an:05549005
Zbl 1176.47026
Arazy, Jonathan; Engli??, Miroslav; Kaup, Wilhelm
Holomorphic retractions and boundary Berezin transforms
EN
Ann. Inst. Fourier 59, No. 2, 641-657 (2009).
00248927
2009
j
47B38 17C27 53C35 46J15 46E22 32M15
Berezin transform; Cartan domain; convolution operator
Summary: In [Ann.\ Inst.\ Fourier 51, No.\,4, 1101--1133 (2001; Zbl 0989.47027)], the first two authors showed that the convolution of a function \(f\) continuous on the closure of a Cartan domain and a \(K\)-invariant finite measure \(\mu\) 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 \(\mu_{F}\) on \(F\) uniquely determined by \(\mu\). In the present article, we give an explicit formula for \(\mu_{ F}\) in terms of \(F\), showing, in particular, that for measures \(\mu\) corresponding to the Berezin transforms the measures \(\mu_{F}\) again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.
Zbl 0989.47027