Commutants of a certain class of Toeplitz operators
Aissa Bouhali, Issam Louhichi, Abdel Rahman Yousef
TL;DR
The paper addresses the commutant problem for Toeplitz operators on the analytic Bergman space with symbols consisting of a cubic angular part plus an anti-analytic tail. It develops Mellin-transform–based techniques and a structured calculation to show a strong rigidity: if a nonzero Toeplitz operator T_f commutes with T_g, then T_f must be a linear polynomial in T_g, i.e., T_f = C_1 T_g + C_0 I. The result identifies the form of f (up to a constant) as proportional to g, and the paper extends the analysis to a special case where the anti-analytic tail is a polynomial of degree at most 4, yielding a slightly higher-degree polynomial relation in T_g. This advances the understanding of commutants for Toeplitz operators on Bergman space and suggests similar rigidity for broader quasihomogeneous symbols.
Abstract
A major open problem in the Theory of Toeplitz operators on the analytic Bergman space over the unit disk is the characterization of the commutant of a given Toeplitz operator--that is, the set of all bounded Toeplitz operators that commute with it. In this paper, we provide a complete description of bounded Toeplitz operators $T_f$, where the symbol $f$ has a truncated polar decomposition, that commute with a Toeplitz operator whose symbol is the sum of a quasihomogeneous function and a bounded analytic function.