Let B-m be the unit ball in the m-dimensional complex plane C-m with the we
ighted measure
d mu (alpha)(z) = (alpha + 1)(alpha + 2)...(alpha + m)/pi (m)(1-\z\(2))(alp
ha)dm(z) (alpha > -1).
From the viewpoint of the Cauchy-Riemann operator we give an orthogonal dir
ect sum decomposition for L-2(B-m, d mu (alpha)(z)), i.e., L-2(B-m,d mu (al
pha)(z)) = circle plus (n is an element ofZ+,sigma is an element of Delta)A
(n)(sigma), where the components A(0)((+,+,...,+)) and A(0)((-,-,...-)) are
just the weighted Bergman and conjugate Bergman spaces, respectively. Usin
g the simplex polynomials from T. H. Koornwinder and A. L. Schwartz (1997,
Constr Approx 13, 537-567), we obtain an orthogonal basis for every subspac
e. As an application of the orthogonal decomposition, we define the Hankel-
and Toeplitz-type operators and discuss S-p-criteria for these kinds of op
erators. (C) 2001 Academic Press.