On the commuting problem for Toeplitz operators with quasi-biharmonic symbols

Al-Naimi, Raja’a; Odetallah, Huda

doi:10.47974/JIM-2309

Open Access Research Article

On the commuting problem for Toeplitz operators with quasi-biharmonic symbols

^*Raja’a Al-NaimiCorresponding authorRajaa.alnaimi@eau.ac.aeDepartment of MathematicsFaculty of Mathematics and Data ScienceEmirates Aviation UniversityDubai, United Arab EmiratesView full profile → , Huda Odetallahh_odetallah@asu.edu.joFaculty of ScienceCollege of ScienceApplied Science Private UniversityAmman, JordanView full profile →

* Corresponding author · click or hover a name for details

pp. 1263–1269Vol. 29Issue 5-AMay 2026DOI: 10.47974/JIM-2309XML

Received:: 01 May 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2309
Pages:: 1263–1269

Abstract

This paper investigates a novel class of bounded symbols for the Toeplitz operator acting on a Bergman space for unit disks. We introduce and rigorously study quasi-biharmonic symbols, which generalize classical biharmonic functions while preserving essential analytical properties for applications in operator theory. Our main theoretical contributions characterize the commutant for the Toeplitz operator with symbol φ(z) = zm + |z|2αg(z) and φ(z) = |z|2ng(z) + h(z), where g is analytic, and h is harmonic. Using advanced techniques from complex analysis and operator theory, including polar decompositions and Mellin transform methods, we prove that under appropriate regularity conditions, any Toeplitz operators that commute with Toeplitz operators having such a quasi-biharmonic symbol must be linear combinations of the identity operator and operators with the quasi-biharmonic symbol. These findings significantly extend existing literature on commute problems for Toeplitz operators and provide new theoretical insights into the algebraic structure of the Toeplitz algebra on the Bergman spaces.

Keywords

Toeplitz operator Bergman spaces Commuting operator Quasi-biharmonic symbols Operator theory

Subject Classifications

Primary 47B35Secondary 32A36

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