Commuting Toeplitz operators and H-Toeplitz operators on Bergman space

In this paper, we consider the H-Toeplitz and Toeplitz operators acting on the Bergman space. First, we describe the characterizations of commutativity of two H-Toeplitz operators with certain harmonic symbols. For the general case, it seems very hard. As an extension to the study of Toeplitz operators on the Bergman space, we present the necessary and sufficient conditions of the commutativity of the H-Toeplitz operator and the Toeplitz operator with non-harmonic symbols.