(A) study on commuting toeplitz operators with pluriharmonic symbols

Abstract

It has been recently proved by Sheldon Axler and Zeljko Cuckovic that on the Bergman space of the unit disc of the complex plane, two Toeplitz operator with harmonic symbols commute only in the obvious cases. In this thesis, we investigate the corresponding problem with pluriharmonic symbols on the higher dimensional situation. We first obtain a necessary condition and a sufficient condition for two commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the unit ball of higher dimensional complex space. As an application we show that if one of symbols in consideration is holomorphic or antiholomorphic, then so is the other. Our proof depends on a recent theorem of P. Ahern and W. Rudin concerning M-harmonic products.