Characterizing functions fixed by a weighted Berezin transform in the bidisc

Abstract

For $c>-1$, let ${\nu }_c$ denote a weighted radial measure on $\mathbb{C}$ normalized so that ${\nu }_c(D)=1.$ For $c_1,c_2>-1$ and $f\in L^1(D^2,{\nu }_{c_1}\times {\nu }_{c_2})$, we define the weighted Berezin transform $B_{c_1,c_2}f$ on $D^2$ by
$$(B_{c_1,c_2})f(z,w)={\int }_D{\int }_{D}f({\phi }_z(x),{\phi }_w(y))d{\nu }_{c_1}(x)d{\nu }_{c_2}(y).$$
This paper is about the space $M_{c_1,c_2}^p$ of function $f\in L^p(D^2,{\nu }_{c_1}\times {\nu }_{c_2})$ satisfying $B_{c_1,c_2}f=f$ for $1\le p<\mathrm{\infty }$. We find the identity operator on $M_{c_1,c_2}^p$ by using invariant Laplacians and we characterize some special type of functions in $M_{c_1,c_2}^p$.