Iterates, fixed-points and the boundary behaviour of the Berezin transform
by
M. Englis
Mathematics Institute, Academy of Sciences, Zitna 25, 11567 Prague 1, Czech Republic
Coauthors: J. Arazy (Haifa)

Let \mu be a measure on a domain \Omega in Cⁿ such that the Bergman space of holomorphic functions in L²(\Omega, \mu) possesses a reproducing kernel K(x, y) and K(x, x) > 0 for allx in \Omega. The Berezin transform associated to \mu is the integral operator

Bf(y) = K(y, y)-1 ó õ \Omega  f(x) |K(x, y)|2  d\mu(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 mean-value property. In this talk we will 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 --> \infty. The best results are obtained for smoothly bounded strictly pseudoconvex domains \Omega with any measure \mu as above, and for bounded symmetric domains \Omega and \mu one of the standard rotation-invariant measures on them. We also carry out similar investigation for convolution operators Bf=f*\mu on a bounded symmetric domain \Omega = G/K with a K-invariant absolutely continuous probability measure \mu, and study the behaviour of the geodesic symmetries \phi_a of \Omega as a tends to the boundary.

Date received: June 13, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-14.