Iterates of weighted Berezin transform under invariant measure in the unit ball

Abstract

We focus on the interations of the weighted Berezin transform $T_{\alpha}$ on $L^{p}(\tau)$, where $\tau$ is the invariant measure on the complex unit ball $B_n$. Iterations of $T_{\alpha}$ on $L^{1}_{R}(\tau)$ the space of radial integrable functions played important roles in proving $\mathcal{M}$-harmonicity of bounded functions with invariant mean value property. Here, we introduce more properties on iterations of $T_{\alpha}$ on $L^{1}_{R}(\tau)$ and observe differences between the iterations of $T_{\alpha}$ on $L^{1}(\tau)$ and $L^{p}(\tau)$ for $1<p<\infty$.