  We study the boundedness of composition operators on the bidisk using
reproducing kernels. We show that a composition operator is bounded on the
Hardy space of the bidisk if some associated function is a positive kernel.
This positivity condition naturally leads to the study of the sub-Hardy Hilbert
spaces of the bidisk, which are analogs of de Branges-Rovnyak spaces on the
unit disk. We discuss multipliers of those spaces and obtain some classes of
bounded composition operators on the bidisk.
