On the Commutants of Toeplitz Operators on the Harmonic Bergman Space

Abstract

A Master of Science in Mathematics by Hasan Iqtaish entitled, “On the Commutants of Toeplitz Operators on the Harmonic Bergman Space”, submitted in May 2025. Thesis advisor is Dr. Abdel Rahman Yousef and thesis co-advisor is Dr. Issam Louhichi. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).A key issue in the theory of Toeplitz operators is the commutant problem, which concerns describing the collection of all bounded Toeplitz operators that commute with a given one. In this thesis, we deal with Toeplitz operators acting on the harmonic Bergman space of the unit disk in the complex plane. Specifically, we characterize the commutants Tƒ of the Toeplitz operator Tz+g, where g is an analytic function and where ƒ possesses a Fourier series truncated above, meaning that ƒ takes the form ƒ(re ͥᶿ) = ∑_(k=-∞)^N▒〖e ͥᵏᶿfk(r). Our main result states that such commutants Tf must be polynomials in Tz+g of degree at most one.College of Arts and SciencesDepartment of Mathematics and StatisticsMaster of Science in Mathematics (MSMTH