Hankel- and Toeplitz-type operators on the unit ball

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.