Translation-invariant operators in reproducing kernel Hilbert spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916706511749120 |
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| author | Herrera-Yañez, Crispin Maximenko, Egor A. Ramos-Vazquez, Gerardo |
| author_facet | Herrera-Yañez, Crispin Maximenko, Egor A. Ramos-Vazquez, Gerardo |
| contents | Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $\mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $\mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $\widehat{H}_ξ$, $ξ\in\widehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $\mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $\mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_05879 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Translation-invariant operators in reproducing kernel Hilbert spaces Herrera-Yañez, Crispin Maximenko, Egor A. Ramos-Vazquez, Gerardo Functional Analysis Operator Algebras Representation Theory 22D25, 46E22, 42A38, 47B35 Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $\mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $\mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $\widehat{H}_ξ$, $ξ\in\widehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $\mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $\mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors. |
| title | Translation-invariant operators in reproducing kernel Hilbert spaces |
| topic | Functional Analysis Operator Algebras Representation Theory 22D25, 46E22, 42A38, 47B35 |
| url | https://arxiv.org/abs/2109.05879 |