  Compared with harmonic Bergman spaces, this paper introduces a new function
space which is called the pluriharmonic Hardy space $h^{2}(\mathbb{T}^{2})$. We
character (semi-) commuting Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ 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 $h^{2}(\mathbb{T}^{2})$ gives new insight into the study of commuting
Toeplitz operators on harmonic Bergman spaces.
