Integral representation of vertical operators on the Bergman space over the upper half-plane
Let $\Pi $ denote the upper half-plane. In this article, we prove that every vertical operator on the Bergman space $\mathcal{A}^2(\Pi )$ over the upper half-plane can be uniquely represented as an integral operator of the form \begin{equation*} \left(S_\varphi f\right)(z) = \int _{\Pi } f(w) \varp...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.477/ |
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| Summary: | Let $\Pi $ denote the upper half-plane. In this article, we prove that every vertical operator on the Bergman space $\mathcal{A}^2(\Pi )$ over the upper half-plane can be uniquely represented as an integral operator of the form
\begin{equation*}
\left(S_\varphi f\right)(z) = \int _{\Pi } f(w) \varphi (z-\overline{w}) d\mu (w),~~\forall f\in \mathcal{A}^2(\Pi ),~z\in \Pi ,
\end{equation*}
where $\varphi $ is an analytic function on $\Pi $ given by
\begin{equation*}
\varphi (z) = \int _{\mathbb{R}_+}\xi \sigma (\xi ) e^{iz\xi } d\xi , \ \forall z\in \Pi
\end{equation*}
for some $\sigma \in L^\infty (\mathbb{R}_+)$. Here $d\mu (w)$ is the Lebesgue measure on $\Pi $. Later on, with the help of above integral representation, we obtain various operator theoretic properties of the vertical operators.Also, we give integral representation of the form $S_\varphi $ for all the operators in the $C^\ast $-algebra generated by Toeplitz operators $T_{\mathbf{a}}$ with vertical symbols ${\mathbf{a}}\in L^\infty (\Pi )$. |
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| ISSN: | 1778-3569 |