 
Summary: Iterates and the boundary behaviour
of the Berezin transform
J. Arazy, M. EngliŸ s
Abstract. Let ¯ be a measure on a
domain\Omega in C n such that the Bergman space
of holomorphic functions in L
2(\Omega ; ¯) possesses a reproducing kernel K(x;y) and
K(x;x) ? 0 8x 2 \Omega\Gamma The Berezin transform associated to ¯ is the integral operator
Bf(y) = K(y; y) \Gamma1
Z
\Omega
f(x)jK(x;y)j 2 d¯(x):
The number Bf(y) can be interpreted as a certain mean value of f around y, and
functions satisfying Bf = f as functions having a certain meanvalue property. In
this paper we investigate the boundary behaviour of Bf , the existence of functions f
satisfying Bf = f and having prescribed boundary values, and the convergence of the
iterates B k f , k ! 1. The best results are obtained for smoothly bounded strictly
pseudoconvex
domains\Omega with any measure ¯ as above, and for bounded symmetric
domains\Omega and ¯ one of the standard rotationinvariant measures on them. We also
