Commuting Toeplitz operators on the harmonic Bergman space.

Abstract

We investigate the characterizing problem of harmonic symbols of commuting Toeplitz operators acting on the harmonic Bergman space of the unit disk. We first derive a necessary condition for a certain class of harmonic symbols. As a consequence, we show that Toeplitz operators with holomorphic symbols can commute only in the obvious cases. We prove the same in case at least one of symbols is a (harmonic) polynomial. Also we give a characterization of harmonic symbols of normal Toeplitz operators. The case of general harmonic symbols remains open.