Generalized de Branges-Rovnyak spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915064576999424 |
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| author | Aleman, Alexandru Dahlin, Frej |
| author_facet | Aleman, Alexandru Dahlin, Frej |
| contents | Given the reproducing kernel $k$ of the Hilbert space $\mathcal{H}_k$ we study spaces $\mathcal{H}_k(b)$ whose reproducing kernel has the form $k(1-bb^*)$, where $b$ is a row-contraction on $\mathcal{H}_k$. In terms of reproducing kernels this it the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces in one or several variables. We study some general properties of $\mathcal{H}_k(b)$ e.g. when the inclusion map into $\mathcal{H}$ is compact. Our main result provides a model for $\mathcal{H}_k(b)$ reminiscent of the Sz.-Nagy-Foiaş model for contractions. As an application we obtain sufficient conditions for the containment and density of the linear span of $\{k_w:w\in\mathcal{X}\}$ in $\mathcal{H}_k(b)$. In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture regarding polynomial approximation in spaces with kernel $\frac{(1-b(z)b(w)^*)^m}{(1-z\overline w)^β}, 1\leq m<β, m\in\mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized de Branges-Rovnyak spaces Aleman, Alexandru Dahlin, Frej Functional Analysis Complex Variables Given the reproducing kernel $k$ of the Hilbert space $\mathcal{H}_k$ we study spaces $\mathcal{H}_k(b)$ whose reproducing kernel has the form $k(1-bb^*)$, where $b$ is a row-contraction on $\mathcal{H}_k$. In terms of reproducing kernels this it the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces in one or several variables. We study some general properties of $\mathcal{H}_k(b)$ e.g. when the inclusion map into $\mathcal{H}$ is compact. Our main result provides a model for $\mathcal{H}_k(b)$ reminiscent of the Sz.-Nagy-Foiaş model for contractions. As an application we obtain sufficient conditions for the containment and density of the linear span of $\{k_w:w\in\mathcal{X}\}$ in $\mathcal{H}_k(b)$. In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture regarding polynomial approximation in spaces with kernel $\frac{(1-b(z)b(w)^*)^m}{(1-z\overline w)^β}, 1\leq m<β, m\in\mathbb{N}$. |
| title | Generalized de Branges-Rovnyak spaces |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2405.07016 |