The Pluriharmonic Hardy Space and Toeplitz Operators

Compared with harmonic Bergman spaces, this paper introduces a new function space which is called the pluriharmonic Hardy space $h^2\left(T^2\right)$. We character (semi-) commuting Toeplitz operators on $h^2\left(T^2\right)$ with bounded pluriharmonic symbols. Interestingly, these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces, Bergman spaces and harmonic Bergman spaces. Our method for Toeplitz operators on $h^2\left(T^2\right)$ gives new insight into the study of commuting Toeplitz operators on harmonic Bergman spaces.