Algebraic and ergodicity properties of the Berezin transform

Abstract

In this paper we derive certain algebraic and ergodicity properties of the Berezin transform defined on $L^2(\mathbb {B}_N, d\eta')$ where $\mathbb {B}_{N}$ is the open unit ball in $\mathbb {C}^N, N\geq 1, N \in \mathbb Z,$ $d\eta'(z)=K_{\mathbb {B}_N}(z, z)d\nu(z)$ is the Mobius invariant measure, $K_{\mathbb {B}_N}$ is the reproducing kernel of the Bergman space $L_a^2(\mathbb {B}_N, d\nu)$ and $d\nu$ is the Lebesgue measure on $\mathbb C^N$, normalized so that $\nu (\mathbb {B}_N)=1$. We establish that the Berezin transform $B$ is a contractive linear operator on each of the spaces $L^p(\mathbb {B}_N, d\eta'(z)), 1\leq p\leq \infty,$ $B^n\to 0$ in norm topology and $B$ is similar to a part of the adjoint of the unilateral shift. As a consequence of these results we also derive certain algebraic and asymptotic properties of the integral operator defined on $L^2[0,1]$ associated with the Berezin transform.