More properties of weighted Berezin transform in the unit ball of Cn

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 $L_R^1(\tau )$ 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 $1<p<\mathrm{\infty }$ and $p=1,\mathrm{\infty }$ 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.