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Holomorphic retractions and boundary Berezin transforms. (English) Zbl 1176.47026
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.

##### MSC:
 47B38 Linear operators on function spaces (general) 17C27 Idempotents, Peirce decompositions 53C35 Differential geometry of symmetric spaces 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
##### Keywords:
Berezin transform; Cartan domain; convolution operator
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##### References:
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