The Heisenberg group action on the Siegel domain and the structure of Bergman spaces
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| Format: | Preprint |
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2023
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| author | Barrera-Reyes, Julio A. Quiroga-Barranco, Raul |
| author_facet | Barrera-Reyes, Julio A. Quiroga-Barranco, Raul |
| contents | We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$). Through the use of symplectic geometry we construct a natural set of coordinates for $D_{n+1}$ adapted to $\mathbb{H}_n$. This yields a useful decomposition of the domain $D_{n+1}$. The latter is then used to compute a decomposition of the Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group $\mathbb{H}_n$. As an application, we consider $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) generated by Toeplitz operators whose symbols belong to $L^\infty(D_{n+1})^{\mathbb{H}_n}$ (essentially bounded and $\mathbb{H}_n$-invariant). We prove that $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is commutative and isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ (very slowly oscillating functions on $\mathbb{R}_+$), for every $λ> -1$ and $n \geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_02540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Heisenberg group action on the Siegel domain and the structure of Bergman spaces Barrera-Reyes, Julio A. Quiroga-Barranco, Raul Functional Analysis Operator Algebras Primary 32A36, 30H20, 22E25, Secondary 47B35, 53D20 We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$). Through the use of symplectic geometry we construct a natural set of coordinates for $D_{n+1}$ adapted to $\mathbb{H}_n$. This yields a useful decomposition of the domain $D_{n+1}$. The latter is then used to compute a decomposition of the Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group $\mathbb{H}_n$. As an application, we consider $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) generated by Toeplitz operators whose symbols belong to $L^\infty(D_{n+1})^{\mathbb{H}_n}$ (essentially bounded and $\mathbb{H}_n$-invariant). We prove that $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is commutative and isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ (very slowly oscillating functions on $\mathbb{R}_+$), for every $λ> -1$ and $n \geq 1$. |
| title | The Heisenberg group action on the Siegel domain and the structure of Bergman spaces |
| topic | Functional Analysis Operator Algebras Primary 32A36, 30H20, 22E25, Secondary 47B35, 53D20 |
| url | https://arxiv.org/abs/2309.02540 |