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 $L^{\infty}(B_{n})$ and on $L^{1}(B_{n})$. As the main result, we prove that if $f \in L^{\infty}(B_{n})$ with $\lim_{k\rightarrow\infty} T_{\alpha}^{k}f$ exists, then there exist unique $\mathcal{M}$-harmonic function $g$ and $h \in \overline{(I-T_{\alpha})L^{\infty}(B_{n})}$ 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.