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Iterates and the boundary behavior of the Berezin transform

Jonathan ArazyMiroslav Engliš — 2001

Annales de l’institut Fourier

Let μ be a measure on a domain Ω in n such that the Bergman space of holomorphic functions in L 2 ( Ω , μ ) possesses a reproducing kernel K ( x , y ) and K ( x , x ) > 0 x Ω . The Berezin transform associated to μ is the integral operator B f ( y ) = K ( y , y ) - 1 Ω f ( x ) | K ( x , y ) | 2 d μ ( x ) . The number B f ( y ) can be interpreted as a certain mean value of f around y , and functions satisfying B f = f as functions having a certain mean-value property. In this paper we investigate the boundary behavior of B f , the existence of functions f satisfying B f = f and having...

Holomorphic retractions and boundary Berezin transforms

Jonathan ArazyMiroslav EnglišWilhelm Kaup — 2009

Annales de l’institut Fourier

In an earlier paper, the first two authors have shown that the convolution of a function f continuous on the closure of a Cartan domain and a K -invariant finite measure μ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face F depends only on the restriction of f to F and is equal to the convolution, in  F , of the latter restriction with some measure μ F on F uniquely determined by  μ . In this article, we give an explicit formula for μ F in terms of  F ,...

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