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Berezin transform on real bounded symmetric domains

Authors:	Genkai Zhang

Institution:	Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden

Abstract:	Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$ with a real form and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.

Keywords:	Real bounded symmetric domains Jordan triples Siegel domains Berezin transform invariant differential operators unitary representations of Lie groups irreducible decomposition

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