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 $$\big(B_{c_1, c_2}\big)f(z,w) = \int_{D}\int_{D} f\big( \varphi_{z}(x) , \varphi_{w}(y) \big)\ d\nu_{c_1}(x) d\nu_{c_2}(y).$$ 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 $1 \le p< \infty$. 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 $M_{c_1, c_2}^{p}$.