Brown-Halmos type theorems for Toeplitz operators on the Bergman space of the upper half-plane (Q6635217)

The famous Brown-Halmos theorem [\textit{A. Brown} and \textit{P. R. Halmos}, J. Reine Angew. Math. 213, 89--102 (1963; Zbl 0116.32501)] says that a necessary and sufficient condition for the product \(T_fT_g\) of two Hardy space Toeplitz operators to be a Toeplitz operator is that either \(f\) is co-analytic or \(g\) is analytic and in both cases \(T_fT_g=T_{fg}\). The paper under review is devoted to the same problem for Toeplitz operators acting on Bergman spaces \(\mathcal{A}^2(\Pi^+)\) over the upper half-plane \(\Pi^+\) in the complex plane. Let \(\mathcal{F}^1(\Pi^+)=L^1(\Pi^+)+L^\infty(\Pi^+)\) and \(\mathcal{G}(\Pi^+)=(\mathcal{A}^1(\Pi^+)\cap\mathcal{A}^2(\Pi^+)) +\mathcal{H}^\infty(\Pi^+)\), where \(\mathcal{A}^i(\Pi^+)\) with \(i=1,2\) are the Bergman spaces and \(\mathcal{H}^\infty(\Pi^+)\) is the Hardy space. The Berezin transform of \(u\in\mathcal{F}^1(\Pi^+)\) is denoted by \(\mathcal{B}(u)\). For \(b,z\in\Pi^+\), let \(\phi_b^{-1}(z)=(z-\operatorname{Re}b)/\operatorname{Im}b\) and \(\varphi(z)=\frac{z-i}{z+i}\).\N\NThe main result of the paper is the following. \N\NTheorem. Let \(f=f_1+\overline{f_2}\) and \(g=g_1+\overline{g_2}\) be two bounded harmonic functions with \(f_j\cdot g_j\in\mathcal{G}(\Pi^+)\) for \(j=1,2\) such that neither \(\overline{f}\) nor \(g\) is holomorphic, and let \(h\in\mathcal{F}^1(\Pi^+)\). Then \(T_fT_g=T_h\) holds on \(\mathcal{A}^2(\Pi^+)\) if and only if there are non-constant holomorphic polynomials \(p\) and \(q\) in \(\varphi\) with \(\operatorname{deg}(pq)\le 3\), satisfying \(\mathcal{B}(u)=p\overline{q}\) such that \(f_1=p\circ\phi_b^{-1}\) and \(g_2=q\circ\phi_b^{-1}\), and \(h=u\circ\phi_b^{-1}+\overline{f_2}g_1+f_1g_1+\overline{f_2}\overline{g_2}\). In particular, if \(h\) is bounded or it is in \(\mathcal{F}^1(\Pi^+)\cap C(\Pi^+)\), then either \(\overline{f}\) or \(g\) is holomorphic. In either case \(h=fg\).