Kernels and ranks of Hankel operators on the Dirichlet spaces

We consider Hankel operators and little Hankel operators on the Dirichlet space and pluriharmonic Dirichlet space of the unit ball. We first describe kernels of Hankel operators with pluriharmonic symbol. Next, we prove that there are no non-trivial finite rank Hankel operators with pluriharmonic symbol. Finally, we characterize finite rank little Hankel operators with pluriharmonic symbol. Our results show that there are several differences between the results on the Dirichlet space and pluriharmonic Dirichlet space.