Commutants of a certain class of Toeplitz operators

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.