Hua operators and Poisson transform for bounded symmetric domains ✩

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary.We consider the Poisson transform Psf (z) for a hyperfunction f on S defined by the Poisson kernel Ps (z, u) = (h(z, z)n/r /|h(z, u)n/r |2)s , (z, u)×Ω×S, s ∈ C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image. © 2006 Elsevier Inc. All rights reserved.