Théorie des opérateurs

Integral representation of vertical operators on the Bergman space over the upper half-plane

Bais, Shubham R.¹ ; Venku Naidu, D.¹  ; Mohan, Pinlodi¹

¹ Department of Mathematics, Indian Institute of Technology – Hyderabad, Kandi, Sangareddy, Telangana, India 502 284.

Comptes Rendus. Mathématique, Tome 361 (2023) no. G10, pp. 1593-1604

Résumé

Let $\Pi$ denote the upper half-plane. In this article, we prove that every vertical operator on the Bergman space ${𝒜}^2\left(\Pi \right)$ over the upper half-plane can be uniquely represented as an integral operator of the form

$$\left(S_{\phi }f\right)\left(z\right)={\int }_{\Pi }f\left(w\right)\phi (z-\overline{w})d\mu \left(w\right),\forall f\in {𝒜}^2\left(\Pi \right),z\in \Pi ,$$

where $\phi$ is an analytic function on $\Pi$ given by

$$\phi \left(z\right)={\int }_{{ℝ}_{+}}\xi \sigma \left(\xi \right)e^{iz\xi }d\xi ,\forall z\in \Pi$$

for some $\sigma \in L^{\infty }\left({ℝ}_{+}\right)$. Here $d\mu \left(w\right)$ 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 $S_{\phi }$ for all the operators in the $C^{*}$-algebra generated by Toeplitz operators $T_{\mathbf{a}}$ with vertical symbols $\mathbf{a}\in L^{\infty }\left(\Pi \right)$.

Détail

Reçu le : 2022-07-12
Révisé le : 2022-12-09
Accepté le : 2023-02-08
Publié le : 2023-11-17

DOI : 10.5802/crmath.477

Classification : 30H20, 47A15, 47B35, 47G10
Keywords: Bergman space, multiplication operator, reducing subspace, Toeplitz operator

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits