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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)
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References:
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