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

Shubham R. Bais; D. Venku Naidu; Pinlodi Mohan

doi:10.5802/crmath.477

Operator theory

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

Shubham R. Bais ¹; D. Venku Naidu ¹ ; Pinlodi Mohan ¹

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

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1593-1604.

Abstract

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

(S_{φ} f) (z) = \int_{Π} f (w) φ (z - \bar{w}) d μ (w), \forall f \in 𝒜^{2} (Π), z \in Π,

where $φ$ is an analytic function on $Π$ given by

φ (z) = \int_{ℝ_{+}} ξ σ (ξ) e^{i z ξ} d ξ, \forall z \in Π

for some $σ \in L^{\infty} (ℝ_{+})$ . Here $d μ (w)$ is the Lebesgue measure on $Π$ . 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 $S_{φ}$ for all the operators in the $C^{*}$ -algebra generated by Toeplitz operators $T_{a}$ with vertical symbols $a \in L^{\infty} (Π)$ .

Received: 2022-07-12
Revised: 2022-12-09
Accepted: 2023-02-08
Published online: 2023-11-17

DOI: 10.5802/crmath.477

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

Author's affiliations:

Shubham R. Bais ¹; D. Venku Naidu ¹; Pinlodi Mohan ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Shubham R. Bais and D. Venku Naidu and Pinlodi Mohan},
     title = {Integral representation of vertical operators on the {Bergman} space over the upper half-plane},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1593--1604},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.477},
     language = {en},
}

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Shubham R. Bais; D. Venku Naidu; Pinlodi Mohan. Integral representation of vertical operators on the Bergman space over the upper half-plane. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1593-1604. doi : 10.5802/crmath.477. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.477/

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