Hua operators and Poisson transform for bounded symmetric domains

Journal article, 2006

Let Ω be a bounded symmetric domain of non-tube type in C n with rank r and S its Shilov boundary. We consider the Poisson transform P s f ( z ) for a hyperfunction f on S defined by the Poisson kernel P s ( 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 M n, m ( C ) ( n {less-than or slanted equal to} m), we prove that an eigenvalue equation for the second order M n, n -valued Hua operator characterizes the image. © 2006 Elsevier Inc. All rights reserved.