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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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.477/ |
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| author | Bais, Shubham R. Venku Naidu, D. Mohan, Pinlodi |
| author_facet | Bais, Shubham R. Venku Naidu, D. Mohan, Pinlodi |
| author_sort | Bais, Shubham R. |
| collection | DOAJ |
| description | 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 ${\mathbf{a}}\in L^\infty (\Pi )$. |
| format | Article |
| id | doaj-art-ecff1af3a67f42b7aa2e3db77607d5cc |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-ecff1af3a67f42b7aa2e3db77607d5cc2025-02-07T11:11:47ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G101593160410.5802/crmath.47710.5802/crmath.477Integral representation of vertical operators on the Bergman space over the upper half-planeBais, Shubham R.0Venku Naidu, D.1Mohan, Pinlodi2Department of Mathematics, Indian Institute of Technology – Hyderabad, Kandi, Sangareddy, Telangana, India 502 284.Department of Mathematics, Indian Institute of Technology – Hyderabad, Kandi, Sangareddy, Telangana, India 502 284.Department of Mathematics, Indian Institute of Technology – Hyderabad, Kandi, Sangareddy, Telangana, India 502 284.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 ${\mathbf{a}}\in L^\infty (\Pi )$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.477/Bergman spacemultiplication operatorreducing subspaceToeplitz operator |
| spellingShingle | Bais, Shubham R. Venku Naidu, D. Mohan, Pinlodi Integral representation of vertical operators on the Bergman space over the upper half-plane Comptes Rendus. Mathématique Bergman space multiplication operator reducing subspace Toeplitz operator |
| title | Integral representation of vertical operators on the Bergman space over the upper half-plane |
| title_full | Integral representation of vertical operators on the Bergman space over the upper half-plane |
| title_fullStr | Integral representation of vertical operators on the Bergman space over the upper half-plane |
| title_full_unstemmed | Integral representation of vertical operators on the Bergman space over the upper half-plane |
| title_short | Integral representation of vertical operators on the Bergman space over the upper half-plane |
| title_sort | integral representation of vertical operators on the bergman space over the upper half plane |
| topic | Bergman space multiplication operator reducing subspace Toeplitz operator |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.477/ |
| work_keys_str_mv | AT baisshubhamr integralrepresentationofverticaloperatorsonthebergmanspaceovertheupperhalfplane AT venkunaidud integralrepresentationofverticaloperatorsonthebergmanspaceovertheupperhalfplane AT mohanpinlodi integralrepresentationofverticaloperatorsonthebergmanspaceovertheupperhalfplane |