Berezin transform on real bounded symmetric domains

Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$is then unitarily equivalent to the restriction to $G$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 $L^2$-space.