
doi: 10.5802/aif.2444
handle: 11104/0178956
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 , showing in particular that for measures μ corresponding to the Berezin transforms the measures μ 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.
Berezin transform, Idempotents, Peirce decompositions, convolution operator, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Cartan domain, Differential geometry of symmetric spaces
Berezin transform, Idempotents, Peirce decompositions, convolution operator, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Linear operators on function spaces (general), Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Cartan domain, Differential geometry of symmetric spaces
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