More properties of weighted Berezin transform in the unit ball of $\mathbb C^n$

Abstract

We exhibit various properties of the weighted Berezin operator $T_{\alpha}$ and its iteration $T_{\alpha}^{k}$ on $L^{p}(\tau)$, where $\alpha > -1$ and $\tau$ 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 have performed important roles in proving $\mathcal{M}$-harmonicity of bounded functions with invariant mean value property. We show differences between the case of $1<p<\infty$ and $p=1, \infty$ under the infinite iteration of $T_{\alpha}$ or the infinite summation of iterations, most of which are extensions or related assertions to the propositions of the previous results.