Bounded function on which infinite iterations of weighted Berezin transform exist

Abstract

We exhibit some properties of the weighted Berezin transform $T_{\alpha }f$ on $L^{\mathrm{\infty }}(B_n)$ and on $L^1(B_n)$. As the main result, we prove that if $f\in L^{\mathrm{\infty }}(B_n)$ with $\underset{k\to \mathrm{\infty }}{lim}{T}_{\alpha }^{k}f$ exists, then there exist unique $\mathcal{M}$-harmonic function $g$ and $h\in \overline{(I-T_{\alpha })L^{\mathrm{\infty }}(B_n)}$ such that $f=g+h$. We also show that of the norm of weighted Berezin operator $T_{\alpha }$ on $L^1(B_n,\nu )$ converges to 1 as $\alpha$ tends to infinity, where $\nu$ is an ordinary Lebesgue measure.