  In this paper we study the boundedness of Bergman projectors on weighted
Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a
natural approach in the special case of tube domains over irreducible symmetric
cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue
spaces.
  The sharp range for the boundedness of such operators is essentially known
only in the case of tube domains over Lorentz cones.
  In this paper we prove that the boundedness of such Bergman projectors is
equivalent to variuos notions of atomic decomposition, duality, and
characterization of boundary values of the mixed-norm weighted Bergman spaces,
extending results moslty known only in the case of tube domains over
irreducible symmetric cones. Some of our results are new even in the latter
simpler context.
  We also study the simpler, but still quite interesting, case of the
"positive" Bergman projectors, the integral operator in which the Bergman
kernel is replaced by its absolute value. We provide a useful characterization
which was previously known for tube domains.
