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 $L_R^1(\tau )$ 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 $L_R^1(\tau )$ and observe differences between the iterations of $T_{\alpha }$ on $L^1(\tau )$ and $L^p(\tau )$ for $1<p<\mathrm{\infty }$.