Range of Berezin Transform

Abstract: Let $\ds dA=\frac{dxdy}\pi$ denote the normalized Lebesgue area measure on the unit disk $\disk$ and $u$, a summable function on $\disk$. $$B(u)(z)=\int_\disk u(\zeta)\frac{(1-|z|^2)^2}{|1-\zeta\oln z|^4}dA(\zeta)$$ is called the Berezin transform of $u$. Ahern \cite{a} described all the possible triples ${u,f,g}$ for which $$B(u)(z)=f(z)\oln g(z)$$ where both $f,g$ are holomorphic in $\disk$. This result was crucial in solving a version of the zero product problem for Toeplitz operators on the Bergman space. The natural next question was to describe all functions in the range of Berezin Transform which are of the form $$\sum_{i=1}^Nf_i\oln g_i$$ where $f_i,g_i$ are all holomorphic in $\disk$. We shall give a complete description of all such $u$ and the corresponding $f_i,g_i,1\leq i\leq N$. Further we give very simple proof of the result of Ahern \cite{a} and the recent results of \v{C}u\v{c}kovi\'c and Li \cite{bz} where they tackle the special case when N=2 and $g_2=z^n$.