On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916306338447360 |
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| author | Luo, Shuaibing Rydhe, Eskil |
| author_facet | Luo, Shuaibing Rydhe, Eskil |
| contents | This paper studies the function spaces $\mathcal{D}(μ)$ by Richter and Aleman, and $\mathcal{D}_{\vecμ}$ by the second author.
It is known that the forward shift $M_z$ is bounded and expansive on $\mathcal{D}(μ)$, and therefore $\mathcal{D}(μ)$ coincides with a de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $μ$ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of $\mathcal{D}(μ)$ for finitely atomic $μ$.
Similarly, we characterize the allowable tuples $\vecμ = (\frac{|dz|}{2π}, μ_1, \ldots, μ_{n-1})$ such that $M_z$ on $\mathcal{D}_{\vecμ}$ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples $\vecμ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19393 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces Luo, Shuaibing Rydhe, Eskil Functional Analysis This paper studies the function spaces $\mathcal{D}(μ)$ by Richter and Aleman, and $\mathcal{D}_{\vecμ}$ by the second author. It is known that the forward shift $M_z$ is bounded and expansive on $\mathcal{D}(μ)$, and therefore $\mathcal{D}(μ)$ coincides with a de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $μ$ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of $\mathcal{D}(μ)$ for finitely atomic $μ$. Similarly, we characterize the allowable tuples $\vecμ = (\frac{|dz|}{2π}, μ_1, \ldots, μ_{n-1})$ such that $M_z$ on $\mathcal{D}_{\vecμ}$ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples $\vecμ$. |
| title | On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2310.19393 |