Bicommutants of Toeplitz operators
A Toeplitz operator \(T_f\) is said to be quasihomogeneous (of order \(p\)) if its symbol \(f\) is of the form \(f(r e^{i \theta})=e^{i p \theta} \varphi (r)\), where \(\varphi\) is a radial function and \(p\) is an integer. Within the context of Toeplitz operators on the Bergman space on the unit disk, it is proven that if two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. This is an interesting unexpected phenomenon, quite different from the well-known theorem by Brown and Halmos on commuting Toeplitz operators on the Hardy space.
scientific article; zbMATH DE number 5500573