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
Mots clés : 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

@article{CRMATH_2023__361_G10_1593_0,
     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},
}

TY  - JOUR
AU  - Shubham R. Bais
AU  - D. Venku Naidu
AU  - Pinlodi Mohan
TI  - Integral representation of vertical operators on the Bergman space over the upper half-plane
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1593
EP  - 1604
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.477
LA  - en
ID  - CRMATH_2023__361_G10_1593_0
ER  -

%0 Journal Article
%A Shubham R. Bais
%A D. Venku Naidu
%A Pinlodi Mohan
%T Integral representation of vertical operators on the Bergman space over the upper half-plane
%J Comptes Rendus. Mathématique
%D 2023
%P 1593-1604
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.477
%G en
%F CRMATH_2023__361_G10_1593_0

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/

References
Cited by

[1] Shubham R. Bais; Dogga Venku Naidu Study of twisted Bargmann transform via Bargmann transform, Forum Math., Volume 33 (2021) no. 6, pp. 1659-1670 | DOI | MR | Zbl

[2] Valentine Bargmann On a Hilbert space of analytic functions and an associated integral transform, Commun. Pure Appl. Math., Volume 14 (1961), pp. 187-214 | DOI | MR | Zbl

[3] Rajendra Bhatia Notes on functional analysis, Texts and Readings in Mathematics, 50, Hindustan Book Agency, 2009 | DOI | Zbl

[4] John B. Conway A course in operator theory, Graduate Studies in Mathematics, 21, American Mathematical Society, 2000 | Zbl

[5] Ronald G. Douglas Banach algebra techniques in operator theory, Pure and Applied Mathematics, 49, Academic Press, New Yor, 1972 | Zbl

[6] Kevin Esmeral; Egor A. Maximenko; Nikolai L. Vasilevski $C^{*}$ -algebra generated by angular Toeplitz operators on the weighted Bergman spaces over the upper half-plane, Integral Equations Oper. Theory, Volume 83 (2015) no. 3, pp. 413-428 | DOI | MR | Zbl

[7] Kevin Esmeral; Nikolai L. Vasilevski $C *$ -algebra generated by horizontal Toeplitz operators on the Fock space, Bol. Soc. Mat. Mex., Volume 22 (2016) no. 2, pp. 567-582 | DOI | MR | Zbl

[8] Gerald B. Folland Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, 1989 | DOI | Zbl

[9] Gerald B. Folland Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992 | Zbl

[10] Sergei M. Grudsky; Raul Quiroga-Barranco; Nikolai L. Vasilevski Commutative $C^{*}$ -algebras of Toeplitz operators and quantization on the unit disk, J. Funct. Anal., Volume 234 (2006) no. 1, pp. 1-44 | DOI | MR | Zbl

[11] Sergei M. Grudsky; Nikolai L. Vasilevski Bergman-Toeplitz operators: radial component influence, Integral Equations Oper. Theory, Volume 40 (2001) no. 1, pp. 16-33 | DOI | MR | Zbl

[12] Sergei M. Grudsky; Nikolai L. Vasilevski Toeplitz operators on the Fock space: radial component effects, Integral Equations Oper. Theory, Volume 44 (2002) no. 1, pp. 10-37 | DOI | MR | Zbl

[13] Crispin Herrera Yañez; Ondrej Hutník; Egor A. Maximenko Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions, C. R. Math. Acad. Sci. Paris, Volume 352 (2014) no. 2, pp. 129-132 | DOI | Numdam | MR | Zbl

[14] Crispin Herrera Yañez; Egor A. Maximenko; Nikolai L. Vasilevski Vertical Toeplitz operators on the upper half-plane and very slowly oscillating functions, Integral Equations Oper. Theory, Volume 77 (2013) no. 2, pp. 149-166 | DOI | MR | Zbl

[15] Raj K. Singh; Ashok Kumar Compact composition operators, J. Aust. Math. Soc., Volume 28 (1979) no. 3, pp. 309-314 | DOI | MR | Zbl

[16] Nikolai L. Vasilevski On Bergman-Toeplitz operators with commutative symbol algebras, Integral Equations Oper. Theory, Volume 34 (1999) no. 1, pp. 107-126 | DOI | MR | Zbl

[17] Nikolai L. Vasilevski Commutative algebras of Toeplitz operators on the Bergman space, Operator Theory: Advances and Applications, 185, Birkhäuser, 2008 | MR | Zbl

[18] Kehe Zhu Analysis on Fock spaces, Graduate Texts in Mathematics, 263, Springer, 2012 | Zbl

[19] Kehe Zhu Towards a dictionary for the Bargmann transform, Handbook of analytic operator theory (Chapman and Hall Handbooks in Mathematics Series), CRC Press, 2019, pp. 319-349 | Zbl

Cited by Sources:

Comments - Politique