The range of the Berezin transform
Let \(H(\mathbb{D})\) be the space of holomorphic functions on the open unit disk \(\mathbb{D}\subset\mathbb{C}\) and let \(B(u)\) be the Berezin transform of a function \(u\in L^1(\mathbb{D})\). In this paper the authors give a description of the functions \(\varphi=\sum_{j=1}^N f_j\overline{g}_j\), \(f_j, g_j\in H(\mathbb{D})\), in the range of \(B\). The case \(N=1\) was proved in [\textit{P. Ahern}, J. Funct. Anal. 215, No. 1, 206--216 (2004; Zbl 1088.47014)]. The main result of this paper proves that if \(B(u)=\sum_{j=1}^N f_j\overline{g}_j\) for some \(u\in L^1(\mathbb{D})\), then there exist a harmonic function \(h\), a set \(\{a_j\}_{j=1}^N\subset\mathbb{D}\) and constants \(A_j, B_j, C_j\), \(j=1,\dots,N\), such that \[ u(z)=h(z)+\sum_{j=1}^N A_j\log|z-a_j|+\frac{B_j}{z-a_j}+\frac{C_j} {\overline{z}-\overline{a}_j}. \] Therefore, \[ B(u)=\tilde h+\sum_{j=1}^N(D_j+E_j\varphi_{a_j} +F_j\overline\varphi_{a_j})\, \varphi_{a_j}\overline\varphi_{a_j}, \] where \(\tilde h\) is a summable harmonic function and \(\varphi_a(z)=\frac{z-a}{1-z\overline a}\).
scientific article; zbMATH DE number 6844037