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 L2(BN,dη') where B N is the open unit ball in CN,N ≥ 1,N ε Z, dη'(z) = KBN (z, z)dν(z) is the Mobius invariant measure, KBN is the reproducing kernel of the Bergman space L2a(BN,dν) and dν is the Lebesgue measure on CN, normalized so that ν(BN) = 1. We establish that the Berezin transform B is a contractive linear operator on each of the spaces Lp(BN,dη'(z)),1 ≤ p ≤ ∞, Bn → 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 L2[0,1] associated with the Berezin transform.