On products and partial isometry of Toeplitz operators with operator-valued symbols
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909633739751424 |
|---|---|
| author | Sarkar, Srijan |
| author_facet | Sarkar, Srijan |
| contents | We solve the following problems associated with Toeplitz operators $T_Φ$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$. $(I)$ Given operator-valued bounded analytic functions $Γ, Ψ$ on $\mathbb{D}^n$, we completely characterize when the product $M_ΓM_Ψ^*$ becomes a Toeplitz operator by identifying tractable conditions on the functions. Furthermore, these conditions can be used to explicitly write the product into a sum of simple Toeplitz operators. $(II)$ We prove that partially isometric Toeplitz operators admit the following factorization: \[ T_Φ = M_Γ M_Ψ^*, \] where, $Γ, Ψ$ are operator-valued inner functions on $\mathbb{D}^n$. A few of the immediate consequences are: $(a)$ every partially isometric Toeplitz operator has a partially isometric symbol almost everywhere on $\mathbb{T}^n$ (distinguished boundary of $\mathbb{D}^n$), $(b)$ any partially isometric analytic Toeplitz operator is of the form $M_{ΓV^*}$, where $Γ$ is an operator-valued inner function and $V$ is an constant isometry. In connection with the result $(ii)$, we establish and use a crucial phenomenon: the range of partially isometric Toeplitz operators is always a Beurling-type invariant subspace of $H_{\mathcal{E}}^2(\mathbb{D}^n)$. Our results are new even in the case of Hardy spaces over the unit disc and extend the work of Brown--Douglas, Deepak--Pradhan--Sarkar on scalar-valued spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On products and partial isometry of Toeplitz operators with operator-valued symbols Sarkar, Srijan Functional Analysis Complex Variables 47B35, 46E40, 47A56, 30H10, 30J05 We solve the following problems associated with Toeplitz operators $T_Φ$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$. $(I)$ Given operator-valued bounded analytic functions $Γ, Ψ$ on $\mathbb{D}^n$, we completely characterize when the product $M_ΓM_Ψ^*$ becomes a Toeplitz operator by identifying tractable conditions on the functions. Furthermore, these conditions can be used to explicitly write the product into a sum of simple Toeplitz operators. $(II)$ We prove that partially isometric Toeplitz operators admit the following factorization: \[ T_Φ = M_Γ M_Ψ^*, \] where, $Γ, Ψ$ are operator-valued inner functions on $\mathbb{D}^n$. A few of the immediate consequences are: $(a)$ every partially isometric Toeplitz operator has a partially isometric symbol almost everywhere on $\mathbb{T}^n$ (distinguished boundary of $\mathbb{D}^n$), $(b)$ any partially isometric analytic Toeplitz operator is of the form $M_{ΓV^*}$, where $Γ$ is an operator-valued inner function and $V$ is an constant isometry. In connection with the result $(ii)$, we establish and use a crucial phenomenon: the range of partially isometric Toeplitz operators is always a Beurling-type invariant subspace of $H_{\mathcal{E}}^2(\mathbb{D}^n)$. Our results are new even in the case of Hardy spaces over the unit disc and extend the work of Brown--Douglas, Deepak--Pradhan--Sarkar on scalar-valued spaces. |
| title | On products and partial isometry of Toeplitz operators with operator-valued symbols |
| topic | Functional Analysis Complex Variables 47B35, 46E40, 47A56, 30H10, 30J05 |
| url | https://arxiv.org/abs/2407.10609 |