On the range of the Berezin transform
Let \(\mathbb{D}\) be the open unit disk in the complex plane. The Bergman space \(B^2\) is the subspace of \(L^2\) consisting of the holomorphic functions on \(\mathbb{D}\). For a bounded function \(u\) on \(\mathbb{D}\), the Toeplitz operator \(T_u\) defined on \(B^2\) is given by \(T_uf= P(uf)\), where \(P: L^2\to B^2\) is the orthogonal projection. It is easy to see that if \(\overline f\) or \(g\) is holomorphic, then \(T_fT_g= T_{fg}\). A. Brown and P. Halmos showed that in the Hardy space case the converse is true: if \(T_fT_g= T_h\), then one of the two symbols \(\overline f\) or \(g\) must be holomorphic and in this case \(h= fg\). In the previous paper by the author and the reviewer [J.\ Funct.\ Anal.\ 187, No.~1, 200--210 (2001; Zbl 0996.47037)], it was shown that, in general, the Brown-Halmos theorem is not true for Bergman space Toeplitz operators. It was also shown that there is a theorem of the Brown-Halmos type if we assume that \(f\) and \(g\) are bounded harmonic functions and that \(h\) is a bounded \(C^2\) function whose invariant Laplacian \(\widetilde\Delta h\) is bounded on \(\mathbb{D}\). One of the key ingredients in the proof of this result is a result about the range of the Berezin transform. For any integrable function \(f\) on \(\mathbb{D}\), the Berezin transform is defined by \(Bf(z)= (1-|z|^2)^2 \int_{\mathbb{D}}{f(\zeta)\over|1-\overline\zeta|^4}\,dA(\zeta)\). In the present paper, the author continues this line of investigation. At first, he characterizes all triples \((f,g,u)\) where \(f\) and \(g\) are nonconstant holomorphic functions on \(\mathbb{D}\) and \(u\) is integrable on \(\mathbb{D}\) such that \(Bu= f\overline g\). He starts with an example: \(Bu(z)= z\overline z^2\) where \(u(\zeta)= 2\overline\zeta-{1\over\zeta}\). The main result is very interesting: if \(f\) and \(g\) are holomorphic in \(\mathbb{D}\) and neither is constant and \(Bu= f\overline g\) for some \(u\in L^1(\mathbb{D})\), then there are nonconstant polynomials \(p\) and \(q\) with \(\deg(pq)\leq 3\) and an \(a\in\mathbb{D}\) such that \(f= p\circ\phi_a\) and \(g= q\circ\phi_a\), where \(\phi_a(z)= {a-z\over 1-\overline az}\). The proof is very nontrivial. This result again has consequences for products of Toeplitz operators on \(B^2\). As a special case, the author shows that if \(f\) and \(g\) are bounded harmonic and \(h\in L^1(\mathbb{D})\) is locally bounded such that \(T_fT_g= T_h\), then \(\overline f\) or \(g\) is holomorphic. This gives an improvement of the above mentioned theorem in the previous paper.
scientific article; zbMATH DE number 2117852