of the corresponding result for Toeplitz operators on $H^2(\mathbb{D})$ by Goor in [13]. We provide finer characterizations for analytic Toeplitz operators by finding the correspondence between the unitary parts of $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $Φ(0)$ on $\mathcal{E}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unitary parts of Toeplitz operators with operator-valued symbols Narayanan, E. K. Sarkar, Srijan Functional Analysis 47B35, 30H10, 46E40, 47A56, 30J05 Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is unitary. We show that for a Hilbert space $\mathcal{E}$ and operator-valued symbol $Φ\in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T})$, the Toeplitz operator $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ has such a unitary subspace if and only if there exists a Hilbert space $\mathcal{F}$, an inner function $Θ(z) \in H_{\mathcal{B}(\mathcal{F}, \mathcal{E})}^{\infty}(\mathbb{D})$, and a unitary $U:\mathcal{F} \rightarrow \mathcal{F}$ such that \[ Φ(e^{it}) Θ(e^{it}) = Θ(e^{it}) U \quad \text{and} \quad Φ(e^{it})^* Θ(e^{it}) = Θ(e^{it}) U^* \quad (\text{ a.e. on }\mathbb{T}). \] This result can be seen as a generalization of the corresponding result for Toeplitz operators on $H^2(\mathbb{D})$ by Goor in [13]. We provide finer characterizations for analytic Toeplitz operators by finding the correspondence between the unitary parts of $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $Φ(0)$ on $\mathcal{E}$. |
| title | Unitary parts of Toeplitz operators with operator-valued symbols |
| topic | Functional Analysis 47B35, 30H10, 46E40, 47A56, 30J05 |
| url | https://arxiv.org/abs/2402.00529 |